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Color term

2023-06-28T16:56:28Z

Crasshopper: /* Stage VII+ */shorter

{{short description|Word or phrase that refers to a specific color}}
{{Use American English|date=October 2020}}
[[File:RBG color wheel.svg|300x300px|thumb|[[Color wheel]] with [[English language|English]] color terms]]A '''color term''' (or color name) is a word or [[phrase]] that refers to a specific color. The color term may refer to human perception of that color (which is affected by visual context) which is usually defined according to the [[Munsell color system]], or to an underlying physical property (such as a specific [[wavelength]] of [[visible light]]). There are also numerical systems of color specification, referred to as [[color space]]s.

An important distinction must be established between color and shape, as these two attributes usually are used in conjunction with one another when describing in language. For example, they are labeled as alternative parts of speech terms color term and shape term.<ref name=":0">{{Cite book|title=Colour Categories in Thought and Language; "The neuropsychology of color" |last=Davidoff |first=Jules |publisher=Press Syndicate of the University of Cambridge |year=1997 |isbn=9780521498005 |location=Cambridge, England |pages=118–120}}</ref>

Psychological conditions for recognition of colors exist, such as those who cannot discern colors in general or those who see colors as sound (a variety of [[synesthesia]]).

== Color dimensions ==
Typical human [[color vision]] is [[trichromatic]], meaning it is based on a three-dimensional color [[gamut]]. These three dimensions can be defined in different ways, but often the most intuitive definition are the dimensions of the [[HSL and HSV|HSL/HSV color space]]:
[[File:Colours in Irish.png|thumb|[[Color wheel]] with [[Irish language|Irish]] color terms, explaining that the difference between ''glas'' ("light blue/gray/green") and ''gorm'' ("deep blue/gray/green") is based on intensity (luminosity) rather than hue. Similarly, ''rua'' refers to deep reds while ''dearg'' refers to bright reds, and ''geal'', ''bán'' and ''fionn'' all refer to varying degrees of brightness or "fairness", without mapping clearly only the English "white."]]
* '''[[Hue]]''': representing the different colors of the [[rainbow]] or [[color wheel]] (e.g. 'red', 'orange', 'yellow', etc.); roughly analogous to the color's [[wavelength]] or [[frequency]].
* '''[[saturation (color theory)|Saturation]]''': the [[colorfulness]] of the color, i.e. a measure of vibrant vs. pale.
* '''[[Luminosity]]''': a measurement of intensity or 'brightness'.

==In natural languages==

===Lexicology===
[[wikt:monolexemic|Monolexemic]] color words are composed of individual [[lexeme]]s, or ''root words'', such as 'red', 'brown', 'fuchsia' or 'olive'. The root words generally describe the hue of the color, but some root words - namely brown - can also describe the other dimensions. Compound color words make use of prefix adjectives (e.g. 'light brown', 'sea green'), that generally describe the saturation or luminosity or compounded basic color words (e.g. 'yellow-green'), which refine the hue of the color relative to root words. 'Vaaleanpunainen', the [[Finnish Language|Finnish]] word for 'pink' is a clear [[agglutination]] of the language's words for 'pale' ('vaalea') and 'red' ('punainen').

{{anchor|Basic}}
=== Basic color terms ===
Basic color terms meet the following criteria:<ref name="BerlinKay1969"/>
* monolexemic ('green', but not 'light green' or 'forest green'),
* high-frequency, and
* agreed upon by speakers of that language.

[[English language|English]] has 11 basic color terms: 'black', 'white', 'red', 'green', 'yellow', 'blue', 'brown', 'orange', 'pink', 'purple', and 'grey'; other languages have between 2 and 12. All other colors are considered by most speakers of that language to be variants of these basic color terms. A useful [[litmus test (politics)|litmus test]] involves replacing each of these basic terms with an approximation of other basic terms, e.g. replacing orange with red-yellow. If the approximation is ''[[wikt:jarring#Adjective|jarring]]'', the replaced term likely meets the requirement for being a basic color term.
An example of a color that comes close to being a basic color term in English is turquoise. It is monolexemic, but is not very high frequency, especially compared to alternatives teal or cyan. It also generally fails the above litmus test in that most people do not find the use of the approximation of other basic color terms (blue-green) to be jarring.

===Color term hierarchy===
In the classic study of Brent Berlin and Paul Kay (1969), ''[[Basic Color Terms|Basic Color Terms: Their Universality and Evolution]]'',<ref name="BerlinKay1969"/> the researchers argued that the differences in number of basic color terms in languages follow a repeatable pattern. Color terms can be organized into a coherent hierarchy and there are a limited number of universal '''basic color terms''' which begin to be used by individual cultures in a relatively fixed order. This order is defined in stages I-VII. Berlin and Kay originally based their analysis on a comparison of color words in 20 languages from around the world. The model is presented below, broken into stages, with stage I on the left and stage VII on the right:<ref>{{cite journal |last1=Kay |first1=Paul |last2=McDaniel |first2=Chad |year=1978 |title=The Linguistic Significance of the Meanings of Basic Color Terms |journal=Language |volume=54 |issue=3 |pages=610–646 |doi=10.1353/lan.1978.0035|s2cid=26961780 }}</ref> 
<math>\begin{Bmatrix}\text{white} \\ \text{black} \end{Bmatrix} < \text{red} < \begin{Bmatrix}\text{green} \\ \text{yellow} \end{Bmatrix} < \text{blue} < \text{brown} <\begin{Bmatrix}\text{purple} \\ \text{pink} \\ \text{orange} \\ \text{grey} \end{Bmatrix}</math>

Berlin and Kay's study identified seven stages of color distinction systems. Each progressive stage features a color term that the previous stages do not.

====Stage I (dark and light)====
{| class="wikitable" style="font-weight:bold; float:right"
|-
! style="background-color:#c0c0c0;" | Stage I<ref name=Kay99>{{cite journal |last1=Kay |first1=Paul |author-link2=Luisa Maffi |last2=Maffi |first2=Luisa |year=1999 |title=Color appearance and the emergence and evolution of basic color lexicons |journal=American Anthropologist |volume=101 |issue=4 |pages=743–760 |doi=10.1525/aa.1999.101.4.743}}</ref>
! style="font-weight:normal;" | light–warm (white / yellow / red) dark–cool (black / blue / green)
|}
Stage I contains two terms, white and black (light and dark); these terms are referenced broadly to describe other undefined color terms. For example, the Jale highland group in New Guinea identify the color of blood as black. This is because blood, as a relatively dark liquid, is grouped into the same color classification as black.

In the [[Bassa language]], there are two terms for classifying colors; ''ziza'' (white, yellow, orange and red) and ''hui'' (black, violet, blue, and green).<ref>{{cite journal |last1=McNeill |first1=N. B. |title=Colour and colour terminology |journal=Journal of Linguistics |date=28 November 2008 |volume=8 |issue=1 |pages=21–33 |doi=10.1017/S002222670000311X|s2cid=26668333 }}</ref>

In the [[Pirahã language]], there appear to be no color terms beyond describing lightness and darkness.<ref>Kay, Paul. (2007). [http://itre.cis.upenn.edu/myl/languagelog/archives/004399.html ''Pirahã Color Terms'']. Retrieved 17 March 2019.</ref>

The [[Grand Valley Dani|Dani language]] of western New Guinea differentiates only two basic colors: ''mili'' for cool/dark shades such as blue, green, and black; and ''mola'' for warm/light colors such as red, yellow, and white.<ref>{{Cite book|url=https://books.google.com/books?id=Fl8gAQAAIAAJ&q=mili+mola++|title=The Invention of Basic Colour Terms|first=Barbara Ann Christine|last=Saunders|date=January 1, 1992|publisher=R.U.U.-I.S.O.R.|isbn=9789051870879 |via=Google Books}}</ref><ref>{{Cite journal|url=http://www.jstor.org/stable/2800917|title=Probabilities, Sampling, and Ethnographic Method: The Case of Dani Colour Names|author=Heider, Eleanor Rosch|year=1972|journal=Man|volume=7|issue=3|pages=448–466|via=JSTOR|doi=10.2307/2800917|jstor=2800917 }}</ref>

====Stage II (red)====
{| class="wikitable" style="font-weight:bold; float:right"
|-
! style="background-color:#c0c0c0;" | Stage II<ref name=Kay99/>
! style="font-weight:normal;" | white red / yellow black / blue / green
|}
Stage II implements a third term for red. Objects begin to rely less on their brightness for classification and in this stage we instead see each term cover a larger scope of colors. Specifically, blue and other darker shades continue to be described as black, yellow and orange colors are classified with red, and other bright colors continue to be classified with white.

In the [[Bambara language]], there are three color terms: ''dyema'' (white, beige), ''blema'' (reddish, brownish) and ''fima'' (dark green, indigo and black).

====Stage III/IV (yellow + green)====
{| class="wikitable" style="font-weight:bold; float:right"
|-
! style="background-color:#c0c0c0;" | Stage III<ref name=Kay99/>
! style="font-weight:normal;" |white red yellow black / blue / green
! style="font-weight:normal;" |white red yellow / green / blue black
! style="font-weight:normal;" |white red / yellow green / blue black
|}
Stage III identifies a third term referring either to green (IIIa) or yellow (IIIb). Most languages in the study with this system identify yellow over green, such as the [[Komi language]], where green is considered a shade of yellow ({{lang|kv|виж}}, {{lang|kv|vizh}}), called {{lang|kv|турун виж}} ({{lang|kv|turun vizh)}}: 'grass yellow'.<ref>Rueter, Jack M. (1996), Komia-anglisköĭ-finsköĭ</ref> However, the Nigerian [[Ibibio language]] and the Philippine [[Hanunoo language]] both identify green instead of yellow.

The [[Himba people|Ova-Himba]] use [[Himba people#Color perception and vision|four color names]]: ''zuzu'' stands for dark shades of blue, red, green, and purple; ''vapa'' is white and some shades of yellow; ''buru'' is some shades of green and blue; and ''dambu'' is some other shades of green, red, and brown.<ref name="himba colour">{{cite book |last1=Roberson |first1=Debi |last2=Davidoff |first2=Jules |last3=Davies |first3=Ian R.L. |last4=Shapiro |first4=Laura R. |date=January 2006 |chapter=Colour categories and category acquisition in Himba and English |pages=159–172 |publisher=John Benjamins Publishing Company |doi=10.1075/z.pics2.14rob |editor1-first=Nicola |editor1-last=Pitchford |editor2-first=Carole P. |editor2-last=Biggam |title=Progress in Colour Studies |volume=II Psychological aspects |isbn=978-90-272-3240-3 |via=ResearchGate.net |chapter-url=https://www.researchgate.net/publication/43627151 |access-date=2012-05-28}}</ref> It is thought that this may [[Stroop effect|increase the time it takes]] for the Ova-Himba to distinguish between two colors that fall under the same Herero color category, compared to people whose language separates the colors into two different color categories.<ref>{{cite journal |last1=Reiger |first1=Terry |first2=Paul |last2=Kay |title=Language, thought, and color: Whorf was half right |journal=[[Trends (journals)|Trends in Cognitive Sciences]] |date=28 August 2009 |doi=10.1016/j.tics.2009.07.001 |url=http://www1.icsi.berkeley.edu/~kay/tics2.pdf |access-date=2012-08-29 |volume=13 |issue=10 |pages=439–446 |pmid=19716754|s2cid=2564005 }}</ref>

{| class="wikitable" style="font-weight:bold; float:right"
|-
! style="background-color:#c0c0c0;" | Stage IV<ref name=Kay99/>
! style="font-weight:normal;" |white red yellow green black / blue
! style="font-weight:normal;" |white red yellow green / blue black
|}
Stage IV incorporates green or yellow, which ever was not already present, i.e. stage IIIa languages will adopt yellow and stage IIIb languages will adopt green. Most stage IV languages continue to [[colexification|colexify]] blue and green, as listed in ''[[Blue–green distinction in language]]''.

The [[Chinese character]] [[wikt:青|青]] (pronounced {{lang|cmn-Latn|qīng}} in [[Standard Chinese|Mandarin]] and {{lang|ja-Latn|ao}} in Japanese) has a meaning that covers both blue and green. In more contemporary terms, they are [[wikt:藍|藍]] ({{lang|cmn-Latn|lán}}, in Mandarin) and [[wikt:綠|綠]] ({{lang|cmn-Latn|lǜ}}, in Mandarin) respectively. Japanese also has two terms that refer specifically to the color green, {{lang|ja|[[wikt:緑|緑]]}} ({{lang|ja-Latn|midori}}, derived from the classical Japanese descriptive verb {{lang|ja-Latn|midoru}} 'to be in leaf, to flourish' in reference to trees) and {{lang|ja|グリーン}} ({{lang|ja-Latn|guriin}}, which is derived from the English word 'green').{{Citation needed|date=January 2017}}

====Stage V (blue)====
{| class="wikitable" style="font-weight:bold; float:right"
|-
! style="background-color:#c0c0c0;" | Stage V<ref name=Kay99/>
! style="font-weight:normal;" |white red yellow green blue black
|}
Stage V introduces blue as its own color term, differentiating from black or from green.

====Stage VI (brown)====
The seventh basic color term is likely to be brown.

In English, this is the first basic color term (other than black and white) that is not differentiated on hue, but rather on lightness. English splits some hues into several distinct colors according to lightness: such as red and pink or orange and brown. To English speakers, these pairs of colors, which are objectively no more different from one another than light green and dark green, are conceived of as belonging to different categories.<ref name="BerlinKay1969">{{cite book |author1-link=Brent Berlin |last1=Berlin |first1=Brent |author2-link=Paul Kay |first2=Paul |last2=Kay |year=1969 |title=Basic Color Terms: Their universality and evolution |title-link=Basic Color Terms: Their Universality and Evolution}}</ref>

====Stage VII====
Stage VII adds additional terms for orange, pink, purple or [[gray|grey]], but these do not exhibit the same hierarchy as the previous seven colors.<ref>{{cite book |editor-last=Varley |editor-first=Helen |title=Color |location=London |date=1980 |publisher=Marshall Editions |isbn=0-89535-037-8 |chapter=The Vocabulary of Color |pages=50–51}}</ref>

English contains eleven basic color terms: 'black', 'white', 'red', 'green', 'yellow', 'blue', 'brown', 'orange', 'pink', 'purple', and 'grey'.

====Stage VII+====
[[File:Moscow-metro-light-blue-line.png|thumb|Using light blue (goluboi) and dark blue (sinii) colors for different lines of the Moscow Metro.]]
Languages with further color distinction use relativistic light / dark terms like light blue / [[Navy blue|dark blue]] (in comparison to blue sky / blue ocean), or [[Pink|pale red]] / [[Maroon|deep red]].

[[Italian language|Italian]], [[Russian language|Russian]] and [[Hebrew language|Hebrew]] have twelve basic color terms, each distinguishing blue and light blue. A Russian will make the same red / pink and orange / brown distinctions, but will also make a further distinction between {{lang|ru-Latn|sinii}} and {{lang|ru-Latn|goluboi}}, which English speakers would call dark and light blue. To Russian speakers, {{lang|ru-Latn|sinii}} and {{lang|ru-Latn|goluboi}} are as separate as red and pink, or orange and brown.<ref>{{cite news |title=Seeing the blues |magazine=Nature |series=News |date=2007-04-30 |url=http://www.nature.com/news/2007/070430/full/news070430-2.html}}</ref>

[[Hungarian language|Hungarian]] and [[Turkish language|Turkish]] [[Hungarian language#Two words for "red"|distinguish multiple words]] for 'red': {{lang|hu|piros}} and {{lang|hu|vörös}} (Hungarian; {{lang|hu|vörös}} is a darker red), and {{lang|tr|kırmızı}}, {{lang|tr|al}}, and {{lang|tr|kızıl}} (Turkish); ''kırmızı'' now includes all reds but originally referred to crimson, to which it is cognate, while ''kızıl'' mainly refers to scarlet and other orange-tinted or brownish reds. Two words for 'red' are also found in Irish and [[Scottish Gaelic]]: ({{lang|ga|dearg}} for light, bright red and {{lang|ga|rua}} or {{lang|ga|ruadh}} respectively for dark, brownish red). Turkish also has two words for 'white' ({{lang|tr|beyaz}} and {{lang|tr|ak}}) and 'black' ({{lang|tr|siyah}} and {{lang|tr|kara}}). ''Ak'' and ''beyaz'' have the same meaning, while ''kara'' is a broader term than ''siyah'' and also includes dark browns; which word is used also depends on the kind of object being described. Both ''Ak'' and ''kara'' are of turkic origin, while ''siyah'' is borrowed from [[Persian language|Persian]], and ''beyaz'' from Arabic ''bayāḍ'' ''بياض''.

In [[Serbo-Croatian|Serbian/Croatian]] language there are differences in dark brown (''mrk''), brown (''smeđ'' & ''kestenjast''), red (''crven''), pink (''ružičast'') and orange (''narandžast''), as well as in blue hues: very dark blue or blue-green (''teget''), dark blue (''modar''), blue (''plav'') and ash blue (''sinj'').

An interesting case that deviates from this pattern is [[Irish language|Irish]]'s two words for green:
* {{lang|ga|glas}} denotes the green color of plants
* {{lang|ga|uaine}} denotes artificial greens of dyes, paints etc.
This distinction is made even if two shades are identical. ''Glas'' is also used for "natural" greys, such as the [[Eastern grey squirrel|grey squirrel]], ''iora glas''.<ref>{{Cite web|url=https://www.thejournal.ie/readme/the-irish-for-colours-you-may-have-forgotten-lots-of-your-schooling-but-you-should-remember-the-word-bandearg-4447647-Jan2019/|title=The Irish For: How many colours can you remember?|first=Darach Ó|last=Séaghdha|date=January 20, 2019|website=TheJournal.ie}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=gnQ9AAAAYAAJ&q=uaine+glas+distinction|title=Transactions of the Gaelic Society of Glasgow ...|first=Gaelic Society of|last=Glasgow|via=Google Books}}</ref>

====Linguistic relativity====
{{main|Linguistic relativity and the color naming debate}}
These colors roughly correspond to the sensitivities of the retinal ganglion cells, leading Berlin and Kay to argue that color naming is not merely a cultural phenomenon, but is one that is also constrained by biology—that is, language is shaped by perception.<ref name="BerlinKay1969"/> A 2012 study<ref>{{cite journal |last1=Loreto |first1=Vittorio |last2=Mukherjee |first2=Animesh |last3=Tria |first3=Francesca |year=2012 |title=On the origin of the hierarchy of color names |journal=Proceedings of the National Academy of Sciences |volume=109 |issue=18 |pages=6819–6824 |doi=10.1073/pnas.1113347109 |pmc=3344991 |pmid=22509002|bibcode=2012PNAS..109.6819L |doi-access=free }}</ref> suggested that the origin of this hierarchy may be tied to human vision and the time ordering in which these color names get accepted or agreed upon in a population perfectly matches the order predicted by the hierarchy.

=== Non-hue terms ===
This article mostly describes the color terms that define the ''hue'' of a color, since hue is considered the most innate dimension of the three. However, other terms are often used to describe the other two dimensions, which can be seen as common prefixes to the root terms that generally describe hue. Adding prefixes to root color terms generates [[wikt:multilexemic|multilexemic]] colors. Examples of common prefix adjectives can be seen in a [[List_of_colors_(compact)|list of color names]] and are described:
* [[Brightness]]: can describe either high luminosity or high saturation, according to the [[Helmholtz–Kohlrausch effect]] and/or [[Color appearance model#Colorfulness appearance|Hunt Effect]].
* [[Lightness]]: describes both a high luminosity ''and'' low saturation
* [[Darkness]]: the opposite of lightness, or low luminosity
* [[Paleness (color)|Paleness]], ''dullness'': a measure of desaturation
* [[wikt:deep#Adjective|Deep]], [[Royal blue|Royal]]: may refer to darkness and/or high saturation; unrelated to [[color depth]].
* [[Colorfulness#Excitation purity|Pure]], ''Bold'', [[wikt:vivid|Vivid]], [[wikt:rich#Adjective|Rich]]: all referring to high saturation
* [[Pastel (color)|Pastel]]: refers to colors with high luminosity and low saturation.
* [[wikt:neon#Adjective|Neon]]: bright, in either of the word's connotations; alluding to the bright glow of [[neon lighting]].
* [[fluorescence|Fluorescent]]: very bright, sometimes also highly saturated. Named after the [[fluorescence]] effect of [[pigment]]s and [[dye]]s, which can produce a luminous glow when viewed under [[Ultraviolet|UV light]], thereby appearing significantly brighter than their surroundings.<ref>{{cite magazine |first=David |last=Schoonmaker |date=May–June 2006 |title=Sunshine on a cloudy day |magazine=American Scientist |volume=94 |number=3 |page=217 |doi=10.1511/2006.59.217 |url=https://www.americanscientist.org/article/sunshine-on-a-cloudy-day}}</ref>

=== Non-dimensional terms ===
Other terms sometimes used to describe color are related to physical phenomenon that do not describe a single color, but describe the dynamic nature of an object's color. These include:
* [[Gloss (optics)|Glossy]]: whether the surface reflects ''diffusely'' or ''specularly'' (sharply)
* [[Metallic color|Metallic]]: distinguishing 'gold' and 'silver' from shades of 'yellow' and 'grey', respectively
* [[Iridescence|Iridescent]]: dependence of color on viewing angle, innate to [[structural coloration]]
* [[opacity (optics)|opacity]]: opaque (solid) vs. translucent (transparent or see-through)

=== Abstract and descriptive color terms ===
Color terms can be classified as ''abstract'' or ''descriptive'', though the distinction is often unclear.

'''Abstract''' color terms only refer to the color they represent and any etymological link to an object of that color is lost. In English white, black, red, yellow, green, blue, brown, and grey are abstract color terms. These terms are also ''basic color terms'' (as described above), though other abstract terms like maroon and [[magenta]] are not considered basic color terms.

'''Descriptive''' color terms are secondarily used to describe a color but primarily refer to an object or phenomenon. 'Salmon', 'rose', 'saffron', and 'lilac' are descriptive color terms in English because their use as color terms is derived in reference to natural colors of [[salmon]] flesh, [[rose]] flowers, infusions of [[saffron]] pistils, and [[lilac]] blossoms respectively.

Abstract color terms in one may be represented by descriptive color terms in another; for example in Japanese pink is {{lang|ja-Latn|momoiro}} ({{lang|ja|桃色}}, lit. 'peach-color') and grey is either {{lang|ja-Latn|haiiro}} or {{lang|ja-Latn|nezumiiro}} ({{lang|ja|灰色}}, {{lang|ja|鼠色}}, lit. 'ash-color' for light greys and 'mouse-color' for dark greys respectively). Nevertheless, as languages evolve they may adopt or invent new abstract color terms, as Japanese has adopted {{lang|ja-Latn|pinku}} ({{lang|ja|ピンク}}) for pink and {{lang|ja-Latn|gurē}} ({{lang|ja|グレー}}) for grey from English.

While most of the 11 basic color terms in English are decidedly abstract, three of them (all stage VII, so understandably the youngest basic color terms) are arguably still descriptive:
* ''Pink'' was originally a descriptive color term derived from the name of a [[dianthus|flower called a 'pink']]. However, because the word 'pink' is rarely used to refer to the flower anymore, relative to its common usage as a color, it is often regarded as an abstract color term.
* ''Purple'' is another example of this shift, as it was originally a word that referred to the [[dye]] named [[Tyrian purple]], which took its name from the latin {{lang|la|purpura}}, which referred to both the dye and the [[sea snail]] from which the dye was derived. However, this etymological link has been lost in translation.
* ''[[orange (word)|Orange]]'' is difficult to categorize as abstract or descriptive because both its uses, as a color term and as a word for an object, are very common and it is difficult to distinguish which of the two is primary. As a basic color term it became established in the early to mid 20th century; before that time artist's palettes called it 'yellow-red'. In English, the use of the word 'orange' for a fruit predates its use as a color term. The word comes from French {{lang|fr|orenge}}, which derives via [[Arabic]] [[wikt:naranj#Arabic|''narand͡ʒ'']] and [[Sanskrit]] {{lang|sa-Latn|narang}} from a [[Dravidian languages|Dravidian language]] such as [[Tamil language|Tamil]] or [[Tulu language|Tulu]].<ref>{{Cite encyclopedia |date=June 2012 |title=orange, n.1 and adj.1 |encyclopedia=[[Oxford English Dictionary]] |publisher=[[Oxford University Press]] |url=http://www.oed.com/view/Entry/132163 |access-date=2012-09-04}}</ref> The derived form ''orangish'' as a color is attested from the late 19th century<ref>''[[Oxford English Dictionary]],'' 'orangish'</ref> by reference to the fruit.

==Struggle in linguistics==
{{Technical|section|date=October 2022}}
Research on color terms is often conducted without reference to common uses of the term or its significance within the context of its original language. In John A. Lucy's article ''The linguistics of 'colour''' he identifies two key categories. One of these is 'characteristic referential range', or the use of a color term to identify or differentiate a referent over a wide context.<ref name=":0" />

==Standardized systems==
In contrast with the color terms of natural language, systematized color terms also exist. Some examples of color naming systems are [[Color Naming System|CNS]]<ref>{{Citation
| last1 = Berk | first1 = T.
| last2 = Brownston | first2 = L.
| last3 = Kaufman | first3 = A.
| contribution = A New Color-Naming System for Graphics Languages
| title = IEEE Computer Graphics and Applications
| publisher = IEEE
| volume = 2
| pages = 37–44
| year = 1982 }}</ref> and [[ISCC-NBS system|ISCC–NBS]] lexicon of color terms. The disadvantage of these systems, however, is that they only specify specific color samples, so while it is possible to, by interpolating, convert any color to or from one of these systems, a lookup table is required. In other words, no simple invertible equation can convert between [[CIE 1931 color space|CIE XYZ]] and one of these systems.

[[Philately|Philatelists]] traditionally use names to identify [[postage stamp color]]s. While the names are largely standardized within each country, there is no broader agreement, and so for instance the US-published [[Scott catalogue]] will use different names than the British [[Stanley Gibbons]] catalogue.

On modern computer systems a standard set of basic color terms is now used across the [[Web colors|web color names]] (SVG 1.0/CSS3), [[HTML color names]], [[X11 color names]] and the [[.NET Framework]] color names, with only a few minor differences.

The [[Crayola]] company is famous for its many [[List of Crayola crayon colors|crayon colors]], often creatively named.

[[Heraldry]] has standardized names for '[[Tincture (heraldry)|tincture]]s', subdivided into 'colors', 'metals', and 'furs'.

==See also==

* [[List of colors (compact)|Lists of colors]]
* [[Color wheel]]
* [[Lazarus Geiger]]
* [[Himba people#Anthropological investigations|How the Himba see green and blue]]

== References ==
{{Reflist}}

== External links ==
* [http://www.worldwidewords.org/articles/colour.htm The Colour of Words] – Article on Color Names
* [http://coloria.net/bonus/colornames.htm Coloria.net: Color names]
* [https://www.lingualift.com/blog/japanese-color-names/ Japanese Colour Names Cheat Sheet]
* [http://www.colordic.org/w/ Japanese Traditional Color Names]
* [http://www.color-guide.com/e_index.htm Japanese colors with English names]
* [http://www.iscc.org/ Inter-Society Color Council]
* The color names in [http://www.w3.org/TR/css3-color/#svg-color CSS 3: Color Module] and [http://www.w3.org/TR/SVG/types.html SVG]
* [http://people.csail.mit.edu/jaffer/Color/Dictionaries Survey of color dictionaries]
* [http://colornaming.net An Online Colour Naming Experiment]
* [http://www.omniglot.com/language/colours/index.php Colour Words in Many Languages]
* [https://en.colour.name/ Test your own color terms]
* [http://www.spoonflower.com/SpoonFlower_ColorMap_2-1.png SpoonFlower color map]
* [http://color.method.ac/ Color Method]
* [http://i.stack.imgur.com/weg6q.png i.stack.imgur basic color terms]
* [https://web.archive.org/web/20140403202023/http://www.w3schools.com/tags/ref_colorpicker.asp?colorhex=F0F8FF HTML Color Picker]

{{color shades|state=collapsed}}
{{color topics|state=collapsed}}

[[Category:Color names| ]]
[[Category:Shades of color]]

Instrumentation (computer programming)

2020-12-30T09:16:42Z

Crasshopper: /* Output */

{{more footnotes|date=December 2013}}
In the context of [[computer programming]], '''instrumentation''' refers to the measure of a product's performance, to diagnose errors, and to write [[Tracing (software)|trace]] information.<ref>[http://pic.dhe.ibm.com/infocenter/rtrthelp/v8r0m0/index.jsp?topic=%2Fcom.ibm.rational.testrt.doc%2Ftopics%2Fcinstruovw.html Source Code Instrumentation Overview at IBM website]</ref> Instrumentation can be of two types: source instrumentation and binary instrumentation.

== Output ==
In programming, instrumentation means:<ref>{{cite web|url=http://www.drdobbs.com/architecture-and-design/commenting-testing-and-instrumenting-cod/229300224|title=Commenting, Testing, and Instrumenting Code|date=January 3, 2011|accessdate=January 29, 2014}}</ref>

* [[Profiling (computer programming)|Profiling]]: measuring dynamic program behaviors during a training run with a representative input. This is useful for properties of a program that cannot be [[static program analysis|analyzed statically]] with sufficient precision, such as [[alias analysis]].
* Inserting timers into functions.
* Logging major events such as crashes.

== Limitations ==

Instrumentation is limited by execution coverage. If the program never reaches a particular point of execution, then instrumentation at that point collects no data. For instance, if a word processor application is instrumented, but the user never activates the print feature, then the instrumentation can say nothing about the routines which are used exclusively by the printing feature.

Some types of instrumentation may cause a dramatic increase in execution time. This may limit the application of instrumentation to debugging contexts.

==See also==
* [[Hooking]] – range of techniques used to alter or augment the behavior of an operating system, of applications, or of other software components either by intercepting function calls or messages or events passed between software components
* [[Instruction set simulator]] – simulation of all instructions at machine code level to provide instrumentation
* [[Runtime intelligence]] – technologies, managed services and practices for the collection, integration, analysis, and presentation of application usage levels, patterns, and practices
* [[Software performance analysis]] – techniques to monitor code performance, including instrumentation
* [[Hardware performance counter]]
* [[DTrace]] – A comprehensive dynamic tracing framework for troubleshooting kernel and application problems on production systems in real time, implemented in [[Solaris (operating system)|Solaris]], [[macOS]], [[FreeBSD]], and many other platforms and products.
* [[Java Management Extensions|''Java Management Extensions'' (JMX)]] – Java technology for managing and monitoring applications, system objects, devices (such as printers), and service-oriented networks
* [[Application Response Measurement]] – standardized instrumentation [[Application programming interface|API]] for [[C (programming language)|C]] and [[Java (programming language)|Java]]
* [[Dynamic recompilation]] – a feature of some emulators and virtual machines where the system may recompile some part of a program during execution

==References==
{{reflist}}
* [http://msdn.microsoft.com/en-us/library/aa983649(VS.71).aspx Introduction to Instrumentation and Tracing: Microsoft Developer Network]
* [https://developer.apple.com/library/content/documentation/DeveloperTools/Conceptual/InstrumentsUserGuide/index.html Apple Developer Tools: Introduction to Instruments]
* [http://sourceware.org/systemtap/ SystemTap] provides free software (GPL) infrastructure to simplify the gathering of information about the running Linux system.
* [https://github.com/corelight/cwrap cwrap] Auto wrap C and C++ functions with instrumentation.

[[Category:Software optimization]]
[[Category:System administration]]
[[Category:Management systems]]
[[Category:Debugging]]

{{Comp-sci-stub}}

Instrumentation (computer programming)

2020-12-30T09:16:21Z

Crasshopper: /* Output */

{{more footnotes|date=December 2013}}
In the context of [[computer programming]], '''instrumentation''' refers to the measure of a product's performance, to diagnose errors, and to write [[Tracing (software)|trace]] information.<ref>[http://pic.dhe.ibm.com/infocenter/rtrthelp/v8r0m0/index.jsp?topic=%2Fcom.ibm.rational.testrt.doc%2Ftopics%2Fcinstruovw.html Source Code Instrumentation Overview at IBM website]</ref> Instrumentation can be of two types: source instrumentation and binary instrumentation.

== Output ==
In programming, instrumentation means:<ref>{{cite web|url=http://www.drdobbs.com/architecture-and-design/commenting-testing-and-instrumenting-cod/229300224|title=Commenting, Testing, and Instrumenting Code|date=January 3, 2011|accessdate=January 29, 2014}}</ref>

* [[Profiling (computer programming)|Profiling]]: measuring dynamic program behaviors during a training run with a representative input. This is useful for properties of a program that cannot be [[static program analysis|analyzed statically]] with sufficient precision, such as [[alias analysis]].
* Inserting timers into functions.
* Logging major events.

== Limitations ==

Instrumentation is limited by execution coverage. If the program never reaches a particular point of execution, then instrumentation at that point collects no data. For instance, if a word processor application is instrumented, but the user never activates the print feature, then the instrumentation can say nothing about the routines which are used exclusively by the printing feature.

Some types of instrumentation may cause a dramatic increase in execution time. This may limit the application of instrumentation to debugging contexts.

==See also==
* [[Hooking]] – range of techniques used to alter or augment the behavior of an operating system, of applications, or of other software components either by intercepting function calls or messages or events passed between software components
* [[Instruction set simulator]] – simulation of all instructions at machine code level to provide instrumentation
* [[Runtime intelligence]] – technologies, managed services and practices for the collection, integration, analysis, and presentation of application usage levels, patterns, and practices
* [[Software performance analysis]] – techniques to monitor code performance, including instrumentation
* [[Hardware performance counter]]
* [[DTrace]] – A comprehensive dynamic tracing framework for troubleshooting kernel and application problems on production systems in real time, implemented in [[Solaris (operating system)|Solaris]], [[macOS]], [[FreeBSD]], and many other platforms and products.
* [[Java Management Extensions|''Java Management Extensions'' (JMX)]] – Java technology for managing and monitoring applications, system objects, devices (such as printers), and service-oriented networks
* [[Application Response Measurement]] – standardized instrumentation [[Application programming interface|API]] for [[C (programming language)|C]] and [[Java (programming language)|Java]]
* [[Dynamic recompilation]] – a feature of some emulators and virtual machines where the system may recompile some part of a program during execution

==References==
{{reflist}}
* [http://msdn.microsoft.com/en-us/library/aa983649(VS.71).aspx Introduction to Instrumentation and Tracing: Microsoft Developer Network]
* [https://developer.apple.com/library/content/documentation/DeveloperTools/Conceptual/InstrumentsUserGuide/index.html Apple Developer Tools: Introduction to Instruments]
* [http://sourceware.org/systemtap/ SystemTap] provides free software (GPL) infrastructure to simplify the gathering of information about the running Linux system.
* [https://github.com/corelight/cwrap cwrap] Auto wrap C and C++ functions with instrumentation.

[[Category:Software optimization]]
[[Category:System administration]]
[[Category:Management systems]]
[[Category:Debugging]]

{{Comp-sci-stub}}

Instrumentation (computer programming)

2020-12-30T09:15:37Z

Crasshopper: Omit needless words.

{{more footnotes|date=December 2013}}
In the context of [[computer programming]], '''instrumentation''' refers to the measure of a product's performance, to diagnose errors, and to write [[Tracing (software)|trace]] information.<ref>[http://pic.dhe.ibm.com/infocenter/rtrthelp/v8r0m0/index.jsp?topic=%2Fcom.ibm.rational.testrt.doc%2Ftopics%2Fcinstruovw.html Source Code Instrumentation Overview at IBM website]</ref> Instrumentation can be of two types: source instrumentation and binary instrumentation.

== Output ==
In programming, instrumentation means:<ref>{{cite web|url=http://www.drdobbs.com/architecture-and-design/commenting-testing-and-instrumenting-cod/229300224|title=Commenting, Testing, and Instrumenting Code|date=January 3, 2011|accessdate=January 29, 2014}}</ref>

* [[Profiling (computer programming)|Profiling]]: measuring dynamic program behaviors during a training run with a representative input. This is useful for properties of a program that cannot be [[static program analysis|analyzed statically]] with sufficient precision, such as [[alias analysis]].
* Function timers.
* Logging major events.

== Limitations ==

Instrumentation is limited by execution coverage. If the program never reaches a particular point of execution, then instrumentation at that point collects no data. For instance, if a word processor application is instrumented, but the user never activates the print feature, then the instrumentation can say nothing about the routines which are used exclusively by the printing feature.

Some types of instrumentation may cause a dramatic increase in execution time. This may limit the application of instrumentation to debugging contexts.

==See also==
* [[Hooking]] – range of techniques used to alter or augment the behavior of an operating system, of applications, or of other software components either by intercepting function calls or messages or events passed between software components
* [[Instruction set simulator]] – simulation of all instructions at machine code level to provide instrumentation
* [[Runtime intelligence]] – technologies, managed services and practices for the collection, integration, analysis, and presentation of application usage levels, patterns, and practices
* [[Software performance analysis]] – techniques to monitor code performance, including instrumentation
* [[Hardware performance counter]]
* [[DTrace]] – A comprehensive dynamic tracing framework for troubleshooting kernel and application problems on production systems in real time, implemented in [[Solaris (operating system)|Solaris]], [[macOS]], [[FreeBSD]], and many other platforms and products.
* [[Java Management Extensions|''Java Management Extensions'' (JMX)]] – Java technology for managing and monitoring applications, system objects, devices (such as printers), and service-oriented networks
* [[Application Response Measurement]] – standardized instrumentation [[Application programming interface|API]] for [[C (programming language)|C]] and [[Java (programming language)|Java]]
* [[Dynamic recompilation]] – a feature of some emulators and virtual machines where the system may recompile some part of a program during execution

==References==
{{reflist}}
* [http://msdn.microsoft.com/en-us/library/aa983649(VS.71).aspx Introduction to Instrumentation and Tracing: Microsoft Developer Network]
* [https://developer.apple.com/library/content/documentation/DeveloperTools/Conceptual/InstrumentsUserGuide/index.html Apple Developer Tools: Introduction to Instruments]
* [http://sourceware.org/systemtap/ SystemTap] provides free software (GPL) infrastructure to simplify the gathering of information about the running Linux system.
* [https://github.com/corelight/cwrap cwrap] Auto wrap C and C++ functions with instrumentation.

[[Category:Software optimization]]
[[Category:System administration]]
[[Category:Management systems]]
[[Category:Debugging]]

{{Comp-sci-stub}}

Rely on Your Beliefs Fund

2020-10-15T09:34:56Z

Crasshopper: /* Notable contributors */

The '''Rely on Your Beliefs Fund''' ('''ROYB Fund''') is an American [[Political action committee#Leadership PACs|Political Action Committee]] associated with [[Roy Blunt|Rep. Roy Blunt]]. Since its inception, the fund has collected and spent more than $1.8 million.

On May 23, 2002, the ROYB Fund was forced to enter into a consent decree with the Missouri [[Ethics Commission]] for failing to comply with Missouri law by filing a statement of organization and required reports in a timely fashion. [http://www.firedupamerica.com/system/files?file=blunt_consent_decree.pdf]

==People==
* [[Roy Blunt]] - honorary chairman. [https://archive.is/20130104142822/http://www.riverfronttimes.com/Issues/2001-10-03/news/feature2.html]
* [[James W. Ellis|Jim Ellis]] - founding staffer [https://web.archive.org/web/20070701113247/http://www.firedupamerica.com/jimellis]

==Notable contributors==
* [[AbbVie]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Altria]] (one of the top corporate contributors in 2001-02 according to [http://www.politicalaccountability.net/Corporate%20Profile%20-%20Altria.pdf]; Blunt's current wife was previously a lobbyist for the company)
* [[American Airlines]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[American Beverage Association]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* American Crystal Sugar Company
* [[American Dental Association]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[Amgen]] [https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]
* [[Anheuser-Busch]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[Anthem Insurance]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[AT&T]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Enron]] [https://web.archive.org/web/20060210125045/http://query.nictusa.com/cgi-bin/com_rcvd/C00344648/]
* [[Facebook]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Koch Industries]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Merck]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Microsoft PAC]] [https://web.archive.org/web/20041011190333/http://redmondmag.com/features/article.asp?EditorialsID=440]
* [[National Association of Broadcasters]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Pfizer]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* Travellers <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[Time-Warner]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[T-Mobile]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Waste Management]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* [[Walmart]] <ref>https://docquery.fec.gov/cgi-bin/com_rcvd/C00344648/</ref>
* In March 2002, the [[Tigua tribe]] began making political contributions on request of [[Jack Abramoff]] in hopes of getting a [[casino]] approved. [http://www.texasobserver.org/showArticle.asp?ArticleID=1830]

==Payees==
* [[James W. Ellis|J.W. Ellis Co.]] (roughly $88,000 in consulting fees) [https://web.archive.org/web/20060207040903/http://www.citizensforethics.org/press/newsrelease.php?view=84]

==References==
* [http://royb.net/ fund's site]
* [http://projects.propublica.org/lpacs/pac/rely-on-your-beliefs Pro Publica's page on the fund]
* [http://herndon1.sdrdc.com/cgi-bin/fecimg/?C00344648 Federal Election Commission documents]
* [http://www.opensecrets.org/pacs/lookup2.php?strID=C00344648 Open Secrets page on the fund]
* [https://www.washingtonpost.com/wp-dyn/content/article/2005/05/16/AR2005051601334_pf.html Washington Post article]

{{DEFAULTSORT:Rely On Your Beliefs Fund}}
[[Category:United States political action committees]]

Rely on Your Beliefs Fund

2020-10-15T09:22:10Z

Crasshopper: /* Notable contributors */

The '''Rely on Your Beliefs Fund''' ('''ROYB Fund''') is an American [[Political action committee#Leadership PACs|Political Action Committee]] associated with [[Roy Blunt|Rep. Roy Blunt]]. Since its inception, the fund has collected and spent more than $1.8 million.

On May 23, 2002, the ROYB Fund was forced to enter into a consent decree with the Missouri [[Ethics Commission]] for failing to comply with Missouri law by filing a statement of organization and required reports in a timely fashion. [http://www.firedupamerica.com/system/files?file=blunt_consent_decree.pdf]

==People==
* [[Roy Blunt]] - honorary chairman. [https://archive.is/20130104142822/http://www.riverfronttimes.com/Issues/2001-10-03/news/feature2.html]
* [[James W. Ellis|Jim Ellis]] - founding staffer [https://web.archive.org/web/20070701113247/http://www.firedupamerica.com/jimellis]

==Notable contributors==
* [[AbbVie]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Altria]] (one of the top corporate contributors in 2001-02 according to [http://www.politicalaccountability.net/Corporate%20Profile%20-%20Altria.pdf]; Blunt's current wife was previously a lobbyist for the company)
* [[Amgen]] [https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]
* [[AT&T]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Enron]] [https://web.archive.org/web/20060210125045/http://query.nictusa.com/cgi-bin/com_rcvd/C00344648/]
* [[Facebook]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Koch Industries]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Merck]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Microsoft PAC]] [https://web.archive.org/web/20041011190333/http://redmondmag.com/features/article.asp?EditorialsID=440]
* [[National Association of Broadcasters]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[Pfizer]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* [[T-Mobile]] <ref>[https://www.opensecrets.org/pacs/pacgave.php?cmte=C00344648&cycle=2020]</ref>
* In March 2002, the [[Tigua tribe]] began making political contributions on request of [[Jack Abramoff]] in hopes of getting a [[casino]] approved. [http://www.texasobserver.org/showArticle.asp?ArticleID=1830]

==Payees==
* [[James W. Ellis|J.W. Ellis Co.]] (roughly $88,000 in consulting fees) [https://web.archive.org/web/20060207040903/http://www.citizensforethics.org/press/newsrelease.php?view=84]

==References==
* [http://royb.net/ fund's site]
* [http://projects.propublica.org/lpacs/pac/rely-on-your-beliefs Pro Publica's page on the fund]
* [http://herndon1.sdrdc.com/cgi-bin/fecimg/?C00344648 Federal Election Commission documents]
* [http://www.opensecrets.org/pacs/lookup2.php?strID=C00344648 Open Secrets page on the fund]
* [https://www.washingtonpost.com/wp-dyn/content/article/2005/05/16/AR2005051601334_pf.html Washington Post article]

{{DEFAULTSORT:Rely On Your Beliefs Fund}}
[[Category:United States political action committees]]

Inscribed angle

2020-08-04T21:26:36Z

Crasshopper: Don’t put a degenerate case in the intro.

[[File:Inscribed angles2.svg|thumb|upright=1.0|The inscribed angle ''θ'' is half of the central angle 2''θ'' that subtends the same arc on the circle. Thus, the angle ''θ'' does not change as its vertex is moved around on the circle.]]

In [[geometry]], an '''inscribed angle''' is the [[angle]] formed in the interior of a [[circle]] when two [[secant line]]s intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.

Equivalently, an inscribed angle is defined by two [[Chord (geometry)|chords]] of the circle sharing an endpoint.

The '''inscribed angle theorem''' relates the [[Angle#Measuring angles|measure]] of an inscribed angle to that of the [[central angle]] subtending the same [[circular arc|arc]].

The inscribed angle theorem appears as Proposition 20 on Book 3 of [[Euclid's Elements|Euclid’s "Elements"]].

==Theorem==

===Statement===

[[File:ArcCapable.gif|thumb|For fixed points ''A'' and ''B'', the set of points ''M'' in the plane for which the angle ''AMB'' is equal to ''α'' is an arc of a circle. The measure of ∠ ''AOB'', where ''O'' is the center of the circle, is 2''α''.]]

The inscribed angle theorem states that an angle ''θ'' inscribed in a circle is half of the central angle 2''θ'' that [[Subtended arc|subtend]]s the same [[Arc (geometry)|arc]] on the circle. Therefore, the angle does not change as its [[Vertex (geometry)|vertex]] is moved to different positions on the circle.

===Proof===

====Inscribed angles where one chord is a diameter====
[[File:InscribedAngle 1ChordDiam.svg|thumb|Case: One chord is a diameter]]
Let ''O'' be the center of a circle, as in the diagram at right. Choose two points on the circle, and call them ''V'' and ''A''. Draw line ''VO'' and extended past ''O'' so that it intersects the circle at point ''B'' which is [[diametrically opposite]] the point ''V''. Draw an angle whose [[Vertex (geometry)|vertex]] is point ''V'' and whose sides pass through points ''A'' and ''B''.

Draw line ''OA''. Angle ''BOA'' is a [[central angle]]; call it ''θ''. Lines ''OV'' and ''OA'' are both [[radius|radii]] of the circle, so they have equal lengths. Therefore, triangle ''VOA'' is [[isosceles]], so angle ''BVA'' (the inscribed angle) and angle ''VAO'' are equal; let each of them be denoted as ''ψ''.

Angles ''BOA'' and ''AOV'' are [[supplementary angle|supplementary]]. They add up to 180°, since line ''VB'' passing through ''O'' is a straight line. Therefore, angle ''AOV'' measures 180° − ''θ''.

It is known that the three angles of a [[triangle]] add up to 180°, and the three angles of triangle ''VOA'' are:

: 180° − ''θ''
: ''ψ''
: ''ψ''.

Therefore,

:<math> 2 \psi + 180^\circ - \theta = 180^\circ. </math>

Subtract 180° from both sides,

:<math> 2 \psi = \theta, </math>

where ''θ'' is the central angle subtending arc ''AB'' and ''ψ'' is the inscribed angle subtending arc ''AB''.

====Inscribed angles with the center of the circle in their interior====
[[File:InscribedAngle CenterCircle.svg|thumb|Case: Center interior to angle]]
Given a circle whose center is point ''O'', choose three points ''V'', ''C'', and ''D'' on the circle. Draw lines ''VC'' and ''VD'': angle ''DVC'' is an inscribed angle. Now draw line ''VO'' and extend it past point ''O'' so that it intersects the circle at point ''E''. Angle ''DVC'' subtends arc ''DC'' on the circle.

Suppose this arc includes point ''E'' within it. Point ''E'' is diametrically opposite to point ''V''. Angles ''DVE'' and ''EVC'' are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.

Therefore,

:<math> \angle DVC = \angle DVE + \angle EVC. </math>

then let

:<math> \psi_0 = \angle DVC, </math>
:<math> \psi_1 = \angle DVE, </math>
:<math> \psi_2 = \angle EVC, </math>

so that

:<math> \psi_0 = \psi_1 + \psi_2. \qquad \qquad (1) </math>

Draw lines ''OC'' and ''OD''. Angle ''DOC'' is a central angle, but so are angles ''DOE'' and ''EOC'', and
:<math> \angle DOC = \angle DOE + \angle EOC. </math>

Let

:<math> \theta_0 = \angle DOC, </math>
:<math> \theta_1 = \angle DOE, </math>
:<math> \theta_2 = \angle EOC, </math>

so that

:<math> \theta_0 = \theta_1 + \theta_2. \qquad \qquad (2) </math>

From Part One we know that <math> \theta_1 = 2 \psi_1 </math> and that <math> \theta_2 = 2 \psi_2 </math>. Combining these results with equation (2) yields

:<math> \theta_0 = 2 \psi_1 + 2 \psi_2 </math>

therefore, by equation (1),

:<math> \theta_0 = 2 \psi_0. </math>

====Inscribed angles with the center of the circle in their exterior====
[[Image:InscribedAngle CenterCircleExtV2.svg|thumb|Case: Center exterior to angle]]
The previous case can be extended to cover the case where the measure of the inscribed angle is the ''difference'' between two inscribed angles as discussed in the first part of this proof.

Given a circle whose center is point ''O'', choose three points ''V'', ''C'', and ''D'' on the circle. Draw lines ''VC'' and ''VD'': angle ''DVC'' is an inscribed angle. Now draw line ''VO'' and extend it past point ''O'' so that it intersects the circle at point ''E''. Angle ''DVC'' subtends arc ''DC'' on the circle.

Suppose this arc does not include point ''E'' within it. Point ''E'' is diametrically opposite to point ''V''. Angles ''EVD'' and ''EVC'' are also inscribed angles, but both of these angles have one side which passes through the center of the circle, therefore the theorem from the above Part 1 can be applied to them.

Therefore,
:<math> \angle DVC = \angle EVC - \angle EVD </math>.
then let
:<math> \psi_0 = \angle DVC, </math>
:<math> \psi_1 = \angle EVD, </math>
:<math> \psi_2 = \angle EVC, </math>
so that
:<math> \psi_0 = \psi_2 - \psi_1. \qquad \qquad (3) </math>

Draw lines ''OC'' and ''OD''. Angle ''DOC'' is a central angle, but so are angles ''EOD'' and ''EOC'', and
:<math> \angle DOC = \angle EOC - \angle EOD. </math>
Let
:<math> \theta_0 = \angle DOC, </math>
:<math> \theta_1 = \angle EOD, </math>
:<math> \theta_2 = \angle EOC, </math>
so that
:<math> \theta_0 = \theta_2 - \theta_1. \qquad \qquad (4) </math>

From Part One we know that <math> \theta_1 = 2 \psi_1 </math> and that <math> \theta_2 = 2 \psi_2 </math>. Combining these results with equation (4) yields
:<math> \theta_0 = 2 \psi_2 - 2 \psi_1 </math>
therefore, by equation (3),
:<math> \theta_0 = 2 \psi_0. </math>

===Corollary===
By a similar argument, the angle between a [[Chord (geometry)|chord]] and the [[tangent]] line at one of its intersection points equals half of the central angle subtended by the chord. See also [[Tangent lines to circles]].

==Applications==

The inscribed angle [[theorem]] is used in many proofs of elementary [[Euclidean geometry of the plane]]. A special case of the theorem is [[Thales' theorem]], which states that the angle subtended by a [[diameter]] is always 90°, i.e., a right angle. As a consequence of the theorem, opposite angles of [[cyclic quadrilateral]]s sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in a circle. As another example, the inscribed angle theorem is the basis for several theorems related to the [[power of a point]] with respect to a circle. Further, it allows one to prove that when two chords intersect in a circle, the products of the lengths of their pieces are equal.

== Inscribed angle theorems for ellipses, hyperbolas and parabolas ==
Inscribed angle theorems exist for ellipses, hyperbolas and parabolas, too. The essential differences are the measurements of an angle. (An angle is considered as a pair of intersecting lines.)
* [[ellipse#Inscribed angles and three-point form|Ellipse]]
* [[hyperbola#Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form|Hyperbola]]
* [[parabola#Inscribed angles and the 3-point form|Parabola]]

==References==
* {{Cite book | authorlink = C. Stanley Ogilvy | last = Ogilvy | first = C. S. | year = 1990 | title = Excursions in Geometry | publisher = Dover | isbn = 0-486-26530-7 | pages = [https://archive.org/details/excursionsingeom0000ogil/page/17 17–23] | postscript =  | url = https://archive.org/details/excursionsingeom0000ogil/page/17 }}
* {{cite book |vauthors=Gellert W, Küstner H, Hellwich M, Kästner H | title = The VNR Concise Encyclopedia of Mathematics | publisher = Van Nostrand Reinhold | location = New York | isbn = 0-442-22646-2 | pages = 172 | year = 1977}}
* {{cite book |first=Edwin E. |last=Moise |authorlink=Edwin E. Moise |title=Elementary Geometry from an Advanced Standpoint |location=Reading |publisher=Addison-Wesley |edition=2nd |year=1974 |isbn=0-201-04793-4 |pages=192–197 }}

==External links==
* {{MathWorld |urlname=InscribedAngle |title=Inscribed Angle}}
* [http://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-angle Relationship Between Central Angle and Inscribed Angle]
* [http://www.cut-the-knot.org/pythagoras/Munching/inscribed.shtml Munching on Inscribed Angles] at [[cut-the-knot]]
* [https://web.archive.org/web/20061030174939/http://www.mathopenref.com/arccentralangle.html Arc Central Angle] With interactive animation
* [http://www.mathopenref.com/arcperipheralangle.html Arc Peripheral (inscribed) Angle] With interactive animation
* [http://www.mathopenref.com/arccentralangletheorem.html Arc Central Angle Theorem] With interactive animation
* [https://www.bookofproofs.org/branches/inscribed-angle-theorem/ At bookofproofs.org]

{{Ancient Greek mathematics}}

[[Category:Euclidean plane geometry]]
[[Category:Angle]]
[[Category:Circles]]
[[Category:Articles containing proofs]]

Intercept theorem

2020-07-29T01:27:19Z

Crasshopper:

{{short description|On ratios of line segments formed when 2 intersecting lines are cut by a pair of parallels}}
The '''intercept theorem''', also known as '''Thales' theorem''' (not to be confused with [[Thales' theorem|another theorem with the same name]]) or '''basic proportionality theorem''', is about ratios of various [[line segment]]s that are created if two intersecting [[line (geometry)|lines]] are intercepted by a pair of [[Parallel (geometry)|parallels]]. It is equivalent to the theorem about ratios in [[similar triangles]]. Traditionally it is attributed to Greek mathematician [[Thales]].<ref name="mactutor"/>

==Formulation==
Suppose S is the intersection point of two lines and A, B are the intersections of the first line with the two parallels, such that B is further away from S than A, and similarly C, D are the intersections of the second line with the two parallels such that D is further away from S than C.

# The ratios of any two segments on the first line equals the ratios of the according segments on the second line: <math>| SA | : | AB | =| SC | : | CD | </math>, <math>| SB | : | AB | =| SD | : | CD | </math>, <math>| SA | : | SB | =| SC | : | SD | </math>
# The ratio of the two segments on the same line starting at S equals the ratio of the segments on the parallels: <math>| SA |:| SB | = | SC | :| SD | =| AC | : | BD | </math>
#The converse of the first statement is true as well, i.e. if the two intersecting lines are intercepted by two arbitrary lines and <math>| SA | : | AB | =| SC | : | CD | </math> holds then the two intercepting lines are parallel. However the converse of the second statement is not true.
# If you have more than two lines intersecting in S, then ratio of the two segments on a parallel equals the ratio of the according segments on the other parallel: <math>| AF | : | BE | =| FC | : | ED | </math> , <math>| AF | : | FC | =| BE | : | ED | </math>

::An example for the case of three lines is given in the second graphic below.

The first intercept theorem shows the ratios of the sections from the lines, the second the ratios of the sections from the lines as well as the sections from the parallels, finally the third shows the ratios of the sections from the parallels.

[[File:Intercept theorem.svg|center|700px|none]]
[[File:Intercept2.svg|center|700px|none]]

==Related concepts==

===Similarity and similar triangles===
[[File:Intercept theorem- Triangles.svg|thumb|right|400px|Arranging two similar triangles, so that the intercept theorem can be applied]]
The intercept theorem is closely related to [[Similarity (geometry)|similarity]]. It is equivalent to the concept of [[similar triangles]], i.e. it can be used to prove the properties of similar triangles and similar triangles can be used to prove the intercept theorem. By matching identical angles you can always place two similar triangles in one another so that you get the configuration in which the intercept theorem applies; and [[converse (logic)|conversely]] the intercept theorem configuration always contains two similar triangles.

===Scalar multiplication in vector spaces===
In a normed [[vector space]], the [[axiom]]s concerning the [[scalar multiplication]] (in particular <math> \lambda \cdot (\vec{a}+\vec{b})=\lambda \cdot \vec{a}+ \lambda \cdot \vec{b} </math> and <math> \|\lambda \vec{a}\|=|\lambda|\cdot\ \|\vec{a}\| </math>) ensure that the intercept theorem holds. One has
<math>
\frac{ \| \lambda \cdot \vec{a} \| }{ \| \vec{a} \|}
=\frac{\|\lambda\cdot\vec{b}\|}{\|\vec{b}\|}
=\frac{\|\lambda\cdot(\vec{a}+\vec{b}) \|}{\|\vec{a}+\vec{b}\|}
=|\lambda|
</math>

[[File:Intercept theorem vectors 2.svg|600px]]

==Applications==

===Algebraic formulation of compass and ruler constructions===
There are three famous problems in elementary geometry which were posed by the Greeks in terms of [[compass and straightedge constructions]]:<ref>{{citation|first=Nicholas D.|last=Kazarinoff|title=Ruler and the Round|year=2003|origyear=1970|publisher=Dover|page=3|isbn=0-486-42515-0}}</ref><ref name=Kunz/>
# [[Trisecting the angle]]
# [[Doubling the cube]]
# [[Squaring the circle]]

It took more than 2000 years until all three of them were finally shown to be impossible with the given tools in the 19th century, using algebraic methods that had become available during that period of time.
In order to reformulate them in algebraic terms using [[field extension]]s, one needs to match [[field (mathematics)|field operations]] with compass and straightedge constructions (see [[constructible number]]). In particular it is important to assure that for two given line segments, a new line segment can be constructed such that its length equals the product of lengths of the other two. Similarly one needs to be able to construct, for a line segment of length <math> a </math>, a new line segment of length <math> a^{-1} </math>. The intercept theorem can be used to show that in both cases such a construction is possible.

{| class="wikitable"
|-
|style="width:330px" valign="top" |
'''Construction of a product'''
[[File:Number construction multiplication.svg|330px]]
|style="width:330px" valign="top" |
'''Construction of an inverse'''
[[File:Number construction inverse.svg|330px]]
|-
|}

===Dividing a line segment in a given ratio===
{| class="wikitable"
|-
|style="width:300px" valign="top" |

To divide an arbitrary line segment <math>\overline{AB}</math> in a <math>m:n </math> ratio, draw an arbitrary angle in A with <math>\overline{AB}</math> as one leg. On the other leg construct <math>m+n </math> equidistant points, then draw the line through the last point and B and parallel line through the ''m''th point. This parallel line divides <math>\overline{AB}</math> in the desired ratio. The graphic to the right shows the partition of a line segment <math>\overline{AB}</math> in a <math>5:3</math> ratio.<ref name=Ostermann/>
|| [[File:Dividing segment.svg|300px|right]]
|-
|}

===Measuring and survey===

====Height of the Cheops pyramid====
[[File:Thales Theorem 6.svg|thumb|300px|right|measuring pieces]]
[[File:Thales Theorem 7.svg|thumb|250px|right|computing C and D]]

According to some historical sources the Greek mathematician [[Thales]] applied the intercept theorem to determine the height of the [[Great Pyramid of Giza|Cheops' pyramid]].<ref name="mactutor"/> The following description illustrates the use of the intercept theorem to compute the height of the pyramid. It does not however recount Thales' original work, which was lost.

Thales measured the length of the pyramid's base and the height of his pole. Then at the same time of the day he measured the length of the pyramid's shadow and the length of the pole's shadow. This yielded the following data:
* height of the pole (A): 1.63 m
* shadow of the pole (B): 2 m
* length of the pyramid base: 230 m
* shadow of the pyramid: 65 m
From this he computed
:<math> C = 65~\text{m}+\frac{230~\text{m}}{2}=180~\text{m} </math>
Knowing A,B and C he was now able to apply the intercept theorem to compute
:<math> D=\frac{C \cdot A}{B}=\frac{1.63~\text{m} \cdot 180~\text{m}}{2~\text{m}}=146.7~\text{m}</math>

==== Measuring the width of a river ====
{| class="wikitable"
|-
|style="width:300px" valign="top" |
The intercept theorem can be used to determine a distance that cannot be measured directly, such as the width of a river or a lake, the height of tall buildings or similar. The graphic to the right illustrates measuring the width of a river. The segments <math>|CF|</math>,<math>|CA|</math>,<math>|FE|</math> are measured and used to compute the wanted distance <math> |AB|=\frac{|AC||FE|}{|FC|} </math>.
|| [[File:River Chart.svg|400px|right]]
|-
|}

===Parallel lines in triangles and trapezoids===
The intercept theorem can be used to prove that a certain construction yields parallel line (segment)s.

{| class="wikitable"
|-
|style="width:300px" valign="top" |
If the midpoints of two triangle sides are connected then the resulting line segment is parallel to the third triangle side (Midpoint theorem of triangles).
[[File:Triangle midpoints.svg|210px|none]]
|style="width:300px" valign="top"|
If the midpoints of the two non-parallel sides of a trapezoid are connected, then the resulting line segment is parallel to the other two sides of the trapezoid.
[[File:Trapezoid midpoint.svg|275px|none]]
|-
|}

==Proof of the theorem==
An elementary proof of the theorem uses triangles of equal area to derive the basic statements about the ratios (claim 1). The other claims then follow by applying the first claim and contradiction.<ref name=Schupp/>

=== Claim 1 ===
{| class="wikitable"
|-
|style="width:800px" valign="top" |
[[File:Intercept theorem proof 2.svg|300px|right]]
Since <math>CA\parallel BD</math>, the altitudes of <math> \triangle CDA </math> and <math> \triangle CBA </math> are of equal length. As those triangles share the same baseline, their areas are identical. So we have <math>| \triangle CDA|=| \triangle CBA|</math> and therefore <math>| \triangle SCB|=| \triangle SDA|</math> as well. This yields

<math>\frac{| \triangle SCA|}{|\triangle CDA|}=\frac{|\triangle SCA|}{|\triangle CBA|}</math> and <math>\frac{| \triangle SCA|}{|\triangle SDA|}=\frac{|\triangle SCA|}{|\triangle SCB|}</math>

Plugging in the formula for triangle areas (<math> \tfrac{\text{baseline} \cdot \text{altitude}}{2}</math>) transforms that into

<math>\frac{|SC||AF|}{|CD||AF|}=\frac{|SA||EC|}{|AB||EC|}</math> and <math>\frac{|SC||AF|}{|SD||AF|}=\frac{|SA||EC|}{|SB||EC|}</math>

Canceling the common factors results in:

(a) <math> \, \frac{|SC|}{|CD|}=\frac{|SA|}{|AB|}</math> and (b) <math> \, \frac{|SC|}{|SD|}=\frac{|SA|}{|SB|}</math>

Now use (b) to replace <math> |SA| </math> and <math> |SC| </math> in (a):
<math> \frac{\frac{|SA||SD|}{|SB|}}{|CD|}=\frac{\frac{|SB||SC|}{|SD|}}{|AB|}</math>

Using (b) again this simplifies to:
(c) <math> \, \frac{|SD|}{|CD|}=\frac{|SB|}{|AB|}</math>
<math>\, \square </math>
|-
|}

=== Claim 2 ===
{| class="wikitable"
|-
|style="width:800px" valign="top" |
[[File:Intercept theorem proof2.svg|300px|right]]
Draw an additional parallel to <math> SD</math> through A. This parallel intersects <math> BD</math> in G. Then one has <math> |AC|=|DG| </math> and due to claim 1 <math>\frac{|SA|}{|SB|}=\frac{|DG|}{|BD|}</math>
and therefore
<math>\frac{|SA|}{|SB|}=\frac{|AC|}{|BD|} </math>

<math>\square </math>
|-
|}

=== Claim 3 ===
{| class="wikitable"
|-
|style="width:800px" valign="top" |

[[File:Intercept Theorem - proof 3.svg|300px|right]]
Assume <Math> AC </math> and <Math> BD</math> are not parallel. Then the parallel line to <math>AC</math> through <math> D</math> intersects <math> SA </math> in <math> B_{0}\neq B </math>. Since <math> |SB|:|SA|=|SD|:|SC|</math> is true, we have <math>|SB|=\frac{|SD||SA|}{|SC|}</math> and on the other hand from claim 2 we have <math>|SB_{0}|=\frac{|SD||SA|}{|SC|}</math>. So <math> B </math> and <math> B_{0} </math> are on the same side of <math> S </math> and have the same distance to <math> S </math>, which means <math> B=B_{0} </math>. This is a contradiction, so the assumption could not have been true, which means <Math> AC </math> and <Math> BD</math> are indeed parallel <math> \square </math>
|-
|}

=== Claim 4 ===
Claim 4 can be shown by applying the intercept theorem for two lines.

== Notes ==
<references>
<ref name="mactutor">No original work of Thales has survived. All historical sources that attribute the intercept theorem or related knowledge to him were written centuries after his death. [[Diogenes Laërtius|Diogenes Laertius]] and [[Pliny the Elder|Pliny]] give a description that strictly speaking does not require the intercept theorem, but can rely on a simple observation only, namely that at a certain point of the day the length of an object's shadow will match its height. Laertius quotes a statement of the philosopher [[Hieronymus of Rhodes|Hieronymus]] (3rd century BC) about Thales: "''Hieronymus says that [Thales] measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves (i.e. as our own height).''". Pliny writes: "''Thales discovered how to obtain the height of pyramids and all other similar objects, namely, by measuring the shadow of the object at the time when a body and its shadow are equal in length.''". However [[Plutarch]] gives an account, that may suggest Thales knowing the intercept theorem or at least a special case of it:"''.. without trouble or the assistance of any instrument [he] merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the intercept of the sun's rays, ... showed that the pyramid has to the stick the same ratio which the shadow [of the pyramid] has to the shadow [of the stick]''". (Source: [http://www-history.mcs.st-and.ac.uk/Biographies/Thales.html ''Thales biography''] of the [[MacTutor History of Mathematics archive|MacTutor]], the (translated) original works of
Plutarch and Laertius are: [http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Plutarch/Moralia/Dinner_of_the_Seven*.html#ref5 ''Moralia, The Dinner of the Seven Wise Men'', 147A] and [http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0258%3Abook%3D1%3Achapter%3D1 ''Lives of Eminent Philosophers'', Chapter 1. Thales, para.27])</ref>
<ref name=Schupp>{{Cite book |last=Schupp |first=H. |title=Elementargeometrie |location= |publisher=UTB Schöningh |year=1977 |isbn=3-506-99189-2 |pages=124–126 |language=de}}</ref>
<ref name=Kunz>{{Cite book |last=Kunz |first=Ernst|title=Algebra|location= |publisher=Vieweg |year=1991 |isbn=3-528-07243-1 |pages=5–7 |language=de}}</ref>
*<ref name=Ostermann>{{cite book |title=Geometry by Its History|url=https://archive.org/details/geometrybyitshis00oste|url-access=limited|first1=Alexander|last1=Ostermann|first2=Gerhard|last2=Wanner
|publisher=Springer|year=2012|isbn=978-3-642-29163-0|pages=[https://archive.org/details/geometrybyitshis00oste/page/n20 7]}} ({{Google books|eOSqPHwWJX8C|online copy|page=7}})</ref>
</references>

== References ==
* {{Cite book |last=Schupp |first=H. |title=Elementargeometrie |location= |publisher=UTB Schöningh |year=1977 |isbn=3-506-99189-2 |pages=124–126 |language=de}}
* {{Cite book |first=Manfred |last=Leppig |title=Lernstufen Mathematik |location= |publisher=Girardet |year=1981 |isbn=3-7736-2005-5 |pages=157–170 |language=de}}
* {{cite book |title=Elementary Geometry|first1=Ilka|last1=Agricola|author1-link= Ilka Agricola |first2=Thomas|last2=Friedrich
|publisher=AMS|year=2008|isbn=0-8218-4347-8|pages= 10–13, 16–18}} ({{Google books|LLXxBwAAQBAJ|online copy|page=10}})
* {{Cite book |first=John |last=Stillwell |title=The Four Pillars of Geometry |location= |publisher=Springer |year=2005 |isbn=978-0-387-25530-9 |page=[https://archive.org/details/fourpillarsofgeo0000stil/page/34 34] |url=https://archive.org/details/fourpillarsofgeo0000stil/page/34 }} ({{Google books|fpAjJ6VJ3y8C|online copy|page=34}})
* {{cite book |title=Geometry by Its History|url=https://archive.org/details/geometrybyitshis00oste|url-access=limited|first1=Alexander|last1=Ostermann|first2=Gerhard|last2=Wanner
|publisher=Springer|year=2012|isbn=978-3-642-29163-0|pages=[https://archive.org/details/geometrybyitshis00oste/page/n16 3]–7}} ({{Google books|eOSqPHwWJX8C|online copy|page=3}})

== External links ==
{{commons category}}
* [http://planetmath.org/intercepttheorem ''Intercept Theorem''] at [[PlanetMath]]
*Alexander Bogomolny: [http://www.cut-the-knot.org/pythagoras/ThalesTheorems.shtml ''Thales' Theorems''] and in particular [http://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/ThalesTheorem.shtml ''Thales' Theorem''] at [[Cut-the-Knot]]

{{Ancient Greek mathematics}}

{{DEFAULTSORT:Intercept Theorem}}
[[Category:Euclidean geometry]]
[[Category:Theorems in plane geometry]]

[[ca:Teorema de Tales]]
[[es:Teorema de Tales]]
[[eu:Talesen teorema (elkarketa)]]
[[he:משפט תאלס]]
[[nl:Stelling van Thales]]
[[sv:Transversalsatsen]]

Natsume Sōseki

2020-06-29T08:55:56Z

Crasshopper: who are important? what is "have an effect"? etc.

{{Japanese name|Natsume}}
{{Infobox writer 
| name = Natsume Sōseki
| image = Natsume_Soseki_photo.jpg
| caption = Sōseki in 1912
| alt = Sōseki in 1912
| native_name = 夏目金之助
| native_name_lang = Japanese
| birth_name = Natsume Kin'nosuke
| birth_date = {{Birth date|1867|2|9|df=y}}
| birth_place = [[Edo]], [[Tokugawa Shogunate]]
| death_date = {{death date and age|1916|12|9|1867|2|9|df=y}}
| death_place = [[Tokyo]], [[Empire of Japan]]
| occupation = Writer
| genre = [[Fiction]], [[Poetry]], [[Essays]]
| movement =
| notableworks = ''[[Kokoro]]'', ''[[Botchan]]'', ''[[I Am a Cat]]''
}}
{{nihongo|'''Natsume Sōseki'''|夏目漱石|extra=9 February 1867 – 9 December 1916}}, born '''{{nihongo|Natsume Kin'nosuke|夏目金之助}}''', was a [[Japanese people|Japanese]] novelist. He is best known around the world for his novels ''[[Kokoro]]'', ''[[Botchan]]'', ''[[I Am a Cat]]'' and his unfinished work ''[[Light and Darkness (novel)|Light and Darkness]]''. He was also a scholar of [[British literature]] and composer of [[haiku]], ''[[Kanshi (poetry)|kanshi]]'', and [[fairy tale]]s. From 1984 until 2004, his portrait appeared on the front of the Japanese [[Banknotes of the Japanese yen|1000 yen note]]. In Japan, he is often considered the greatest writer in modern Japanese history.<ref>{{cite web|url=https://www.ucl.ac.uk/library/sites/library/files/soseki-pamphlet.pdf|title=Natsume Sōseki, the Greatest Novelist in Modern Japan}}</ref> He has had a profound effect on almost all important Japanese writers since.{{fact}}

==Early years==
Born in 1867 as Natsume Kinnosuke in the town of Babashita in the [[Edo]] region of Ushigome (present Kikui, [[Shinjuku, Tokyo|Shinjuku]]), Sōseki began his life as an unwanted child, born to his mother late in her life, forty years old and his father then fifty-three.<ref name="edwin">{{cite book | author=McClellan, Edwin | title= Two Japanese Novelists: Sōseki & Tōson | publisher= Tuttle Publishing | year=2004 | isbn=978-0-8048-3340-0}}</ref> When he was born, he already had five siblings. Having five children and a toddler had created family insecurity and was in some ways a disgrace to the Natsume family.<ref name="edwin"/> A childless couple, Shiobara Masanosuke and his wife, adopted him in 1868 and raised him until the age of nine, when the couple divorced.<ref name="edwin"/> He returned to his family and was welcomed by his mother although regarded as a nuisance by his father. His mother died when he was fourteen, and his two eldest brothers died in 1887, intensifying his sense of insecurity.{{Citation needed|date=April 2009}}

Sōseki attended the First Tokyo Middle School (now [[Hibiya High School]]),<ref>{{cite book|title=新書で入門漱石と鴎外 (A pocket paperback == introduction: Natsume and Ōgai)|last=Takahashi|first=Akio|isbn=978-4-10-610179-3|year=2006|publisher=Shinchosha}}</ref> where he became enamored with [[Chinese literature]], and fancied that he might someday become a writer. His desire to become an author arose when he was about fifteen when he told his older brother about his interest in literature.<ref name="edwin"/> However, his family disapproved strongly of this course of action, and when Sōseki entered the [[University of Tokyo|Tokyo Imperial University]] in September 1884, it was with the intention of becoming an architect. Although he preferred [[Chinese classics]], he started studying English at that time, feeling that it might prove useful to him in his future career, as English was a necessity in Japanese college.<ref name="edwin"/>

In 1887, Sōseki met [[Masaoka Shiki]], a friend who would give him encouragement on the path to becoming a writer, which would ultimately be his career. Shiki tutored him in the art of composing [[haiku]]. From this point on, he began signing his poems with the name Sōseki, which is a Chinese idiom meaning "stubborn". In 1890, he entered the English Literature department, and quickly mastered the English language. In 1891 he produced a translation into English of the classical work ''[[Hōjōki]]''.<ref name="keene 308">Keene 1998 : 308.</ref> Sōseki graduated in 1893, and enrolled for some time as a graduate student and part-time teacher at the Tokyo Normal School.{{Citation needed|date=April 2009}}

In 1895, Sōseki began teaching at [[Matsuyama Middle School]] in [[Shikoku]], which became the setting of his novel ''[[Botchan]]''. Along with fulfilling his teaching duties, Sōseki published haiku and Chinese poetry in a number of newspapers and periodicals. He resigned his post in 1896, and began teaching at the Fifth High School in [[Kumamoto, Kumamoto|Kumamoto]]. On June 10 of that year, he married Nakane Kyoko.<ref>{{cite web|url=http://www.library.tohoku.ac.jp/en/collections/soseki/life.html|title=Soseki's Life {{!}} Tohoku University Library|website=www.library.tohoku.ac.jp|language=en|access-date=2017-11-03}}</ref>

[[File:Natsume Soseki house Clapham.jpg|thumb|right|Natsume Sōseki's lodgings in [[Clapham]], South London]]

==In the United Kingdom, 1901–1903==
In 1900, the Japanese government sent Sōseki to study in Great Britain as "Japan's first Japanese English literary scholar".<ref>Brodey and Tsunematsu p.7</ref> He visited [[Cambridge]] and stayed a night there, but gave up the idea of studying at the university because he could not afford it on his government scholarship.<ref>Brodey and Tsunematsu p.8</ref> He studied instead at [[University College London]] (UCL). He had a miserable time in [[London]], spending most of his days indoors buried in books, and his friends feared that he might be losing his mind.<ref>Introduction, p.V {{cite book | author=Natsume Soseki | title=I Am A Cat | publisher= Tuttle Publishing | year=2002 | isbn=978-0-8048-3265-6}}</ref> He also visited [[Pitlochry]] in Scotland, where he lodged with John Henry Dixon at the Dundarach Hotel.

He lived in four different lodgings: 76 Gower Street, near the British Museum; 85 Priory Road, West Hampstead; 6 Flodden Road, Camberwell; and 81 The Chase, Clapham (see the photograph). Only the last of these addresses, where he lodged with Priscilla Leale and her sister Elizabeth, proved satisfactory. Five years later, in his preface to ''Bungakuron'' (''The Criticism of Literature''), he wrote about the period:
{{quote|The two years I spent in London were the most unpleasant years in my life. Among English gentlemen I lived in misery, like a poor dog that had strayed among a pack of wolves.<ref>''Theory of Literature'', May 1907, introduction</ref>}}

He got along well with Priscilla, who shared his love of literature, notably Shakespeare and Milton (his tutor at UCL was the Shakespeare scholar [[William James Craig|W. J. Craig]]),<ref>{{cite book|last1=Natsume|first1=Sōseki|last2=Tsunematsu|first2=Ikuo|title=Spring miscellany and London essays|year=2002|publisher=Tuttle|location=Rutland, VT|isbn=978-0-8048-3326-4|page=80}}</ref> and who also spoke fluent French, much to his admiration. The Leales were a Channel Island family, and Priscilla had been born in France. The sisters worried about Sōseki's incipient paranoia and successfully urged him to get out more and take up cycling.

Despite his poverty, loneliness, and mental problems, he solidified his knowledge of English literature during this period and returned to the [[Empire of Japan]] in January 1903.<ref name="McClellan_164">McClellan (1959) p.164</ref> In April he was appointed to the First National College in Tokyo. Also, he was given the lectureship in [[English literature]], subsequently replacing Koizumi Yakumo ([[Lafcadio Hearn]]) and ultimately becoming a professor of English literature at the [[University of Tokyo|Tokyo Imperial University]],<ref name="McClellan_164"/> where he taught literary theory and [[literary criticism]].

==Literary career==

Sōseki's literary career began in 1903, when he began to contribute haiku, ''[[renku]]'' (haiku-style linked verse), ''haitaishi'' (linked verse on a set theme) and literary sketches to [[literary magazine]]s, such as the prominent ''[[Hototogisu (magazine)|Hototogisu]],'' edited by his former mentor [[Masaoka Shiki]], and later by [[Kyoshi Takahama|Takahama Kyoshi]]. However, it was the public success of his satirical novel ''[[I Am a Cat]]'' in 1905 that won him wide public admiration as well as critical acclaim.<ref>Mostow, Joshua S. ''The Columbia Companion to modern East Asian literature'', Columbia University Press, 2003. {{ISBN|978-0-231-11314-4}} p88</ref>

He followed on this success with short stories, such as "''Rondon tō''" ("Tower of London") in 1905 and the novels ''[[Botchan]]'' ("Little Master"), and ''[[Kusamakura (novel)|Kusamakura]]'' ("Grass Pillow") in 1906, which established his reputation, and which enabled him to leave his post at the university for a position with ''[[Asahi Shimbun]]'' in 1907, and to begin writing full-time. Much of his work deals with the relation between [[Japanese culture]] and [[Western culture]]. His early works in particular are influenced by his studies in London; his novel ''[[Kairo-kō]]'' was the earliest and only major prose treatment of the [[Arthurian legend]] in Japanese.<ref>Takamiya, Toshiyuki (1991). "Natsume Sōseki". In [[Norris J. Lacy]], ''The New Arthurian Encyclopedia'', p. 424. (New York: Garland, 1991). {{ISBN|0-8240-4377-4}}.</ref> He began writing one novel a year before his death from a [[stomach ulcer]] in 1916.

[[File:1000 yen Natsume Soseki.jpg|thumb|right|Obverse of a 1984 series 1000 [[Japanese yen banknote]] ]]

Major themes in Sōseki's works include ordinary people fighting against economic hardship, the conflict between duty and desire (a traditional Japanese theme; see [[Giri (Japanese)|giri]]), loyalty and group mentality versus freedom and individuality, personal isolation and estrangement, the rapid [[industrialization of Japan]] and its social consequences, contempt of Japan's aping of Western culture, and a pessimistic view of human nature. Sōseki took a strong interest in the writers of the ''[[Shirakaba]]'' (White Birch) literary group. In his final years, authors such as [[Ryūnosuke Akutagawa|Akutagawa Ryūnosuke]] and [[Masao Kume|Kume Masao]] became close followers of his literary style as his disciples.<ref>{{Cite web|url=https://www.japantimes.co.jp/culture/2017/08/19/books/book-reviews/ryunosuke-akutagawa-writing-shadows-japans-literary-giants/#.XWTF2ugzZEY|title=Ryunosuke Akutagawa: Writing in the Shadow of Japan's Literary Giants|last=Laflamme|first=Martin|date=19 August 2017|website=The Japan Times|access-date=27 August 2019}}</ref><ref>{{Cite web|url=https://www.britannica.com/biography/Kume-Masao|title=Kume Masao|date=2018|website=Britannica Online Encyclopedia|access-date=27 August 2019}}</ref>

==Legacy==
In the 21st century, there has been a global emergence of interest in Sōseki.<ref name=shimbun>{{cite web |url=http://ajw.asahi.com/article/cool_japan/style/AJ201404200016 |title=Meiji-Taisho Era novelist Natsume becoming trendy across the world 100 years later |work=[[The Asahi Shimbun]] |author1=Yusuke Takatsu |author2=Mariko Nakamura |date=April 20, 2014 |accessdate=April 28, 2014 |url-status=dead |archiveurl=https://web.archive.org/web/20140428051434/http://ajw.asahi.com/article/cool_japan/style/AJ201404200016 |archivedate=April 28, 2014 }}</ref> Sōseki's ''Kokoro'' has been newly published in 10 languages, such as Arabic, Slovenian and Dutch, since 2001.<ref name=shimbun/> In South Korea, the complete collection of Sōseki's long works began to be published in 2013.<ref name=shimbun/> In English-speaking countries there has been a succession of English translations since 2008.<ref name=shimbun/> About 60 of his works have been translated into more than 30 languages. Reasons for this emergence of global interest have been attributed in part to [[Haruki Murakami]] who said Sōseki was his favorite writer.<ref name=shimbun/> Political scientist Kang Sang-jung, who is the principal of [[Seigakuin University]], said, "Soseki predicted the problems we are facing today. He had a long-term view of civilization." He also said, "His popularity will become more global in the future".<ref name=shimbun/>

In 2016, the centennial of Sōseki's death, [[Nishogakusha University]] in Tokyo collaborated with [[Hiroshi Ishiguro]], robotics researcher at [[Osaka University]], to create a robotic android version of Sōseki. Sōseki's grandson, [[Fusanosuke Natsume]], voiced the 130 cm figure which depicted Sōseki at age 45. The robot gave lectures and recitations of Sōseki's works at the university, as a way to engage students' interest in literature.<ref>{{Cite web|url=https://www.japantimes.co.jp/life/2016/12/19/language/lets-discuss-soseki-robot/#.XWSwi-gzZEY|title=Let's Discuss the Soseki Robot|last=Otake|first=Tomoko|date=9 December 2016|website=Japan Times|access-date=26 August 2019}}</ref>

In 2017, as part of the 150 year commemoration of Sōseki's birth, the [[Asahi beer Oyamazaki Villa Museum of Art]] displayed the letter Sōseki had written suggesting names for the villa itself.<ref>[https://www.asahibeer-oyamazaki.com/english/ ]</ref> Although he died before its completion in 1917, Sōseki remained on good terms with the owner, Shotaro Kaga. Sōseki's diary was also on display during the exhibition.<ref>{{Cite web|url=https://www.asahibeer-oyamazaki.com/english/tokubetu/33770/|title=Soseki, Kyoto and the Oyamazaki Villa|date=March 2017|website=Asahi Beer Oyamazaki Villa Museum of Art|access-date=27 August 2019}}</ref><ref>{{Cite web|url=https://www.japantimes.co.jp/culture/2017/03/14/arts/openings-outside-tokyo/soseki-kyoto-oyamazaki-villa-commemorating-150th-anniversary-novelists-birth/#.XWSwhegzZEY|title=Commemorating the 150th Anniversary of the Novelist's Birth|last=Tanaka|first=Yukari|date=14 March 2017|website=Japan Times|access-date=27 August 2019}}</ref> In June 2019, retired professor Ikuo Tsunematsu reopened the Sōseki Museum, in Surrey, dedicated to the writer's life in the United Kingdom. The museum originally opened in 1982 in London, but closed in 2016 due to high maintenance costs and a decreased rate of attendance.<ref>{{Cite web|url=https://www.japantimes.co.jp/news/2019/07/08/national/museum-chronicling-novelist-natsume-sosekis-life-u-k-begins-new-chapter/#.XWSwZegzZEY|title=Museum Chronicling Novelist Natsume Soseki's Life in U.K. Begins New Chapter|date=July 8, 2019|website=Japan Times}}</ref> The collection includes over 10,000 items including works in translation, collected books and magazines from Sōseki's stay in London, and census records.<ref>{{Cite web|url=https://www.culture24.org.uk/am31865|title=Soseki Museum|date=2017|website=Culture 24|access-date=12 August 2019}}</ref>

Sōseki appears as a character in ''[[Dai Gyakuten Saiban: Naruhodō Ryūnosuke no Bōken]]'', where he is charged with stabbing a woman in the back during his stay in London, and defended by the protagonist. In the game, he has a pet cat called Wagahai, a reference to ''[[I Am a Cat]]''. He also appears in the sequel, ''[[Dai Gyakuten Saiban 2: Naruhodō Ryūnosuke no Kakugo]]'', where he is further charged with a man's poisoning in London, as well as appearing as a witness to a murder that occurs in Japan.<ref>{{cite web|title=Dai Gyakuten Saiban/Great Ace Attorney scans from Weekly Famitsu 07/02|url=http://www.japanese3ds.com/post/121759368809|website=japanese3ds.com|publisher=japanese3ds.com|url-status=dead|archiveurl=https://web.archive.org/web/20150619112314/http://www.japanese3ds.com/post/121759368809|archivedate=2015-06-19}}</ref> In the manga and anime [[Bungo Stray Dogs|''Bungou Stray Dogs'']], a character is named and based around Sōseki. In homage to his novel of the same name, Sōseki's character uses the ability 'I Am a Cat' which allows him to transform into a calico cat.<ref>{{Cite book|title=文豪ストレイドッグス (Bungou Stray Dogs) Volume 12|last=Kafka|first=Asagiri|publisher=Kadokawa Shoten|year=2017|isbn=978-4-04-104287-8|location=|pages=|chapter=Chp. 50}}</ref>

==Major works==
Sōseki's major works include:
{| class="wikitable"
|-
! Year
! colspan="2" | Japanese title
! English title
! Comments
|-
|rowspan="3"| [[1905 in literature|1905]] || 吾輩は猫である || ''Wagahai wa Neko dearu'' || ''[[I Am a Cat]]''||
|-
| 倫敦塔 || ''Rondon Tō'' || ''The Tower of London''||
|-
| 薤露行 || ''Kairo-kō''|| ''[[Kairo-kō]]''||
|-
|rowspan="4"| [[1906 in literature|1906]] || 坊っちゃん || ''Botchan'' || ''[[Botchan]]''||
|-
| 草枕 || ''Kusamakura'' || ''[[Kusamakura (novel)|The Three-Cornered World]]'' (lit. ''The Grass Pillow'') || latest translation uses Japanese title
|-
| 趣味の遺伝 || ''Shumi no Iden'' || ''[[The Heredity of Taste]]'' ||
|-
| 二百十日 || ''Nihyaku-tōka'' || ''The 210th Day'' ||
|-
|rowspan="2"| [[1907 in literature|1907]] ||野分 ||''Nowaki''||''[[Nowaki (novel)|Nowaki]]''||Translated in 2011
|-
|| 虞美人草 || ''Gubijinsō'' || ''The Poppy''||
|-
|rowspan="3"| [[1908 in literature|1908]] || 坑夫 || ''Kōfu'' || ''[[The Miner]]'' ||
|-
| 夢十夜 || ''Yume Jū-ya'' || ''[[Ten Nights of Dreams]]'' ||
|-
| 三四郎 || ''Sanshirō'' || ''[[Sanshirō (novel)|Sanshirō]]'' ||
|-
| [[1909 in literature|1909]] || それから || ''Sorekara'' || ''[[Sorekara|And Then]]'' ||
|-
|rowspan="3"| [[1910 in literature|1910]] || 門 || ''Mon'' || ''[[The Gate (novel)|The Gate]]'' ||
|-
| 思い出す事など || ''Omoidasu Koto nado'' || literally ''Random Memories'' || Translated in 1997 as ''Recollections'' by Maria Flutsch
|-
| 永日小品 || ''Eijitsu shōhin'' || literally ''Long (Spring) Days, Small Pieces'' || Translated in 2005 as ''Spring Miscellany''
|-
| rowspan="2"| [[1912 in literature|1912]] || 彼岸過迄 || ''Higan Sugi Made'' || ''To the Spring Equinox and Beyond'' ||
|-
| 行人 || ''Kōjin'' || ''[[The Wayfarer (novel)|The Wayfarer]]'' ||
|-
| rowspan="2"| [[1914 in literature|1914]] || こころ || ''Kokoro'' || ''[[Kokoro]]'' ||
|-
| 私の個人主義 || ''Watakushi no Kojin Shugi'' || ''My Individualism'' || Speech
|-
| rowspan="2" | [[1915 in literature|1915]] || 道草 || ''Michikusa'' || ''[[Grass on the Wayside]]'' ||
|-
| 硝子戸の中 || ''Garasu Do no Uchi''|| ''Inside My Glass Doors'' || English translation, 2002
|-
| [[1916 in literature|1916]] || 明暗 || ''Meian'' || ''[[Light and Darkness (novel)]]'' ''Light and Dark'' || Unfinished
|}

==See also==
{{Portal|Novels|Japan}}
* [[Anglo-Japanese relations]]
*[[Fukuzawa Yukichi]]
* [[Fusanosuke Natsume]] – Sōseki's grandson
* [[Japanese community of London]]
* [[Japanese literature]]
* [[List of Japanese authors]]
* [[Minae Mizumura]] – finished Natsume's last, unfinished novel, ''Light and Darkness''
*[[Nakae Chōmin]]
*[[Susumu Nishibe]]
*[[Tsuneari Fukuda]]
*[[Yamamoto Tsunetomo]]

==References==
{{Reflist}}

==Sources==
* Bargen, Doris D. ''Suicidal Honor: General Nogi and the Writings of Mori Ogai and Natsume Sōseki''. University of Hawaii Press (2006). {{ISBN|0-8248-2998-0}}
* Brodey, I. S. and S. I. Tsunematsu, ''Rediscovering Natsume Sōseki'', (Kent: Global Oriental, 2000)
* Doi, Takeo, trans. by W. J. Tyler, ''The Psychological World of Natsume Sōseki''. Harvard University Asia Center (1976). {{ISBN|0-674-72116-0}}
* Gessel, Van C. ''Three Modern Novelists: Soseki, Tanizaki, Kawabata.'' Kodansha International, 1993
* {{cite book |last=Keene |first=Donald |authorlink=Donald Keene |year=1998 |title=A History of Japanese Literature, Vol. 3: Dawn to the West – Japanese Literature of the Modern Era (Fiction) |publisher=[[Columbia University Press]] |location=New York, NY |edition=paperback |orig-year=1984 |isbn=978-0-231-11435-6}}
*[[Edwin McClellan|McClellan]], Edwin: An Introduction to Sōseki. In: ''Harvard Journal of Asiatic Studies'', Vol. 22 (Dec., 1959), pp. 150–208.
* [[Peter Milward|Milward]], Peter. ''The Heart of Natsume Sōseki: First Impressions of His Novels''. Azuma Shobo (1981). [[ASIN]]: B000IK2690
* Olson, Lawrence. ''Ambivalent Moderns: Portraits of Japanese Cultural Identity''. Savage, Maryland: Rowman & Littlefield (1992). {{ISBN|0-8476-7739-7}}
* Ridgeway, William N. ''A Critical Study of The Novels of Natsume Sōseki, 1867–1916''. [[Edwin Mellen Press]] (January 28, 2005). {{ISBN|0-7734-6230-9}}
* Yu, Beongchoeon. ''Natsume Sōseki''. Macmillan Publishing Company (1984). {{ISBN|0-8057-2850-3}}

==External links==
{{wikiquote}}
{{wikisource|Author:Natsume Sōseki|Natsume Sōseki}}
*[http://www.natsumesoseki.com/ natsumesoseki.com]
* {{Gutenberg author |id=Natsume,+Soseki}}
* {{Internet Archive author |sname=Natsume Sōseki |sopt=w}}
* {{Librivox author |id=3289}}
*[http://www.ibiblio.org/eldritch/ns/soseki.html Sōseki page] including links to the entire text of ''Kokoro''
*[http://www.aozora.gr.jp/index_pages/person148.html Natsume Sōseki on aozora.gr.jp] (complete texts with [[furigana]])
*[http://www.sosekiproject.org Soseki Project] (resources for reading Sōseki's works in their original Japanese form)
* [https://web.archive.org/web/20080530114849/http://www.horror-house.jp/e/cat4/soseki-natsume-18671916.html Natsume Sōseki's grave]
* [http://www008.upp.so-net.ne.jp/hybiblio/index.html Hiroshi Yamashita: Bibliographical and Textual Studies of Edmund Spenser and Natsume Soseki]
* [https://www.youtube.com/watch?v=jvI5a3kZl0M Glenn Gould reads Natsume Soseki]

{{Natsume Sōseki}}

{{Authority control}}

{{DEFAULTSORT:Natsume, Soseki}}
[[Category:Natsume Sōseki| ]]
[[Category:1867 births]]
[[Category:1916 deaths]]
[[Category:People from Shinjuku]]
[[Category:Writers from Tokyo]]
[[Category:People of Meiji-period Japan]]
[[Category:19th-century Japanese novelists]]
[[Category:20th-century Japanese novelists]]
[[Category:The Asahi Shimbun people]]
[[Category:Japanese male short story writers]]
[[Category:Japanese expatriates in the United Kingdom]]
[[Category:University of Tokyo alumni]]
[[Category:Alumni of University College London]]
[[Category:Pseudonymous writers]]
[[Category:Deaths from ulcers]]
[[Category:19th-century Japanese poets]]
[[Category:19th-century Japanese short story writers]]
[[Category:20th-century Japanese short story writers]]

Clock signal

2020-02-26T14:47:40Z

Crasshopper:

{{about|timing of electronic circuits|setting clocks to the correct time of day|Time signal}}
In [[electronics]] and especially [[Synchronous logic|synchronous]] [[digital circuit]]s, a '''clock signal''' oscillates between a high and a low state and is used like a [[metronome]] to coordinate actions of digital [[Electronic circuit|circuits]].

A clock [[Signal (electrical engineering)|signal]] is produced by a [[clock generator]]. Although more complex arrangements are used, the most common clock signal is in the form of a [[square wave]] with a 50% [[duty cycle]], usually with a fixed, constant frequency. Circuits using the clock signal for synchronization may become active at either the rising edge, falling edge, or, in the case of [[double data rate]], both in the rising and in the falling edges of the clock cycle.

== Digital circuits ==

Most [[integrated circuit]]s (ICs) of sufficient complexity use a clock signal in order to synchronize different parts of the circuit, cycling at a rate slower than the worst-case internal [[propagation delay]]s. In some cases, more than one clock cycle is required to perform a predictable action. As ICs become more complex, the problem of supplying accurate and synchronized clocks to all the circuits becomes increasingly difficult. The preeminent example of such complex chips is the [[microprocessor]], the central component of modern computers, which relies on a clock from a [[crystal oscillator]]. The only exceptions are [[asynchronous circuit]]s such as [[Asynchronous Processor|asynchronous CPUs]].

A clock signal might also be gated, that is, combined with a controlling signal that enables or disables the clock signal for a certain part of a circuit. This technique is often used to save power by effectively shutting down portions of a digital circuit when they are not in use, but comes at a cost of increased complexity in timing analysis.

=== Single-phase clock ===
Most modern [[synchronous circuit]]s use only a "single phase clock" – in other words, all clock signals are (effectively) transmitted on 1 wire.

=== Two-phase clock ===
In [[synchronous circuit]]s, a "two-phase clock" refers to clock signals distributed on 2 wires, each with non-overlapping pulses. Traditionally one wire is called "phase 1" or "φ1", the other wire carries the "phase 2" or "φ2" signal.<ref>[http://www.princeton.edu/~wolf/modern-vlsi/Overheads/CHAP5-2/sld010.htm Two-phase clock] {{webarchive |url=https://web.archive.org/web/20071109090150/http://www.princeton.edu/~wolf/modern-vlsi/Overheads/CHAP5-2/sld010.htm |date=November 9, 2007 }}</ref><ref>{{citation |url=http://tams-www.informatik.uni-hamburg.de/applets/hades/webdemos/12-gatedelay/40-tpcg/two-phase-clock-gen.html |title=Two-phase non-overlapping clock generator |publisher=Tams-www.informatik.uni-hamburg.de |date= |accessdate=2012-01-08 |archive-url=https://web.archive.org/web/20111226073122/http://tams-www.informatik.uni-hamburg.de/applets/hades/webdemos/12-gatedelay/40-tpcg/two-phase-clock-gen.html |archive-date=2011-12-26 |url-status=dead }}</ref><ref>{{citation|url=http://micro.magnet.fsu.edu/primer/digitalimaging/concepts/twophase.html |title=Concepts in Digital Imaging - Two Phase CCD Clocking |publisher=Micro.magnet.fsu.edu |date= |accessdate=2012-01-08}}</ref><ref>{{citation |url=http://www.hpc.msstate.edu/mpl/distributions/scmos/scmos_doc/cells/cgf104.html |title=Cell cgf104: Two phase non-overlapping clock generator |publisher=Hpc.msstate.edu |accessdate=2012-01-08 |url-status=dead |archiveurl=https://web.archive.org/web/20120208054348/http://www.hpc.msstate.edu/mpl/distributions/scmos/scmos_doc/cells/cgf104.html |archivedate=2012-02-08 }}</ref> Because the two phases are guaranteed non-overlapping, [[gated latch]]es rather than [[edge-triggered flip-flop]]s can be used to store [[State (computer science)#Digital logic circuit state|state information]] so long as the inputs to latches on one phase only depend on outputs from latches on the other phase. Since a gated latch uses only four gates versus six gates for an edge-triggered flip-flop, a two phase clock can lead to a design with a smaller overall gate count but usually at some penalty in design difficulty and performance.

MOS ICs typically used dual clock signals (a two-phase clock) in the 1970s. These were generated externally for both the 6800 and 8080 microprocessors.<ref name = "MC6870">{{Cite journal | title = How to drive a microprocessor | journal = Electronics | volume = 49 | issue = 8 | page =159 | publisher = McGraw-Hill | location = New York | date = April 15, 1976 | url = http://commons.wikimedia.org/wiki/File:Motorola_MC6870_ad_April_1976.jpg}} Motorola's Component Products Department sold hybrid ICs that included a quartz oscillator. These IC produced the two-phase non-overlapping waveforms the 6800 and 8080 required. Later Intel produced the 8224 clock generator and Motorola produced the MC6875. The Intel 8085 and the Motorola 6802 include this circuitry on the microprocessor chip.</ref> The next generation of microprocessors incorporated the clock generation on chip. The 8080 uses a 2 MHz clock but the processing throughput is similar to the 1 MHz 6800. The 8080 requires more clock cycles to execute a processor instruction. The 6800 has a minimum clock rate of 100 kHz and the 8080 has a minimum clock rate of 500 kHz. Higher speed versions of both microprocessors were released by 1976.<ref name = "MD Sep 1975 8080A">{{Cite journal | title = Intel's Higher Speed 8080 μP | journal = Microcomputer Digest | volume = 2 | issue = 3 | page =7 | publisher = Microcomputer Associates | location = Cupertino CA | date = September 1975 | url = http://www.bitsavers.org/pdf/microcomputerAssociates/Microcomputer_Digest_v02n03_Sep75.pdf}}</ref>

The [[6501]] requires an external 2-phase clock generator.
The [[MOS Technology 6502]] uses the same 2-phase logic internally, but also includes a two-phase clock generator on-chip, so it only needs a single phase clock input, simplifying system design.

=== 4-phase clock ===
Some early integrated circuits use [[four-phase logic]], requiring a four phase clock input consisting of four separate, non-overlapping clock signals.<ref>{{citation |url=http://micro.magnet.fsu.edu/primer/digitalimaging/concepts/fourphase.html |title=Concepts in digital imaging - Four Phase CCD Clocking |publisher=Micro.magnet.fsu.edu |date= |accessdate=2012-01-08}}</ref>
This was particularly common among early microprocessors such as the [[National Semiconductor]] [[IMP-16]], [[Texas Instruments TMS9900]], and the [[Western Digital]] WD16 chipset used in the DEC LSI-11.

Four phase clocks have only rarely been used in newer CMOS processors such as the DEC WRL MultiTitan microprocessor.<ref>
[[Norman P. Jouppi]] and Jeffrey Y. F. Tang.
[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.85.988&rep=rep1&type=pdf "A 20-MIPS Sustained 32-bit CMOS Microprocessor with High Ratio of Sustained to Peak Performance"].
1989.
{{citeseerx|10.1.1.85.988}}
p. 10.
</ref> and in [[Intrinsity]]'s Fast14 technology. Most modern microprocessors and [[microcontroller]]s use a single-phase clock.

=== Clock multiplier ===
{{main|clock multiplier}}

Many modern [[microcomputer]]s use a "[[clock multiplier]]" which multiplies a lower frequency external clock to the appropriate [[clock rate]] of the microprocessor. This allows the CPU to operate at a much higher frequency than the rest of the computer, which affords performance gains in situations where the CPU does not need to wait on an external factor (like memory or [[input/output]]).

=== Dynamic frequency change ===

The vast majority of digital devices do not require a clock at a fixed, constant frequency.
As long as the minimum and maximum clock periods are respected, the time between clock edges can vary widely from one edge to the next and back again.
Such digital devices work just as well with a clock generator that dynamically changes its frequency, such as [[spread spectrum clock|spread-spectrum clock generation]], [[dynamic frequency scaling]], [[PowerNow!]], [[Cool'n'Quiet]], [[SpeedStep]], etc.
Devices that use [[static logic (digital logic)|static logic]] do not even have a maximum clock period; such devices can be slowed down and paused indefinitely, then resumed at full clock speed at any later time.

== Other circuits ==

Some sensitive [[Mixed-signal integrated circuit|mixed-signal circuits]], such as precision [[analog-to-digital converter]]s, use [[sine wave]]s rather than square waves as their clock signals, because square waves contain high-frequency [[harmonic]]s that can interfere with the analog circuitry and cause [[signal noise|noise]]. Such sine wave clocks are often [[differential signaling|differential signals]], because this type of signal has twice the [[slew rate]], and therefore half the timing uncertainty, of a [[single-ended signalling|single-ended signal]] with the same voltage range. Differential signals radiate less strongly than a single line. Alternatively, a single line shielded by power and ground lines can be used.

In CMOS circuits, gate capacitances are charged and discharged continually. A capacitor does not dissipate energy, but energy is wasted in the driving transistors. In [[reversible computing]], [[inductor]]s can be used to store this energy and reduce the energy loss, but they tend to be quite large. Alternatively, using a sine wave clock, CMOS [[transmission gate]]s and energy-saving techniques, the power requirements can be reduced.{{Citation needed|date=October 2008}}

== Distribution ==
The most effective way to get the clock signal to every part of a chip that needs it, with the lowest skew, is a metal grid. In a large microprocessor, the power used to drive the clock signal can be over 30% of the total power used by the entire chip. The whole structure with the gates at the ends and all amplifiers in between have to be loaded and unloaded every cycle.<ref>{{citation |url=http://www.anandtech.com/showdoc.aspx?i=3276&p=14 |title=Intel's Atom Architecture: The Journey Begins |author=Anand Lal Shimpi |year=2008}}</ref><ref>{{citation |url=http://alasir.com/articles/alpha_history/alpha_21264.html |title=Alpha: The history in facts and comments |author=Paul V. Bolotoff |year=2007 |quote=power consumed by the clock subsystem of EV6 was about 32% of the total core power. To compare, it was about 25% for EV56, about 37% for EV5 and about 40% for EV4. |access-date=2012-01-03 |archive-url=https://web.archive.org/web/20120218005309/http://alasir.com/articles/alpha_history/alpha_21264.html |archive-date=2012-02-18 |url-status=dead }}</ref> To save energy, [[clock gating]] temporarily shuts off part of the tree.

The '''clock distribution network''' (or '''clock tree''', when this network forms a tree) distributes the clock signal(s) from a common point to all the elements that need it. Since this function is vital to the operation of a synchronous system, much attention has been given to the characteristics of these clock signals and the [[electrical network]]s used in their distribution. Clock signals are often regarded as simple control signals; however, these signals have some very special characteristics and attributes.

Clock signals are typically loaded with the greatest [[fanout]] and operate at the highest speeds of any signal within the synchronous system. Since the data signals are provided with a temporal reference by the clock signals, the clock [[waveform]]s must be particularly clean and sharp. Furthermore, these clock signals are particularly affected by technology scaling (see [[Moore's law]]), in that long [[global interconnect]] lines become significantly more resistive as line dimensions are decreased. This increased line resistance is one of the primary reasons for the increasing significance of clock distribution on synchronous performance. Finally, the control of any differences and uncertainty in the arrival times of
the clock signals can severely limit the maximum performance of the entire system and create catastrophic [[Race hazard|race conditions]] in which an incorrect data signal may latch within a register.

Most synchronous ''[[Digital data|digital]]'' systems consist of cascaded banks of sequential [[Flip-flop (electronics)|registers]] with [[combinational logic]] between each set of registers. The [[functional requirements]] of the digital system are satisfied by the logic stages. Each logic stage introduces delay that affects timing performance, and the timing performance of the digital design can be evaluated relative to the timing requirements by a timing analysis. Often special consideration must be made to meet the timing requirements. For example, the global performance and local timing requirements may be satisfied by
the careful insertion of [[Pipeline (computing)|pipeline registers]] into equally spaced time windows to satisfy critical worst-case ''[[timing constraints]]''. The proper design of the clock distribution network helps ensure that critical timing requirements are satisfied and that no race conditions exist (see also [[clock skew]]).

The delay components that make up a general synchronous system are composed of the following three individual subsystems: the memory storage elements, the logic elements, and the clocking circuitry and distribution network.

Novel structures are currently under development to ameliorate these issues and provide effective solutions. Important areas of research include resonant clocking techniques, on-chip optical interconnect, and local synchronization methodologies.

== See also ==
* [[Clock rate]]
* [[Electronic design automation]]
* [[Design flow (EDA)]]
* [[Integrated circuit design]]
* [[Self-clocking signal]]
* [[Four-phase logic]]
* [[Jitter]]
* [[Bit-synchronous operation]]
* [[Pulse-per-second signal]]
* [[Clock domain crossing]]

== References ==
{{reflist|30em}}
* [[Eby Friedman|Eby G. Friedman]] (Ed.), ''Clock Distribution Networks in VLSI Circuits and Systems'', {{ISBN|0-7803-1058-6}}, IEEE Press. 1995.
* [[Eby Friedman|Eby G. Friedman]], {{doi-inline|10.1109/5.929649|"Clock Distribution Networks in Synchronous Digital Integrated Circuits"}}, ''Proceedings of the IEEE'', Vol. 89, No. 5, pp. 665–692, May 2001.
* [http://archive.sigda.org/ispd/contests/10/ispd10cns.html "ISPD 2010 High Performance Clock Network Synthesis Contest"], International Symposium on Physical Design, Intel, IBM, 2010.
* D.-J. Lee, [http://www.eecs.umich.edu/~imarkov/pubs/diss/DJdiss.pdf "High-performance and Low-power Clock Network Synthesis in the Presence of Variation"], Ph.D. dissertation, University of Michigan, 2011.
* I. L. Markov, D.-J. Lee, [http://www.eecs.umich.edu/~imarkov/pubs/conf/iccad11-tuto.pdf "Algorithmic Tuning of Clock Trees and Derived Non-Tree Structures"], in Proc. Int'l. Conf. Comp.-Aided Design (ICCAD), 2011.
* V. G. Oklobdzija, V. M. Stojanovic, D. M. Markovic, and N. M. Nedovic, ''Digital System Clocking: High-Performance and Low-Power Aspects'', {{ISBN|0-471-27447-X}}, IEEE Press/Wiley-Interscience, 2003.
* Mitch Dale, [https://web.archive.org/web/20131224102708/http://chipdesignmag.com/display.php?articleId=915 "The power of RTL Clock-gating"], ''Electronic Systems Design Engineering Incorporating Chip Design'', January 20, 2007.
----
Adapted from [http://www.ece.rochester.edu/users/friedman/ Eby Friedman]'s column in the ACM [http://www.sigda.org SIGDA] [https://web.archive.org/web/20070208034716/http://www.sigda.org/newsletter/index.html e-newsletter] by [http://www.eecs.umich.edu/~imarkov/ Igor Markov] 
Original text is available at https://web.archive.org/web/20100711135550/http://www.sigda.org/newsletter/2005/eNews_051201.html

[[Category:Clock signal| ]]

AND gate

2020-02-25T01:08:01Z

Crasshopper: /* Alternatives */

{{short description|Logical gate}}
{| class="wikitable floatright" style="text-align: center"
|- bgcolor="#ddeeff"
!colspan=2|INPUT !! OUTPUT
|- bgcolor="#ddeeff"
| A || B || A AND B
|- bgcolor="#ddffdd"
|0 || 0 || 0
|- bgcolor="#ddffdd"
|0 || 1 || 0
|- bgcolor="#ddffdd"
|1 || 0 || 0
|- bgcolor="#ddffdd"
|1 || 1 || 1
|}
The '''AND gate''' is a basic digital [[logic gate]] that implements [[logical conjunction]] - it behaves according to the [[truth table]] to the right. A HIGH output (1) results only if all the inputs to the AND gate are HIGH (1). If none or not all inputs to the AND gate are HIGH, a LOW output results. The function can be extended to any number of inputs.

==Symbols==

There are three symbols for AND gates: the American ([[American National Standards Institute|ANSI]] or 'military') symbol and the [[International Electrotechnical Commission|IEC]] ('European' or 'rectangular') symbol, as well as the deprecated [[Deutsches Institut für Normung|DIN]] symbol. Additional inputs can be added as needed. For more information see [[Logic gate#Symbols|Logic Gate Symbols]]. It can also be denoted as symbol "^" or "&".

{| align=center style="text-align:center"
|[[Image:AND ANSI.svg]]
|[[Image:AND IEC.svg]]
|[[image:AND DIN.svg]]
|-
|''MIL/ANSI Symbol''
|''IEC Symbol''
|''DIN Symbol''
|}

The AND gate with inputs ''A'' and ''B'' and output ''C'' implements the logical expression <math>C = A \cdot B</math>. This expression also may be denoted as '''C=A^B''' or '''C=A&B'''.

==Implementations==
{{Gallery
|File:DiodeANDgate.png|AND gate using diodes
|File:TransistorANDgate.png|AND gate using transistors
|File:NMOS AND gate.png|NMOS AND gate
}}
An AND gate is usually designed using N-channel (pictured) or P-channel [[MOSFET]]s. The digital inputs '''a''' and '''b''' cause the output '''F''' to have the same result as the AND function.

=== Analytical representation ===
<math>f(a,b)=a*b</math> is the analytical representation of AND gate:

* <math>f(0,0)=0*0=0</math>
* <math>f(0,1)=0*1=0</math>
* <math>f(1,0)=1*0=0</math>
* <math>f(1,1)=1*1=1</math>

=== Alternatives ===
{{further|NAND logic|NOR logic}}
If no specific AND gates are available, one can be made from [[NAND gate|NAND]] or [[NOR gate|NOR]] gates, because NAND and NOR gates are "universal gates," <ref>
Mano, M. Morris and Charles R. Kime. ''Logic and Computer Design Fundamentals, Third Edition.'' Prentice Hall, 2004. p. 73.</ref> meaning that they can be used to make all the others.

{| align=center style="text-align:center"
! width="250" |Desired gate!! width="150" |NAND construction!! width="150" |NOR construction
|-
|[[Image:AND ANSI Labelled.svg]]||[[Image:AND from NAND.svg]]||[[File:AND from NOR.svg]]
|}

== IC package ==
AND gates are available in IC packages. The 7408 IC is a well known QUAD 2-Input AND GATES and contains four independent gates each of which performs the logic AND function.
[[File:QUAD 2-Input AND GATE IC.png|thumb|IC 7408]]

== See also ==
{{commons category|AND gates}}

*[[OR gate]]
*[[Inverter (logic gate)|NOT gate]]
*[[NAND gate]]
*[[NOR gate]]
*[[XOR gate]]
*[[XNOR gate]]
*[[IMPLY gate]]
*[[Boolean algebra]]
*[[Logic gate]]

== References ==
{{reflist}}

{{Logical connectives}}

[[Category:Logic gates]]

Dishna Papers

2019-06-27T05:26:15Z

Crasshopper: /* Overview */

{{short description|A collection of biblical manuscripts from 200 AD until the 6th century}}
{{about|22 papyri discovered in Egypt in 1952, purchased by Martin Bodmer|a list of the 58 papyri in Bodmer's collection|List of Bodmer Papyri}}
[[File:Papyrus66.jpg|thumb|[[Papyrus 66]] of the Bodmer Papyri]]

The '''Bodmer Papyri''' are a group of twenty-two [[papyri]] discovered in Egypt in 1952. They are named after [[Martin Bodmer]] who purchased them. The papyri contain segments from the Old and New Testaments, early Christian literature, [[Homer]], and [[Menander]]. The oldest, [[Papyrus 66|P66]] dates to ''c.'' 200 AD. Most of the papyri are kept at the [[Bibliotheca Bodmeriana]], in [[Cologny]], [[Switzerland]] outside [[Geneva]].

In 2007 the [[Vatican Library]] acquired Bodmer Papyrus 14-15 (known as [[Papyrus 75|P75]]).

==Overview==
The Bodmer Papyri were found in 1952 at Pabau near Dishna, [[Egypt]], the ancient headquarters of the [[Pachomius|Pachomian order]] of monks; the discovery site is not far from [[Nag Hammadi]], where the secreted [[Nag Hammadi library]] had been found some years earlier. The [[manuscript]]s were covertly assembled by a [[Cypriote]], [[Phokio Tano]] of Cairo, then smuggled to Switzerland,<ref>A. H. M. Kessels and P. W. Van Der Horst, "The Vision of Dorotheus (Pap. Bodmer 29): Edited with Introduction, Translation and Notes", ''Vigiliae Christianae'' '''41'''.4 (December 1987, pp. 313-359, p 313.</ref> where they were bought by Martin Bodmer (1899–1971). The series ''Papyrus Bodmer'' began to be published in 1954, giving transcriptions of the texts with note and introduction in French and a French translation. The Bodmer Papyri, now conserved in the Bibliotheca Bodmeriana, in Cologny, outside [[Geneva]],<ref>Some papyri from the same provenance escaped Martin Bodmer and are conserved elsewhere. Sir [[Alfred Chester Beatty]] acquired some of the material, and further material is at [[Oxford, Mississippi|Oxford]], Mississippi, [[Cologne]] and [[Barcelona]]. For convenience scholars also refer to these as "Bodmer Papyri". (''Anchor Bible Dictionary'').</ref> are not a [[gnostic]] cache, like the Nag Hammadi Library: they bear some pagan as well as Christian texts, parts of some thirty-five books in all, in [[Coptic language|Coptic]]<ref>Texts in the [[Bohairic]] dialect of Coptic had not previously been known older than the ninth century (6. p 51.</ref> and in [[Ancient Greek|Greek]]. With fragments of correspondence, the number of individual texts represented reaches to fifty.<ref>''Anchor Bible Dictionary''.</ref> Most of the works are in [[codex]] form, a few in [[scroll]]s. Three are written on [[parchment]].

Books V and VI of Homer's ''[[Iliad]]'' (P1), and three comedies of [[Menander]] (''[[Dyskolos]]'' (P4), ''[[Samia (play)|Samia]]'' and ''[[Aspis (Menander)|Aspis]]'') appear among the Bodmer Papyri, as well as gospel texts: [[Papyrus 66]] (P66), is a text of the [[Gospel of John]],<ref>John 1:1-6:11, 6:35b-14:26 and fragments of forty other pages of John 14-21.</ref> dating around 200 AD, in the manuscript tradition called the [[Alexandrian text-type]]. Aside from the papyrus fragment in the [[Rylands Library Papyrus P52]], it is the oldest testimony for John; it omits the passage concerning the moving of the waters (John 5:3b-4) and the [[pericope adulterae|pericope of the woman taken in adultery]] (John 7:53-8:11). [[Papyrus 72|<math>\mathfrak{P}</math>72]] is the earliest known copy of the [[Epistle of Jude]], and 1 and 2 Peter. [[Papyrus 75]] (P75) is a partial codex containing most of Luke and John. Comparison of the two versions of John in the Bodmer Papyri with the third-century [[Chester Beatty Papyri]] convinced Floyd V. Filson that "...there was no uniform text of the Gospels in Egypt in the third century."<ref>"A comparison of all three, which had their origins in Egypt, shows that there was no uniform text of the Gospels in Egypt in the third century." (Filson 1962: 52).</ref>

There are also Christian texts that were declared [[apocrypha]]l in the fourth century, such as the ''[[Infancy Gospel of James]]''. There is a Greek-Latin [[lexicon]] to some of Paul's letters, and there are fragments of [[Melito of Sardis]]. Among the works is ''[[The Vision of Dorotheus]]'', one of the earliest examples of Christian hexametric poem, attributed to a Dorotheus, son of "Quintus the poet" (assumed to be the pagan poet [[Quintus Smyrnaeus]]). ([[Papyrus 29|P29]]). The earliest extant copy of the [[Third Epistle to the Corinthians]] is published in ''Bodmer Papryri X''.

The collection includes some non-literary material, such as a collection of letters from the abbots of the monastery of Saint Pachomius, raising the possibility that the unifying circumstance in the collection is that all were part of a monastic library.<ref>Kessels and Van der Horst 1987:214.</ref>

The latest of the Bodmer Papyri (P74) dates to the sixth or seventh century.<ref>Filson 1962:52.</ref>

== Vatican acquisition ==
Plans announced by the Foundation Bodmer in October 2006<ref>[http://archiv.twoday.net/stories/2866554/ Sale of Bodmer Papyri]</ref> to sell two of the manuscripts for millions of dollars, to capitalize the library, which opened in 2003, drew consternation from scholars around the world, fearing that the unity of the collection would be broken.

Then, in March 2007 it was announced the Vatican had acquired the Bodmer Papyrus XIV-XV (P75), which is believed to contain the world's oldest known written fragment from the [[Gospel of Luke]], the earliest known [[Lord's Prayer]], and one of the oldest written fragments from the [[Gospel of John]].<ref>{{cite web|url=http://www.ewtn.com/library/SCRIPTUR/bodmerpapyrus.HTM |title=Bodmer Papyrus: History Becomes Reality |publisher=Ewtn.com |date= |accessdate=2013-06-04}}</ref>

The papyri had been sold for an undisclosed "significant" price to [[Frank Hanna III]], of Atlanta, Georgia. In January 2007, Hanna presented the papyri to the Pope. They are kept in the Vatican Library and will be made available for scholarly review, and in the future, excerpts may be put on display for the general public. They were transported from Switzerland to the Vatican in "An armed motorcade surrounded by people with machine guns."<ref>[http://dsc.discovery.com/news/2007/03/05/gospel_arc.html "Earliest Gospels Acquired by Vatican"], by Jennifer Viegas, ''Discovery News'', March 5, 2007</ref>

== Bible related manuscripts ==
=== Greek ===
* [[Papyrus 66|Papyrus Bodmer II]] (<math>\mathfrak{P}</math>66)
* Bodmer V — Nativity of Mary, Apocalypse of James; fourth century
* [[Papyrus 72|Papyrus Bodmer VII-IX]] (<math>\mathfrak{P}</math>72) — Epistle of Jude, 1-2 Peter, Psalms 33-34
* Bodmer X — Epistle of Corinthians to Paul and Third Epistle of Paul to the Corinthians; third/fourth century
* Bodmer XI — Ode of Solomon 1; fourth century
* [[Papyrus 75|Papyrus Bodmer XIV-XV]] (<math>\mathfrak{P}</math>75)
* [[Papyrus 74|Papyrus Bodmer XVII]] (<math>\mathfrak{P}</math>74)
* Bodmer XXIV — Psalms 17:46-117:44; third/fourth century
* Bodmer XLVI — Daniel 1:1-20
* [[Papyrus 73|Papyrus Bodmer L]] — Matthew 25-26; seventh century

=== Coptic ===
* [[Papyrus Bodmer III|Bodmer III]] — John 1:1-21:25; Genesis 1:1-4:2; fourth century; Bohairic
* Bodmer VI — Proverbs 1:1-21:4; fourth/fifth century; Paleo-Theban ("Dialect P")
* Bodmer XVI — Exodus 1:1-15:21; fourth century
* Bodmer XVIII — Deuteronomium 1:1-10:7; fourth century
* [[Papyrus Bodmer XIX|Bodmer XIX]] — Matthew 14:28-28:20; Romans 1:1-2:3; fourth/fifth century; Sahidic
* Bodmer XXI — Joshua 6:16-25; 7:6-11:23; 22:1-2; 22:19-23:7; 23:15-24:2; fourth century
* Bodmer XXII (''Mississippi Codex II'') — Jeremiah 40:3-52:34; Lamentations; Epistle of Jeremiah; Book of Baruch; fourth/fifth century
* Bodmer XXIII — Isaiah 47:1-66:24; fourth century
* Bodmer XL — Song of Songs
* Bodmer XLI — Acta Pauli; fourth century; sub-Achmimic
* Bodmer XLII — 2 Corinthians; dialect reported by Wolf-Peter Funk to be [[Coptic versions of the Bible|Sahidic]]
* Bodmer XLIV — Book of Daniel; Bohairic

==See also==

*[[List of New Testament papyri]]
* [[Bodmer Library]]

==Notes==

{{reflist|2}}

==References==
*''Anchor Bible Dictionary'' 1:766-77 "Bodmer Papyri".
*Robinson, James M. 1987. ''The Story of the Bodmer Papyri, the First Christian Monastic Library'' (Nashville) Includes an inventory of the Bodmer Papyri.

==External links==
* [https://commons.wikimedia.org/wiki/Category:Bodmer_Papyri Category:Bodmer Papyri on Wikimedia Commons]
* [http://www.earlham.edu/~seidti/iam/tc_pap75.html A folio of Bodmer codex containing parts of Luke and John]

[[Category:New Testament papyri]]
[[Category:Septuagint manuscripts]]

Closed manifold

2019-06-11T20:03:01Z

Crasshopper: /* Examples */

{{seealso|Classification of manifolds#Point-set}}

In [[mathematics]], a '''closed manifold''' is a type of [[topological space]], namely a [[compact space|compact]] [[manifold]] without boundary. (Compactness means that every open cover has a finite subcover.)

Compact manifolds are, in an intuitive sense, "finite". A closed manifold is the [[disjoint union]] of a finite number of connected closed manifolds. It’s not fully known what the supply of possible closed manifolds is.

==Examples==
The only one-dimensional example is a [[circle]].
The [[torus]] and the [[Klein bottle]] are closed.
A [[real line|line]] is not a closed because it is not compact: it can’t be covered by finitely many segments. A [[Disk (mathematics)|disk]] is compact (coverable), but is not a closed manifold because it has a boundary.

==Properties==
All compact topological manifolds can be embedded into <math>\mathbf{R}^n</math> for some ''n'', by the [[Whitney embedding theorem]].

==Contrasting terms==
A '''compact manifold''' means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty).
By contrast, a closed manifold is compact ''without'' boundary.

An '''open manifold''' is a manifold without boundary with no compact component.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger.
For instance, the [[disjoint union]] of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.

The notion of closed manifold is unrelated with that of a [[closed set]]. A disk with its boundary is a closed subset of the plane, but not a closed manifold.

==Use in physics==
The notion of a "[[Shape of the universe|closed universe]]" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive [[Ricci curvature]].

== References ==
* [[Michael Spivak]]: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, {{ISBN|0-914098-70-5}}.

[[Category:Geometric topology]]
[[Category:Manifolds]]

Closed manifold

2019-06-11T20:01:13Z

Crasshopper: it may or may not be “basic” — it’s not simple, and it’s not what all geometric topologists are interested in ………

{{seealso|Classification of manifolds#Point-set}}

In [[mathematics]], a '''closed manifold''' is a type of [[topological space]], namely a [[compact space|compact]] [[manifold]] without boundary. (Compactness means that every open cover has a finite subcover.)

Compact manifolds are, in an intuitive sense, "finite". A closed manifold is the [[disjoint union]] of a finite number of connected closed manifolds. It’s not fully known what the supply of possible closed manifolds is.

==Examples==
The simplest example is a [[circle]], which is a compact one-dimensional manifold.
Other examples of closed manifolds are the [[torus]] and the [[Klein bottle]].
As a counterexample, the [[real line]] is not a closed manifold because it is not compact. A [[Disk (mathematics)|disk]] is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.

==Properties==
All compact topological manifolds can be embedded into <math>\mathbf{R}^n</math> for some ''n'', by the [[Whitney embedding theorem]].

==Contrasting terms==
A '''compact manifold''' means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty).
By contrast, a closed manifold is compact ''without'' boundary.

An '''open manifold''' is a manifold without boundary with no compact component.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger.
For instance, the [[disjoint union]] of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.

The notion of closed manifold is unrelated with that of a [[closed set]]. A disk with its boundary is a closed subset of the plane, but not a closed manifold.

==Use in physics==
The notion of a "[[Shape of the universe|closed universe]]" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive [[Ricci curvature]].

== References ==
* [[Michael Spivak]]: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, {{ISBN|0-914098-70-5}}.

[[Category:Geometric topology]]
[[Category:Manifolds]]

Closed manifold

2019-06-11T19:58:25Z

Crasshopper:

{{seealso|Classification of manifolds#Point-set}}

In [[mathematics]], a '''closed manifold''' is a type of [[topological space]], namely a [[compact space|compact]] [[manifold]] without boundary. (Compactness means that every open cover has a finite subcover.)

Compact manifolds are, in an intuitive sense, "finite". By the basic properties of compactness, a closed manifold is the [[disjoint union]] of a finite number of connected closed manifolds. One of the most basic objectives of [[geometric topology]] is to understand what the supply of possible closed manifolds is.

==Examples==
The simplest example is a [[circle]], which is a compact one-dimensional manifold.
Other examples of closed manifolds are the [[torus]] and the [[Klein bottle]].
As a counterexample, the [[real line]] is not a closed manifold because it is not compact. A [[Disk (mathematics)|disk]] is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.

==Properties==
All compact topological manifolds can be embedded into <math>\mathbf{R}^n</math> for some ''n'', by the [[Whitney embedding theorem]].

==Contrasting terms==
A '''compact manifold''' means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty).
By contrast, a closed manifold is compact ''without'' boundary.

An '''open manifold''' is a manifold without boundary with no compact component.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger.
For instance, the [[disjoint union]] of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.

The notion of closed manifold is unrelated with that of a [[closed set]]. A disk with its boundary is a closed subset of the plane, but not a closed manifold.

==Use in physics==
The notion of a "[[Shape of the universe|closed universe]]" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive [[Ricci curvature]].

== References ==
* [[Michael Spivak]]: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, {{ISBN|0-914098-70-5}}.

[[Category:Geometric topology]]
[[Category:Manifolds]]

Closed manifold

2019-06-11T19:57:36Z

Crasshopper: obviously…

{{seealso|Classification of manifolds#Point-set}}

In [[mathematics]], a '''closed manifold''' is a type of [[topological space]], namely a [[compact space|compact]] [[manifold]] without boundary.

Compact manifolds are, in an intuitive sense, "finite". By the basic properties of compactness, a closed manifold is the [[disjoint union]] of a finite number of connected closed manifolds. One of the most basic objectives of [[geometric topology]] is to understand what the supply of possible closed manifolds is.

==Examples==
The simplest example is a [[circle]], which is a compact one-dimensional manifold.
Other examples of closed manifolds are the [[torus]] and the [[Klein bottle]].
As a counterexample, the [[real line]] is not a closed manifold because it is not compact. A [[Disk (mathematics)|disk]] is a compact two-dimensional manifold, but is not a closed manifold because it has a boundary.

==Properties==
All compact topological manifolds can be embedded into <math>\mathbf{R}^n</math> for some ''n'', by the [[Whitney embedding theorem]].

==Contrasting terms==
A '''compact manifold''' means a "manifold" that is compact as a topological space, but possibly has boundary. More precisely, it is a compact manifold with boundary (the boundary may be empty).
By contrast, a closed manifold is compact ''without'' boundary.

An '''open manifold''' is a manifold without boundary with no compact component.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger.
For instance, the [[disjoint union]] of a circle and the line is non-compact, but is not an open manifold, since one component (the circle) is compact.

The notion of closed manifold is unrelated with that of a [[closed set]]. A disk with its boundary is a closed subset of the plane, but not a closed manifold.

==Use in physics==
The notion of a "[[Shape of the universe|closed universe]]" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive [[Ricci curvature]].

== References ==
* [[Michael Spivak]]: ''A Comprehensive Introduction to Differential Geometry.'' Volume 1. 3rd edition with corrections. Publish or Perish, Houston TX 2005, {{ISBN|0-914098-70-5}}.

[[Category:Geometric topology]]
[[Category:Manifolds]]

Forging

2019-05-29T03:06:16Z

Crasshopper: /* History */

{{Redirect|Forged|counterfeiting|forgery|the book about Bible authorship|Forged (book)|the 2010 film|Forged (film)}}
{{About|the [[metalworking]] process|the hearth used in that process|forge}}
[[File:Bochumer Verein-03-50142.jpg|thumb|upright=1.4|Hot metal [[ingot]] being loaded into a hammer forge]]
'''Forging''' is a [[manufacturing process]] involving the shaping of [[metal]] using localized [[compression (physics)|compressive]] forces. The blows are delivered with a [[hammer]] (often a [[power hammer]]) or a [[die (manufacturing)|die]]. Forging is often classified according to the temperature at which it is performed: cold forging (a type of [[cold working]]), warm forging, or hot forging (a type of [[hot working]]). For the latter two, the metal is [[heat]]ed, usually in a [[forge]]. Forged parts can range in weight from less than a kilogram to hundreds of metric tons.<ref name="Degarmo389">Degarmo, p. 389</ref><ref name="wnaForge"/> Forging has been done by [[metalsmith|smith]]s for millennia; the traditional products were [[kitchenware]], [[household hardware|hardware]], [[hand tool]]s, [[edged weapon]]s, [[cymbals]], and [[jewellery]]. Since the [[Industrial Revolution]], forged parts are widely used in [[mechanism (engineering)|mechanism]]s and [[machine]]s wherever a component requires high [[strength of materials|strength]]; such '''forgings''' usually require further processing (such as [[machining]]) to achieve a finished part. Today, forging is a major worldwide industry.<ref>{{cite web|title=Forging: The Early Years|url=http://www.steelforge.com/literature/history-of-forgings/|publisher=All Metals & Forge Group|accessdate=1 October 2013}}</ref>

==History==
[[File:Forging a nail. Valašské muzeum v přírodě.webm|thumb|Forging a nail. Valašské muzeum v přírodě, Czech Republic]]
Forging is one of the oldest known [[metalworking]] processes.<ref name="Degarmo389"/> Traditionally, forging was performed by a [[Blacksmith|smith]] using hammer and [[anvil]], though introducing water power to the production and working of iron in the 12th century allowed the use of large trip hammers or power hammers that increased the amount and size of iron that could be produced and forged. The smithy or [[forge]] has evolved over centuries to become a facility with engineered processes, production equipment, tooling, raw materials and products to meet the demands of modern industry.

In modern times, industrial forging is done either with [[machine press|presses]] or with hammers powered by compressed air, electricity, hydraulics or steam. These hammers may have reciprocating weights in the thousands of pounds. Smaller [[power hammer]]s, {{convert|500|lb|abbr=on}} or less reciprocating weight, and hydraulic presses are common in art smithies as well. Some steam hammers remain in use, but they became obsolete with the availability of the other, more convenient, power sources.

== Advantages and disadvantages ==

Forging can produce a piece that is stronger than an equivalent [[casting (metalworking)|cast]] or [[machining|machined]] part. As the metal is shaped during the forging process, its internal [[Texture (crystalline)|grain texture]] deforms to follow the general shape of the part. As a result, the texture variation is continuous throughout the part, giving rise to a piece with improved strength characteristics.<ref name="Degarmo392"/> Additionally, forgings can achieve a lower total cost than casting or fabrication. Considering all the costs that are incurred in a product’s life cycle from procurement to lead time to rework, and factoring in the costs of scrap, and downtime and other quality considerations, the long-term benefits of forgings can outweigh the short-term cost savings that castings or fabrications might offer.<ref>http://www.scotforge.com/Why-Forging/Casting-Conversions</ref>

Some metals may be forged cold, but [[iron]] and [[steel]] are almost always [[hot working|hot forged]]. Hot forging prevents the [[work hardening]] that would result from [[cold forming]], which would increase the difficulty of performing secondary machining operations on the piece. Also, while work hardening may be desirable in some circumstances, other methods of hardening the piece, such as [[heat treating]], are generally more economical and more controllable. Alloys that are amenable to [[precipitation hardening]], such as most [[aluminium]] alloys and [[titanium]], can be hot forged, followed by hardening.{{Citation needed|date=November 2009}}

Production forging involves significant capital expenditure for machinery, tooling, facilities and personnel. In the case of hot forging, a high-temperature furnace (sometimes referred to as the forge) is required to heat [[ingot]]s or [[Billet (bar stock)|billets]]. Owing to the size of the massive forging hammers and presses and the parts they can produce, as well as the dangers inherent in working with hot metal, a special building is frequently required to house the operation. In the case of drop forging operations, provisions must be made to absorb the shock and vibration generated by the hammer. Most forging operations use metal-forming dies, which must be precisely machined and carefully heat-treated to correctly shape the workpiece, as well as to withstand the tremendous forces involved.

==Processes==
[[Image:ForgedConrodShowingEtchedSection-s.jpg|thumb|upright|A cross-section of a forged [[connecting rod]] that has been [[industrial etching|etched]] to show the grain flow]]
There are many different kinds of forging processes available; however, they can be grouped into three main classes:<ref name="Degarmo389"/>
*Drawn out: length increases, cross-section decreases
*Upset: length decreases, cross-section increases
*Squeezed in closed compression dies: produces multidirectional flow

Common forging processes include: roll forging, [[swage|swaging]], [[cogging]], open-die forging, impression-die forging, press forging, automatic hot forging and upsetting.<ref name="Degarmo389"/>

===Temperature===
{{Main|Hot working|Cold working}}
All of the following forging processes can be performed at various temperatures; however, they are generally classified by whether the metal temperature is above or below the recrystallization temperature. If the temperature is above the material's recrystallization temperature it is deemed ''hot forging''; if the temperature is below the material's recrystallization temperature but above 30% of the recrystallization temperature (on an absolute scale) it is deemed ''warm forging''; if below 30% of the recrystallization temperature (usually room temperature) then it is deemed ''cold forging''. The main advantage of hot forging is that it can be done more quickly and precisely, and as the metal is deformed [[work hardening]] effects are negated by the recrystallization process. Cold forging typically results in work hardening of the piece.<ref name="Degarmo373">Degarmo, p. 373</ref><ref name="Degarmo375">Degarmo, p. 375</ref>

===Drop forging===
[[File:Boat nail production.ogv|thumb|left|Boat nail production in [[Hainan]], China]]
Drop forging is a forging process where a hammer is raised and then "dropped" onto the workpiece to deform it according to the shape of the die. There are two types of drop forging: open-die drop forging and closed-die drop forging. As the names imply, the difference is in the shape of the die, with the former not fully enclosing the workpiece, while the latter does.

====Open-die drop forging====
[[File:Bochumer Verein-08-50124.jpg|thumb|upright|Open-die drop forging (with two dies) of an ingot to be further processed into a wheel]]
Open-die forging is also known as ''smith forging''.<ref name="Degarmo391">Degarmo, p. 391</ref> In open-die forging, a hammer strikes and deforms the workpiece, which is placed on a stationary [[anvil]]. Open-die forging gets its name from the fact that the dies (the surfaces that are in contact with the workpiece) do not enclose the workpiece, allowing it to flow except where contacted by the dies. The operator therefore needs to orient and position the workpiece to get the desired shape. The dies are usually flat in shape, but some have a specially shaped surface for specialized operations. For example, a die may have a round, concave, or convex surface or be a tool to form holes or be a cut-off tool.<ref name="Degarmo390">Degarmo, p. 390</ref>
Open-die forgings can be worked into shapes which include discs, hubs, blocks, shafts (including step shafts or with flanges), sleeves, cylinders, flats, hexes, rounds, plate, and some custom shapes.<ref>{{cite web|title=Forging Shapes|url=http://www.steelforge.com/custom-forged-shapes/forging-capabilities-chart/|publisher=All Metals & Forge Group|accessdate=1 October 2013}}</ref>
Open-die forging lends itself to short runs and is appropriate for art smithing and custom work. In some cases, open-die forging may be employed to rough-shape [[ingot]]s to prepare them for subsequent operations. Open-die forging may also orient the grain to increase strength in the required direction.<ref name="Degarmo390"/>

'''Advantages of open-die forging'''
* Reduced chance of voids
* Better fatigue resistance
* Improved microstructure
* Continuous grain flow
* Finer grain size
* Greater strength<ref>{{cite web|title=Forged Crankshaft Advantages|url=http://www.glforge.com/crankshafts.html|publisher=Great Lakes Forge|accessdate=28 February 2014}}</ref>
"{{visible anchor|Cogging}}" is the successive deformation of a bar along its length using an open-die drop forge. It is commonly used to work a piece of raw material to the proper thickness. Once the proper thickness is achieved the proper width is achieved via "edging".<ref>{{Citation|title=Cast steel: Forging |url=http://steel.keytometals.com/Articles/Art168.htm |accessdate=3 March 2010 |archiveurl=https://www.webcitation.org/5nxpp3qEi?url=http://steel.keytometals.com/Articles/Art168.htm |archivedate=3 March 2010 |deadurl=yes |df= }}</ref>
"{{visible anchor|Edging}}" is the process of concentrating material using a concave shaped open-die. The process is called "edging" because it is usually carried out on the ends of the workpiece. "{{visible anchor|Fullering}}" is a similar process that thins out sections of the forging using a convex shaped die. These processes prepare the workpieces for further forging processes.<ref>{{Citation|last=Kaushish|first=J. P.|title=Manufacturing Processes|page=469|publisher=PHI Learning|year=2008|url=https://books.google.com/?id=1ZOXXV9LdcwC&pg=PA469|isbn=978-81-203-3352-9}}</ref>
<gallery>
File:Forging-edging.svg|Edging
File:Forging-fullering.svg|Fullering
</gallery>

====Impression-die forging====
Impression-die forging is also called "closed-die forging". In impression-die forging, the metal is placed in a die resembling a mold, which is attached to an anvil. Usually, the hammer die is shaped as well. The hammer is then dropped on the workpiece, causing the metal to flow and fill the die cavities. The hammer is generally in contact with the workpiece on the scale of milliseconds. Depending on the size and complexity of the part, the hammer may be dropped multiple times in quick succession. Excess metal is squeezed out of the die cavities, forming what is referred to as "[[flash (manufacturing)|flash]]". The flash cools more rapidly than the rest of the material; this cool metal is stronger than the metal in the die, so it helps prevent more flash from forming. This also forces the metal to completely fill the die cavity. After forging, the flash is removed.<ref name="Degarmo391"/><ref name="Degarmo394"/>

In commercial impression-die forging, the workpiece is usually moved through a series of cavities in a die to get from an ingot to the final form. The first impression is used to distribute the metal into the rough shape in accordance to the needs of later cavities; this impression is called an "edging", "fullering", or "bending" impression. The following cavities are called "blocking" cavities, in which the piece is working into a shape that more closely resembles the final product. These stages usually impart the workpiece with generous bends and large [[fillet (mechanics)|fillets]]. The final shape is forged in a "final" or "finisher" impression cavity. If there is only a short run of parts to be done, then it may be more economical for the die to lack a final impression cavity and instead machine the final features.<ref name="Degarmo392">Degarmo, p. 392</ref>

Impression-die forging has been improved in recent years through increased automation which includes induction heating, mechanical feeding, positioning and manipulation, and the direct heat treatment of parts after forging.<ref name="Degarmo393">Degarmo, p. 393</ref>
One variation of impression-die forging is called "flashless forging", or "true closed-die forging". In this type of forging, the die cavities are completely closed, which keeps the workpiece from forming flash. The major advantage to this process is that less metal is lost to flash. Flash can account for 20 to 45% of the starting material. The disadvantages of this process include additional cost due to a more complex die design and the need for better lubrication and workpiece placement.<ref name="Degarmo392"/>

There are other variations of part formation that integrate impression-die forging. One method incorporates casting a forging preform from liquid metal. The casting is removed after it has solidified, but while still hot. It is then finished in a single cavity die. The flash is trimmed, then the part is quench hardened. Another variation follows the same process as outlined above, except the preform is produced by the spraying deposition of metal droplets into shaped collectors (similar to the [[Osprey process]]).<ref name="Degarmo393"/>

Closed-die forging has a high initial cost due to the creation of dies and required design work to make working die cavities. However, it has low recurring costs for each part, thus forgings become more economical with greater production volume. This is one of the major reasons closed-die forgings are often used in the automotive and tool industries. Another reason forgings are common in these industrial sectors is that forgings generally have about a 20 percent higher strength-to-weight ratio compared to cast or machined parts of the same material.<ref name="Degarmo392"/>

===== Design of impression-die forgings and tooling =====
Forging dies are usually made of [[alloy steel|high-alloy]] or [[tool steel]]. Dies must be impact- and wear-resistant, maintain strength at high temperatures, and have the ability to withstand cycles of rapid heating and cooling. In order to produce a better, more economical die the following standards are maintained:<ref name="Degarmo393"/>
*The dies part along a single, flat plane whenever possible. If not, the parting plane follows the contour of the part.
*The parting surface is a plane through the center of the forging and not near an upper or lower edge.
*Adequate [[Draft (engineering)|draft]] is provided; usually at least 3° for aluminium and 5° to 7° for steel.
*Generous fillets and radii are used.
*Ribs are low and wide.
*The various sections are balanced to avoid extreme difference in metal flow.
*Full advantage is taken of fiber flow lines.
*Dimensional tolerances are not closer than necessary.
The dimensional tolerances of a steel part produced using the impression-die forging method are outlined in the table below. The dimensions across the parting plane are affected by the closure of the dies, and are therefore dependent on die wear and the thickness of the final flash. Dimensions that are completely contained within a single die segment or half can be maintained at a significantly greater level of accuracy.<ref name="Degarmo394">Degarmo, p. 394</ref>
{| class="wikitable"
|+ Dimensional tolerances for impression-die forgings<ref name="Degarmo394"/>
|-
! Mass [kg (lb)]
! Minus tolerance [mm (in)]
! Plus tolerance [mm (in)]
|-
| 0.45 (1)
| 0.15 (0.006)
| 0.46 (0.018)
|-
| 0.91 (2)
| 0.20 (0.008)
| 0.61 (0.024)
|-
| 2.27 (5)
| 0.25 (0.010)
| 0.76 (0.030)
|-
| 4.54 (10)
| 0.28 (0.011)
| 0.84 (0.033)
|-
| 9.07 (20)
| 0.33 (0.013)
| 0.99 (0.039)
|-
| 22.68 (50)
| 0.48 (0.019)
| 1.45 (0.057)
|-
| 45.36 (100)
| 0.74 (0.029)
| 2.21 (0.087)
|}
A lubricant is used when forging to reduce friction and wear. It is also used as a thermal barrier to restrict heat transfer from the workpiece to the die. Finally, the lubricant acts as a parting compound to prevent the part from sticking in the dies.<ref name="Degarmo394"/>

===Press forging===
Press forging works by slowly applying a continuous pressure or force, which differs from the near-instantaneous impact of drop-hammer forging. The amount of time the dies are in contact with the workpiece is measured in seconds (as compared to the milliseconds of drop-hammer forges). The press forging operation can be done either cold or hot.<ref name="Degarmo394"/>

The main advantage of press forging, as compared to drop-hammer forging, is its ability to deform the complete workpiece. Drop-hammer forging usually only deforms the surfaces of the work piece in contact with the hammer and anvil; the interior of the workpiece will stay relatively undeformed. Another advantage to the process includes the knowledge of the new part's strain rate. By controlling the compression rate of the press forging operation, the internal strain can be controlled.

There are a few disadvantages to this process, most stemming from the workpiece being in contact with the dies for such an extended period of time. The operation is a time-consuming process due to the amount and length of steps. The workpiece will cool faster because the dies are in contact with workpiece; the dies facilitate drastically more heat transfer than the surrounding atmosphere. As the workpiece cools it becomes stronger and less ductile, which may induce cracking if deformation continues. Therefore, heated dies are usually used to reduce heat loss, promote surface flow, and enable the production of finer details and closer tolerances. The workpiece may also need to be reheated.

When done in high productivity, press forging is more economical than hammer forging. The operation also creates closer tolerances. In hammer forging a lot of the work is absorbed by the machinery; when in press forging, the greater percentage of work is used in the work piece. Another advantage is that the operation can be used to create any size part because there is no limit to the size of the press forging machine. New press forging techniques have been able to create a higher degree of mechanical and orientation integrity. By the constraint of oxidation to the outer layers of the part, reduced levels of microcracking occur in the finished part.<ref name="Degarmo394"/>

Press forging can be used to perform all types of forging, including open-die and impression-die forging. Impression-die press forging usually requires less draft than drop forging and has better dimensional accuracy. Also, press forgings can often be done in one closing of the dies, allowing for easy automation.<ref name="Degarmo395">Degarmo, p. 395</ref>

===Upset forging===
{{redirect|Upsetting||upset (disambiguation)}}
Upset forging increases the diameter of the workpiece by compressing its length.<ref name="Degarmo395"/> Based on number of pieces produced, this is the most widely used forging process.<ref name="Degarmo395"/> A few examples of common parts produced using the upset forging process are engine valves, couplings, bolts, screws, and other fasteners.

Upset forging is usually done in special high-speed machines called ''crank presses''. The machines are usually set up to work in the horizontal plane, to facilitate the quick exchange of workpieces from one station to the next, but upsetting can also be done in a vertical crank press or a hydraulic press. The initial workpiece is usually wire or rod, but some machines can accept bars up to {{convert|25|cm|in|abbr=on}} in diameter and a capacity of over 1000 tons. The standard upsetting machine employs split dies that contain multiple cavities. The dies open enough to allow the workpiece to move from one cavity to the next; the dies then close and the heading tool, or ram, then moves longitudinally against the bar, upsetting it into the cavity. If all of the cavities are utilized on every cycle, then a finished part will be produced with every cycle, which makes this process advantageous for mass production.<ref name="Degarmo395"/>

These rules must be followed when designing parts to be upset forged:<ref>Degarmo, pp. 395–396</ref>
*The length of unsupported metal that can be upset in one blow without injurious buckling should be limited to three times the diameter of the bar.
*Lengths of stock greater than three times the diameter may be upset successfully, provided that the diameter of the upset is not more than 1.5 times the diameter of the stock.
*In an upset requiring stock length greater than three times the diameter of the stock, and where the diameter of the cavity is not more than 1.5 times the diameter of the stock, the length of unsupported metal beyond the face of the die must not exceed the diameter of the bar.

===Automatic hot forging===
The automatic hot forging process involves feeding mill-length steel bars (typically {{convert|7|m|0|abbr=on}} long) into one end of the machine at room temperature and hot forged products emerge from the other end. This all occurs rapidly; small parts can be made at a rate of 180 parts per minute (ppm) and larger can be made at a rate of 90 ppm. The parts can be solid or hollow, round or symmetrical, up to {{convert|6|kg|lb|abbr=on}}, and up to {{convert|18|cm|in|abbr=on}} in diameter. The main advantages to this process are its high output rate and ability to accept low-cost materials. Little labor is required to operate the machinery.

There is no flash produced so material savings are between 20 and 30% over conventional forging. The final product is a consistent {{convert|1050|°C|°F|abbr=on}} so air cooling will result in a part that is still easily machinable (the advantage being the lack of [[annealing (metallurgy)|annealing]] required after forging). Tolerances are usually ±{{convert|0.3|mm|in|abbr=on}}, surfaces are clean, and draft angles are 0.5 to 1°. Tool life is nearly double that of conventional forging because contact times are on the order of 0.06-second. The downside is that this process is only feasible on smaller symmetric parts and cost; the initial investment can be over $10 million, so large quantities are required to justify this process.<ref>Degarmo, pp. 396–397</ref>

The process starts by heating the bar to {{convert|1200|to|1300|C|F}} in less than 60 seconds using high-power induction coils. It is then descaled with rollers, sheared into blanks, and transferred through several successive forming stages, during which it is upset, preformed, final forged, and pierced (if necessary). This process can also be coupled with high-speed cold-forming operations. Generally, the cold forming operation will do the finishing stage so that the advantages of cold-working can be obtained, while maintaining the high speed of automatic hot forging.<ref>Degarmo, p. 396</ref>

Examples of parts made by this process are: wheel hub unit bearings, transmission gears, tapered roller bearing races, stainless steel coupling flanges, and neck rings for LP gas cylinders.<ref>[http://www.samtech.co.jp/e/precision_hot/index.html Precision Hot Forging]. Samtech. Retrieved 22 November 2007</ref> Manual transmission gears are an example of automatic hot forging used in conjunction with cold working.<ref>[http://www.samtech.co.jp/e/precision_composite/index.html Precision Composite Forging]. Samtech. Retrieved 22 November 2007</ref>

===Roll forging===
Roll forging is a process where round or flat bar stock is reduced in thickness and increased in length. Roll forging is performed using two cylindrical or semi-cylindrical rolls, each containing one or more shaped grooves. A heated bar is inserted into the rolls and when it hits a spot the rolls rotate and the bar is progressively shaped as it is rolled through the machine. The piece is then transferred to the next set of grooves or turned around and reinserted into the same grooves. This continues until the desired shape and size is achieved. The advantage of this process is there is no flash and it imparts a favorable grain structure into the workpiece.<ref>Degarmo, pp. 397–398</ref>

Examples of products produced using this method include [[axle]]s, tapered levers and [[leaf spring]]s.

===Net-shape and near-net-shape forging===
{{See also|Near-net-shape}}
This process is also known as ''precision forging''. It was developed to minimize cost and waste associated with post-forging operations. Therefore, the final product from a precision forging needs little or no final machining. Cost savings are gained from the use of less material, and thus less scrap, the overall decrease in energy used, and the reduction or elimination of machining. Precision forging also requires less of a draft, 1° to 0°. The downside of this process is its cost, therefore it is only implemented if significant cost reduction can be achieved.<ref>Degarmo, p. 398</ref>

====Cold forging====
Near net shape forging is most common when parts are forged without heating the slug, bar or billet. Aluminum is a common material that can be cold forged depending on final shape. Lubrication of the parts being formed is critical to increase the life of the mating dies.

====Cost implications====
To achieve a low-cost net shape forging for demanding applications that are subject to a high degree of scrutiny, i.e. [[non-destructive testing]] by way of a dye-penetrant inspection technique, it is crucial that basic forging process disciplines be implemented. If the basic disciplines are not met, subsequent material removal operations will likely be necessary to remove material defects found at non-destructive testing inspection. Hence low-cost parts will not be achievable.{{Citation needed|date=January 2010}}

Example disciplines are: die-lubricant management (Use of uncontaminated and homogeneous mixtures, amount and placement of lubricant). Tight control of die temperatures and surface finish / friction.{{Citation needed|date=January 2010}}

===Induction forging===
{{Main|Induction forging}}
Unlike the above processes, induction forging is based on the type of heating style used. Many of the above processes can be used in conjunction with this heating method.

===Multidirectional forging===
Multidirectional Forging is forming of a work piece in a single step in several directions. The multidirectional forming takes place through constructive measures of the tool. The vertical movement of the press ram is redirected using wedges which distributes and redirects the force of the forging press in horizontal directions.<ref>Behrens, Stonis, Rüther, Blohm: ''Flash reduced forging of complicated high duty parts using preforming operations'', IPH - Institut für Integrierte Produktion Hannover gGmbH, Hannover, 2014.</ref>

===Isothermal forging===
Isothermal forging is a process by which the materials and the die are heated to the same temperature (''[[wikt:iso-#English|iso-]]'' meaning "equal"). Adiabatic heating is used to assist in the deformation of the material, meaning the strain rates are highly controlled. Commonly used for forging aluminum, which has a lower forging temperature than steels. Forging temperatures for Aluminum are around 800 °F, while steels and super alloys can be 1700-2300 °F.[https://www.forging.org/forging/design/5224-hot-die-and-isothermal-forging.html]

Benefits:
*Near net shapes which lead to lower machining requirements and therefore lower scrap rates
*Reproducibility of the part
*Due to the lower heat loss smaller machines can be used to make the forging

Disadvantages:
*Higher die material costs to handle temperatures and pressures
*Uniform heating systems are required
*Protective atmospheres or vacuum to reduce oxidation of the dies and material
*Low production rates

== Materials and applications ==

=== Forging of steel ===

Depending on the forming temperature steel forging can be divided into:<ref>Doege, E., Behrens, B.-A.: ''Handbuch Umformtechnik: Grundlagen, Technologien, Maschinen'' (in German), Springer Verlag, 2010, p. 7</ref>

* Hot forging of steel
** Forging temperatures above the recrystallization temperature between 950–1250 °C
** Good formability
** Low forming forces
** Constant tensile strength of the workpieces
* Warm forging of steel
** Forging temperatures between 750–950 °C
** Less or no scaling at the workpiece surface
** Narrower tolerances achievable than in hot forging
** Limited formability and higher forming forces than for hot forging
** Lower forming forces than in cold forming
* Cold forging of steel
** Forging temperatures at room conditions, self-heating up to 150 °C due to the forming energy
** Narrowest tolerances achievable
** No scaling at workpiece surface
** Increase of strength and decrease of ductility due to strain hardening
** Low formability and high forming forces are necessary

For industrial processes steel alloys are primarily forged in hot condition. Brass, bronze, copper, precious metals and their alloys are manufactured by cold forging processes, while each metal requires a different forging temperature.

=== Forging of aluminium ===

* Aluminium forging is performed at a temperature range between 350–550 °C
* Forging temperatures above 550 °C are too close to the solidus temperature of the alloys and lead in conjunction with varying effective strains to unfavorable workpiece surfaces and potentially to a partial melting as well as fold formation.<ref>Doege, E.; Behrens, B.-A.: ''Handbuch Umformtechnik: Grundlagen, Technologien, Maschinen'', Springer Verlag, 2010, pp. 671f.</ref>
* Forging temperatures below 350 °C reduce formability by increasing the yield stress, which can lead to unfilled dies, cracking at the workpiece surface and increased die forces

Due to the narrow temperature range and high thermal conductivity, aluminium forging can only be realized in a particular process window. To provide good forming conditions a homogeneous temperature distribution in the entire workpiece is necessary. Therefore, the control of the tool temperature has a major influence to the process. For example, by optimizing the preform geometries the local effective strains can be influenced to reduce local overheating for a more homogeneous temperature distribution.<ref>Stonis, M.: ''Mehrdirektionales Schmieden von flachen Aluminiumlangteilen'' (in German), In: Behrens, B.-A.; Nyhuis, P.; Overmeyer, L. (ed.): Berichte aus dem IPH, Volume 01/2011, PZH Produktionstechnisches Zentrum GmbH, Garbsen 2011.</ref>

==== Application of aluminium forged parts ====

High-strength aluminium alloys have the tensile strength of medium strong steel alloys while providing significant weight advantages. Therefore, aluminium forged parts are mainly used in aerospace, automotive industry and many other fields of engineering especially in those fields, where highest safety standards against failure by abuse, by shock or vibratory stresses are needed. Such parts are for example pistons,<ref>{{Cite news|url=http://blog.wiseco.com/what-is-forging|title=What is Forging? The Ins And Outs of Squishing Aluminum Into Pistons|last=Huizenga|first=Paul|access-date=2018-04-22|language=en}}</ref> chassis parts, steering components and brake parts. Commonly used alloys are AlSi1MgMn ([[6082 aluminium alloy|EN AW-6082]]) and AlZnMgCu1,5 ([[7075 aluminium alloy|EN AW-7075]]). About 80% of all aluminium forged parts are made of AlSi1MgMn. The high-strength alloy AlZnMgCu1,5 is mainly used for aerospace applications.<ref>Richter, J.; Stonis, M.: ''Qualitätsverbesserung beim Aluminiumschmieden'' (in German), In Aluminium Praxis, Giesel Verlag GmbH, Volume 20 (2015), Issue 6/15, p. 20.</ref>

== Equipment ==
[[Image:Forging shop-Gesenkschmiede 1.JPG|thumb|Hydraulic drop-hammer]]
[[Image:impactor flow.png|thumb|(a) Material flow of a conventionally forged disc; (b) Material flow of an impactor forged disc]]
The most common type of forging equipment is the hammer and anvil. Principles behind the hammer and anvil are still used today in ''drop-hammer'' equipment. The principle behind the machine is simple: raise the hammer and drop it or propel it into the workpiece, which rests on the anvil. The main variations between drop-hammers are in the way the hammer is powered; the most common being air and steam hammers. Drop-hammers usually operate in a vertical position. The main reason for this is excess energy (energy that isn't used to deform the workpiece) that isn't released as heat or sound needs to be transmitted to the foundation. Moreover, a large machine base is needed to absorb the impacts.<ref name="Degarmo390"/>

To overcome some shortcomings of the drop-hammer, the ''counterblow machine'' or ''impactor'' is used. In a counterblow machine both the hammer and anvil move and the workpiece is held between them. Here excess energy becomes recoil. This allows the machine to work horizontally and have a smaller base. Other advantages include less noise, heat and vibration. It also produces a distinctly different flow pattern. Both of these machines can be used for open-die or closed-die forging.<ref>Degarmo, pp. 392–393</ref>
===Forging presses===
A ''forging press'', often just called a press, is used for press forging. There are two main types: mechanical and hydraulic presses. Mechanical presses function by using cams, cranks and/or toggles to produce a preset (a predetermined force at a certain location in the stroke) and reproducible stroke. Due to the nature of this type of system, different forces are available at different stroke positions. Mechanical presses are faster than their hydraulic counterparts (up to 50 strokes per minute). Their capacities range from 3 to 160 MN (300 to 18,000 short tons-force). Hydraulic presses use fluid pressure and a piston to generate force. The advantages of a hydraulic press over a mechanical press are its flexibility and greater capacity. The disadvantages include a slower, larger, and costlier machine to operate.<ref name="Degarmo394"/>

The roll forging, upsetting, and automatic hot forging processes all use specialized machinery.

{| class="wikitable"
|+List of large forging presses, by ingot size<ref name="wnaForge">[http://www.world-nuclear.org/info/inf122_heavy_manufacturing_of_power_plants.html Heavy Manufacturing of Power Plants] ''[[World Nuclear Association]]'', September 2010. Retrieved: 25 September 2010.</ref><ref name="neiForge">Kidd, Steve. [http://www.neimagazine.com/story.asp?sectioncode=147&storyCode=2052302 New nuclear build – sufficient supply capability?] {{webarchive |url=https://web.archive.org/web/20110613104418/http://www.neimagazine.com/story.asp?sectioncode=147&storyCode=2052302 |date=June 13, 2011 }} ''Nuclear Engineering International'', 3 March 2009. Retrieved: 25 September 2010</ref>
|-
! [[Force (physics)#Units of measurement|Force]] ([[tonne]]s)
! [[Ingot]] size ([[tonne]]s)
! Company
! Location
|-
| 16,500
| 600
| [[Shanghai Electric Group]]<ref name=erzhong/>
| [[Shanghai]], China
|-
| 16,000
| 600
| [[China National Erzhong Group]]<ref name=erzhong/>
| [[Deyang]], China
|-
| 14,000
| 600
| [[Japan Steel Works]]
| Japan
|-
| 15,000
| 580
| [[China First Heavy Industries Group]]<ref>{{cite news|title=World's Largest 15000MN hydraulic forging press|url=http://www.chinatechgadget.com/worlds-largest-150mn-hydraulic-forging-press.html|accessdate=15 May 2012|newspaper=China Tech Gadget|date=3 November 2011}}</ref>
| [[Heilongjiang]], China
|-
| 13,000
|
| [[Doosan Heavy Industries & Construction|Doosan]]
| South Korea
|-
|}

{| class="wikitable"
|+List of large forging presses, by force
|-
! [[Force (physics)#Units of measurement|Force]] ([[tonne]]s)
! [[Force (physics)#Units of measurement|Force]] ([[ton]]s)
! [[Ingot]] size ([[tonne]]s)
! Company
! Location
|-
| 80,000
| ''(88,200)''
| >150
| [[China Erzhong]]<ref name=erzhong>{{cite news|title=China Building World's Largest Press Forge|url=http://www.chinatechgadget.com/china-building-worlds-largest-press-forge.html|accessdate=12 February 2012|newspaper=China Tech Gadget|date=27 October 2011}}</ref>
| [[Deyang]], China
|-
| 75,000
| ''(82,690)''
|
|[[VSMPO-AVISMA]]
| Russia
|-
| 65,000
| ''(71,660)''
|
|[[Aubert & Duval]]<ref name="Eramet alloys">{{cite web|title=Eramet alloys|url=http://www.eramet.fr/us/Site/Template/T1.aspx?SELECTID=119&ID=106|accessdate=18 May 2012}}</ref><ref name="ADA132398">{{cite book|last=Altan|first=Taylan|title=Feasibility of Using a Large Press (80,000 – 200,000 Ton) for Manufacturing Future Components on Army Systems|year=1983|url=http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA132398|page=12}}</ref>
| [[Issoire]], France
|-
| ''(45,350)''
| 50,000
| 20
| [[Alcoa]],<ref>{{cite news|last=Heffernan|first=Tim|title=Iron Giant|url=https://www.theatlantic.com/magazine/archive/2012/03/iron-giant/8886/|accessdate=12 February 2012|newspaper=The Atlantic|date=8 February 2012}}</ref><ref>{{cite book|publisher=American Society of Mechanical Engineers|title=50,000 Ton Closed Die Forging Press|year=1981|url=http://files.asme.org/asmeorg/communities/history/landmarks/5488.pdf}} History of the Mesta Press at Alcoa</ref> [[Wyman Gordon]]<ref>{{cite book |url=http://files.asme.org/ASMEORG/Communities/History/Landmarks/5662.pdf |title=The Wyman-Gordon 50,000 Ton Forging Press |publisher=American Society of Mechanical Engineers |year=1983 |deadurl=yes |archiveurl=https://web.archive.org/web/20150201213858/http://files.asme.org/ASMEORG/Communities/History/Landmarks/5662.pdf |archivedate=2015-02-01 |df= }} History of the Loewy Press at Wyman-Gordon</ref><ref>{{cite news|last=Edson|first=Peter|title=Revolutionary Metal Press Cuts Cost of Planes and Guns|url=https://news.google.com/newspapers?id=8kIgAAAAIBAJ&sjid=J4oEAAAAIBAJ&pg=6350,424911&dq=forging+aircraft&hl=en|accessdate=12 February 2012|newspaper=Sarasota Journal|date=18 April 1952}}</ref>
| USA
|-
| 40,000
| ''(44,100)''
|
|[[Aubert & Duval]]<ref name="Eramet alloys"/>
| [[Pamiers]], France
|-
| 30,000
| ''(33,080)''
| 8
| [[Wyman Gordon]]<ref name="Wyman Gordon Livingston">{{cite web|title=Wyman Gordon Livingston|url=http://www.wyman-gordon.com/livingston.htm|accessdate=18 May 2012}}</ref>
| [[Livingston, West Lothian|Livingston]], Scotland
|-
| 30,000
| ''(33,070)''
|
| [[Weber Metals, Inc.]]<ref name="Weber Metals, Inc.">{{cite web|title=Weber Metals|url=http://www.webermetals.com|accessdate=18 July 2013}}</ref>
| [[Paramount, California|California]], United States
|-
| 30,000
| ''(30,108)''
|
| [[Firth Rixson]]<ref name="Firth Rixson Forgings">{{cite web|title=Firth Rixson|url=http://www.firthrixson.com|accessdate=18 May 2012}}</ref>
| [[Midway, Georgia|Georgia]], United States
|-
|}

==See also==
*[[Casting]]
*[[Cold sizing]]
*[[Hammerscale]]
*[[Thixoforming]]
*[[Forging temperature]]

==References==
{{reflist}}

===Bibliography===
* {{Cite book|last=Degarmo|first=E. Paul|last2=Black|first2=J. T.|last3=Kohser|first3=Ronald A.|title=Materials and Processes in Manufacturing|publisher=Wiley|year=2011|edition=11th|isbn=978-0-470-92467-9|}}
* Doege, E.; Behrens, B.-A.: ''Handbuch Umformtechnik: Grundlagen, Technologien, Maschinen'' (in German), 2nd Edition, Springer Verlag, 2010, {{ISBN|978-3-642-04248-5}}
* Ostermann, F.: ''Anwendungstechnologie Aluminium'' (in German), 3rd Edition, Springer Verlag, 2014, {{ISBN|978-3-662-43806-0}}

==External links==
{{Commons category|Forging}}
*[http://www.beyondrigging.com/2014/04/hot-forging-vs-cold-forging/ Hot Forging vs. Cold Forging]
*[http://www.qcforge.info Glossary of Forging Terms and Definitions]

{{Metalworking navbox|formopen|tool}}

{{Authority control}}

[[Category:Metal forming]]
[[Category:Articles containing video clips]]

1993 World Trade Center bombing

2019-05-17T20:18:20Z

Crasshopper:

{{about|the 1993 World Trade Center bombing|the terrorist attack in 2001|September 11 attacks}}
{{Infobox civilian attack
| title = 1993 World Trade Center bombing
|partof=[[terrorism in the United States]]
| image = WTC 1993 ATF Commons.jpg
| caption = Underground damage after the bombing
| location = [[World Trade Center (1973–2001)|World Trade Center]] [[New York City]], [[New York (state)|New York]], U.S.
| coordinates = {{coord|40.711452|N|74.011919|W|region:US-NY_type:event|display=title,inline}}
| target = [[World Trade Center (1973-2001)|World Trade Center]]
| date = {{start date and age|1993|02|26}}
| time = 12:17:37 p.m.
| timezone = [[UTC-05:00]]
| type = [[Car bomb|Truck bombing]], [[mass murder]]
| fatalities = 6
| injuries = 1,042
| perpetrators = [[Ramzi Yousef]], [[Eyad Ismoil]], and co-conspirators
| motive = [[American foreign policy]] [[Israel–United States military relations#Military aid and procurement|U.S. support for Israel]]
}}

The '''1993 World Trade Center bombing''' was a [[Terrorism|terrorist]] attack on the [[World Trade Center (1973-2001)|World Trade Center]], carried out on February 26, 1993, when a [[truck bomb]] detonated below the [[One World Trade Center|North Tower]] of the World Trade Center in [[New York City]]. The {{convert|1336|lb|kg|abbr=on}} [[urea nitrate]]–[[hydrogen]] gas enhanced device<ref>{{cite news |url=https://www.washingtonpost.com/wp-dyn/content/article/2007/07/04/AR2007070401814_pf.html |title=Homemade, Cheap and dangerous – Terror Cells Favor from Simple Ingredients In Building Bombs |date=July 5, 2005 |author=Whitlock, Craig |work=The Washington Post |accessdate=September 9, 2009 |archive-url=https://web.archive.org/web/20081006102225/http://www.washingtonpost.com/wp-dyn/content/article/2007/07/04/AR2007070401814_pf.html |archive-date=October 6, 2008 |dead-url=no |df=mdy-all }}</ref> was intended to send the North Tower ([[One World Trade Center#Original building|Tower 1]]) crashing into the South Tower ([[Two World Trade Center#Original building|Tower 2]]), bringing both towers down and killing tens of thousands of people.<ref name='SJC 1998-02-24'>{{cite web |url=http://judiciary.senate.gov/oldsite/childers.htm |title=Senate Judiciary Committee Hearings: Foreign Terrorists in America: Five Years After the World Trade Center |accessdate=January 8, 2008 |last=Childers |first=J. Gilmore |author2=Henry J. DePippo |date=February 24, 1998 |publisher=US Senate Judiciary Committee |archiveurl=https://web.archive.org/web/20071227065444/http://judiciary.senate.gov/oldsite/childers.htm |archivedate=December 27, 2007 |deadurl=yes |df=mdy }}</ref><ref>Wright, Lawrence, ''Looming Tower'', Knopf, (2006) p. 178.</ref> It failed to do so but killed six people and injured over a thousand.<ref name="FBI">{{cite web |url=https://www.fbi.gov/news/stories/2008/february/tradebom_022608 |title=FBI 100 First Strike: Global Terror in America |publisher=FBI.gov |accessdate=September 8, 2011 |archive-url=https://web.archive.org/web/20110903024103/http://www.fbi.gov/news/stories/2008/february/tradebom_022608 |archive-date=September 3, 2011 |dead-url=no |df=mdy-all }}</ref>

The attack was planned by a group of terrorists including [[Ramzi Yousef]], [[Mahmud Abouhalima]], [[Mohammad Salameh]], Nidal A. Ayyad, [[Abdul Rahman Yasin]], and [[Ahmed Ajaj]]. They received{{fact}} financing from [[Khalid Sheikh Mohammed|Khalid Sheikh Mohammed]], Yousef's uncle.{{fact}} In March 1994, four men were convicted of carrying out the bombing: Abouhalima, Ajaj, Ayyad, and Salameh. The charges included conspiracy, explosive destruction of property, and interstate transportation of explosives. In November 1997, two more were convicted: Ramzi Yousef, the mastermind behind the bombings, and [[Eyad Ismoil]], who drove the truck carrying the bomb.

==Planning and organization==
{{moresources|section|date=April 2018}}
Ramzi Yousef, who was born as Abdul Basit Mahmoud Abdul Karim in [[Kuwait]], spent time at an [[al-Qaeda training camp]] in [[Afghanistan]],<ref name="wright">Wright (2006), Chapter 9.</ref> before beginning in 1991 to plan a bombing attack within the [[United States]]. Yousef's uncle [[Khalid Shaikh Mohammed|Khalid Shaikh Mohammed Ali Fadden]], who later was considered the principal architect of the [[September 11 attacks]], gave him advice and tips over the phone, and funded his co-conspirator [[Mohammed Salameh]] with a [[United States dollar|US$]]660 [[wire transfer]].<ref name="ksm">{{cite web|url=http://www.globalsecurity.org/military/world/para/ksm.htm|title=Khalid Sheikh Mohammed|publisher=GlobalSecurity.org|accessdate=October 26, 2008|archiveurl=https://web.archive.org/web/20081021135829/http://www.globalsecurity.org/military/world/para/ksm.htm| archivedate=October 21, 2008|deadurl=no}}</ref>

Yousef arrived illegally in the United States on September 1, 1992, traveling with [[Ahmed Ajaj]] from Pakistan, though both sat apart on the flight and acted as though they were traveling separately. Ajaj tried to enter with a forged Swedish passport, though it had been altered and thus raised suspicions among INS officials at [[John F. Kennedy International Airport]]. When officials put Ajaj through secondary inspection, they discovered bomb-making instructions and other materials in his luggage, and arrested him. The name [[Abu Barra]], an alias of [[Mohammed Jamal Khalifa]], appeared in the manuals. Yousef tried to enter with a false [[Ba'athist Iraq|Iraq]]i passport, claiming [[political asylum]]. Yousef was allowed into the United States, and was given a hearing date.<ref>{{cite web|url=https://fas.org/irp/congress/1998_hr/s980224c.htm|title=Foreign Terrorists in America|work=1998 Congressional Hearings – Intelligence and Security|publisher=Federation of American Scientists|date=February 24, 1998|accessdate=October 27, 2008|archive-url=https://web.archive.org/web/20090112071542/http://www.fas.org/irp/congress/1998_hr/s980224c.htm|archive-date=January 12, 2009|dead-url=no|df=mdy-all}}</ref>

Yousef set up residence in [[Jersey City, New Jersey]], traveled around [[New York (state)|New York]] and New Jersey and called Sheikh [[Omar Abdel Rahman]], a controversial blind [[Muslim]] cleric, via [[cell phone]]. After being introduced to his co-conspirators by Abdel Rahman at the latter's Al-Farooq Mosque in [[Brooklyn]], Yousef began assembling the {{convert|1500|lb|abbr=on}} [[urea nitrate]]–[[hydrogen]] gas enhanced device for delivery to the WTC. He ordered chemicals from his hospital room when injured in a car crash – one of three accidents caused by [[Mohammed A. Salameh|Salameh]] in late 1992 and early in 1993.

[[El Sayyid Nosair]], one of the blind sheikh's men, was arrested in 1991 for the murder of [[Rabbi]] [[Meir Kahane]]. According to prosecutors, "the Red" [[Mahmud Abouhalima]], also convicted in the bombing, told [[Wadih el Hage]] to buy the .357 caliber [[revolver]] used by Nosair in the Kahane shooting. In the initial court case in NYS Criminal Court Nosair was acquitted of murder but convicted of gun charges (in a related and follow-up case in Federal Court, he was convicted). Dozens of [[Arabic language|Arabic]] bomb-making manuals and documents related to terrorist plots were found in Nosair's New Jersey apartment, with manuals from [[John F. Kennedy Special Warfare Center and School|Army Special Warfare Center]] at [[Fort Bragg, North Carolina]], secret memos linked to the Joint Chiefs of Staff, and 1,440 rounds of ammunition. (Lance 2004 26)

According to the transcript of his trial, Yousef hoped that his explosion would topple Tower 1 which would fall into Tower 2, killing the occupants of both buildings, which he estimated to be about 250,000 people<ref>{{Cite book|url=https://books.google.com/books?id=wglOAgAAQBAJ&pg=PT478&dq=1993+bombing+250,000+americans&hl=en&sa=X&ved=0ahUKEwi87Jq6uZjSAhUI5YMKHabzCz4Q6AEIKDAC#v=onepage&q=1993%20bombing%20250,000%20americans&f=false|title=City in the Sky: The Rise and Fall of the World Trade Center|last=Glanz|first=James|last2=Lipton|first2=Eric|date=January 21, 2014|publisher=Times Books|isbn=9781466863071|language=en|access-date=February 18, 2017|archive-url=https://web.archive.org/web/20170218173239/https://books.google.com/books?id=wglOAgAAQBAJ&pg=PT478&dq=1993+bombing+250,000+americans&hl=en&sa=X&ved=0ahUKEwi87Jq6uZjSAhUI5YMKHabzCz4Q6AEIKDAC#v=onepage&q=1993%20bombing%20250,000%20americans&f=false|archive-date=February 18, 2017|dead-url=no|df=mdy-all}}</ref> in vengeance for U.S. support for Israel against Palestine.<ref>{{Cite book|url=https://books.google.com/books?id=8dCnb4uR63EC&printsec=frontcover&dq=towers+1993+bombing+250,000+americans&hl=en&sa=X&ved=0ahUKEwiNz7WfvJjSAhWH3oMKHYHZAlY4FBDoAQhOMAk#v=onepage&q=250,000&f=false|title=The Looming Tower|last=Wright|first=Lawrence|date=August 8, 2006|publisher=Knopf Doubleday Publishing Group|year=|isbn=9780307266088|location=|page=178|access-date=February 18, 2017|archive-url=https://web.archive.org/web/20170218173124/https://books.google.com/books?id=8dCnb4uR63EC&printsec=frontcover&dq=towers+1993+bombing+250,000+americans&hl=en&sa=X&ved=0ahUKEwiNz7WfvJjSAhWH3oMKHYHZAlY4FBDoAQhOMAk#v=onepage&q=250,000&f=false|archive-date=February 18, 2017|dead-url=no|df=mdy-all}}</ref>

According to the journalist [[Steve Coll]], Yousef mailed letters to various New York newspapers just before the attack, in which he claimed he belonged to "Liberation Army, Fifth Battalion".<ref>{{cite book|last=Coll|first=Steve|title=Ghost Wars: The Secret History of the CIA, Afghanistan, and Bin Laden, from the Soviet Invasion to September 10, 2001|publisher=The Penguin Press HC|year=2004|isbn=1-59420-007-6}}</ref>

These letters made three demands: an end to all US aid to [[Israel]], an end to US diplomatic relations with Israel, and a pledge by the United States to end interference "with any of the [[Middle East]] countries' interior affairs." He stated that the attack on the World Trade Center would be merely the first of such attacks if his demands were not met. In his letters, Yousef admitted that the World Trade Center bombing was an act of terrorism, but this was justified because "the terrorism that Israel practices (which America supports) must be faced with a similar one."{{cn|date=April 2018}} Yousef did not make any religious justification for the bombing. When asked about his religious views, he was evasive.<ref>{{Cite web|url=https://www.rand.org/blog/2001/09/religion-isnt-sole-motive-of-terror.html|title=Religion Isn't Sole Motive of Terror|last=Parachini|first=John V.|date=2001-09-16|website=www.rand.org|language=en|access-date=2019-04-13|archive-url=https://web.archive.org/web/20190413034959/https://www.rand.org/blog/2001/09/religion-isnt-sole-motive-of-terror.html|archive-date=April 13, 2019|dead-url=no|df=mdy-all}}</ref>

==Attack==
[[File:1993 World Trade Center Bombing by Eric Ascalon WTC5.jpg|thumb|Image of the procession of rescue vehicles responding to the 1993 World Trade Center bombing. One World Trade Center is on the far right of the frame.]]
[[File:WTC1993 BlastDamage.png|thumb|Depiction of [[Explosion|blast]] damage]]

On Friday, February 26, 1993, Ramzi Yousef and a Jordanian friend, [[Eyad Ismoil]], drove a yellow [[Ryder]] van into [[Lower Manhattan]], and pulled into the public parking garage beneath the World Trade Center around noon. They parked on the underground B-2 level. Yousef ignited the 20-foot fuse, and fled. Twelve minutes later, at 12:17:37 p.m., the bomb exploded in the underground garage, generating an estimated pressure of 150,000 [[pounds per square inch|psi]].<ref>Reeve (1999), p. 10.</ref> The bomb opened a 30-m (98 ft) wide hole through four sublevels of concrete. The detonation velocity of this bomb was about 15,000 ft/s (4.5 km/s), or 10,066.2133 mph. Initial news reports indicated a main transformer might have blown, before it became clear that a bomb had exploded in the basement.

The bomb instantly cut off the World Trade Center's main electrical power line, knocking out the emergency lighting system. The bomb caused smoke to rise to the 93rd floor of both towers, including through the stairwells which were not pressurized, and smoke went up the damaged elevators in the World Trade Center Towers 1 & 2.<ref>{{cite news |url=https://query.nytimes.com/gst/fullpage.html?res=9F0CE5DC103DF934A15751C0A965958260&sec=&spon=&pagewanted=all |title=Tougher Code May Not Have Helped |author=Barbanel, Josh |date=February 27, 1993 |work=The New York Times| accessdate= September 9, 2009 }}</ref> With thick smoke filling the stairwells, evacuation was difficult for building occupants and led to many smoke inhalation injuries. Hundreds were trapped in elevators in the towers when the power was cut, including a group of 17 kindergartners, on their way down from the South Tower observation deck, who were trapped between the 35th and 36th floors for five hours.<ref>{{cite news |url=http://www.newsweek.com/id/111113 |title=A Shaken City's Towering Inferno |author=Mathews, Tom |date=March 8, 1993 |work=Newsweek |accessdate=October 26, 2008| archiveurl= https://web.archive.org/web/20081030073528/http://www.newsweek.com/id/111113| archivedate=October 30, 2008 | deadurl= no}}</ref><ref>{{cite news |title=A major calamity, a lot of fear |author=Stone, Andrea |work=USA Today |date=March 1, 1993}}</ref>

Also as a result of the loss of power most of New York City's radio and television stations lost their over-the-air broadcast signal for almost a week, with television stations only being able to broadcast via cable and satellite via a microwave hookup between the stations and three of the New York area's largest cable companies, [[Cablevision]], [[Comcast]], and [[Time Warner Cable]]. Telephone service for much of Lower Manhattan was also disrupted.

Six people were killed, five Port Authority employees and a businessman whose car was in the parking garage. Additionally, 1,042 people were injured, most during the evacuation that followed the blast.<ref name="Reeve 1999, p. 15">Reeve (1999), p. 15.</ref> A report from the US Fire Administration states that, "Among the scores of people who fled to the roofs of the towers, 28 with medical problems were [[airlift]]ed by [[New York City Police Department|New York City police]] helicopters". It is known that 15 people received traumatic injury from the blast and 20 complained of cardiac problems. One firefighter was hospitalized, while 87 others, 35 police officers, and an EMS worker were also injured in dealing with the fires and other aftermath.<ref name="USFA-TR-076">{{cite news|url=http://www.usfa.fema.gov/downloads/pdf/publications/tr-076.pdf|title=The World Trade Center Bombing: Report and Analysis|date=February 1993|publisher=US Fire Administration, DHS|accessdate=October 25, 2011|archive-url=https://web.archive.org/web/20111216064409/http://www.usfa.fema.gov/downloads/pdf/publications/tr-076.pdf|archive-date=December 16, 2011|dead-url=no|df=mdy-all}}</ref>

The plan was that if the bomb truck was parked at the right place, the North Tower would fall onto the South Tower, collapsing them both. The tower did not collapse according to Yousef's plan, but the garage was severely damaged in the explosion. Had the van been parked closer to the WTC's poured concrete foundations, Yousef's plan might have succeeded.<ref>{{cite news |archiveurl=https://web.archive.org/web/20050316140649/http://msnbc.msn.com/id/3069653/ |archivedate=March 16, 2005 |url=http://msnbc.msn.com/id/3069653/ |publisher=MSNBC |title=An Icon Destroyed |year=2003}}</ref> He escaped to [[Pakistan]] several hours after the bombing.

===Bomb characteristics===
{{refimprove section|date=February 2011}}
Yousef was assisted by Iraqi bomb maker [[Abdul Rahman Yasin]], who helped assemble the complex {{convert|1310|lb|kg|adj=on}} bomb, which was made of a [[urea nitrate]] main charge with [[aluminum]], [[magnesium]] and [[ferric oxide]] particles surrounding the explosive. The charge used [[nitroglycerine]], [[ammonium nitrate]] dynamite, [[smokeless powder]] and [[Fuse (explosives)|fuse]] as booster explosives.<ref>{{cite web |url=https://www.fbi.gov/wanted/terrorists/teryasin.htm |title=Abdul Rahman Yasin |work=Most Wanted Terrorists |publisher=Federal Bureau of Investigation |accessdate=October 26, 2008| archiveurl= https://web.archive.org/web/20081014074948/https://www.fbi.gov/wanted/terrorists/teryasin.htm| archivedate=October 14, 2008 | deadurl= no}}</ref> Three tanks of bottled [[hydrogen]] were also placed in a circular configuration around the main charge, to enhance the fireball and afterburn of the solid metal particles.<ref>{{cite web |url=https://fas.org/irp/congress/1998_hr/s980224c.htm |title=Foreign Terrorists In America |publisher=Federation of American Scientists |accessdate=October 26, 2008 |archive-url=https://web.archive.org/web/20090112071542/http://www.fas.org/irp/congress/1998_hr/s980224c.htm |archive-date=January 12, 2009 |dead-url=no |df=mdy-all }}</ref> The use of compressed gas cylinders in this type of attack closely resembles the [[1983 Beirut barracks bombing]] 10 years earlier. Both of these attacks used compressed gas cylinders to create fuel-air and [[thermobaric]] bombs<ref>Paul Rogers(2000) [http://www.brad.ac.uk/acad/peace/publications/papers/psp1full.pdf Politics in the Next 50 Years: The Changing Nature of International Conflict] {{Webarchive|url=https://web.archive.org/web/20090329054243/http://www.brad.ac.uk/acad/peace/publications/papers/psp1full.pdf |date=March 29, 2009 }}.</ref> that release more energy than conventional high explosives. According to testimony in the bomb trial, only once before the 1993 attack had the FBI recorded a bomb that used [[urea nitrate]].<ref>{{Cite web |url=http://www.public-action.com/SkyWriter/WacoMuseum/death/tscr/whitehur/fw_21.gif |title="Urea nitrate rarely used as explosive." |access-date=June 2, 2006 |archive-url=https://web.archive.org/web/20060630180649/http://public-action.com/SkyWriter/WacoMuseum/death/tscr/whitehur/fw_21.gif |archive-date=June 30, 2006 |dead-url=no |df=mdy-all }}</ref><ref>Alternate link: If you get a 403 server error, try this [http://www.public-action.com/SkyWriter/WacoMuseum/death/tscr/whitehur/fw_test.html link] {{Webarchive|url=https://web.archive.org/web/20060821225002/http://www.public-action.com/SkyWriter/WacoMuseum/death/tscr/whitehur/fw_test.html |date=August 21, 2006 }} and then click on the link for "Page 16335".</ref><ref>
Frederic Whitehurst, FBI Lab Whistleblower
Testifying at the World Trade Center Bombing Trial
August 14, 1995 [http://www.web-ak.com/waco/death/tscr/whitehur/fw_test.html] {{Webarchive|url=https://web.archive.org/web/20160730002833/http://www.web-ak.com/waco/death/tscr/whitehur/fw_test.html |date=July 30, 2016 }}[http://www.web-ak.com/waco/death/tscr/whitehur/fw_21.gif] {{Webarchive|url=https://web.archive.org/web/20160807041039/http://www.web-ak.com/waco/death/tscr/whitehur/fw_21.gif |date=August 7, 2016 }}
</ref>
The [[Ryder]] van used in the bombing had {{convert|295|cuft|m3}} of space, which would hold up to {{convert|2000|lb|kg}} of explosives. However, the van was not filled to capacity. Yousef used four 20 ft (6 m) long [[Fuse (explosives)|fuses]], all covered in surgical tubing. Yasin calculated that the fuse would trigger the bomb in twelve minutes after he had used a cigarette [[lighter]] to light the fuse.

Yousef wanted the smoke to remain in the tower, therefore catching the public eye by smothering people inside, killing them slowly. He anticipated Tower One collapsing onto Tower Two after the blast.

There remains a popular belief that there was [[cyanide]] in the bomb, which is reinforced by Judge Duffy's statement at sentencing, "You had sodium cyanide around, and I'm sure it was in the bomb." However, the bomb's true composition was not able to be ascertained from the crime scene and Robert Blitzer, a senior FBI official who worked on the case, stated that there was "no forensic evidence indicating the presence of sodium cyanide at the bomb site." Furthermore, Yousef is said only to have considered adding cyanide to the bomb, and to have regretted not doing so in [[Peter Lance]]'s book ''[[1000 Years for Revenge]]''.

==Investigation==
Though the cause of the blast was not immediately known, with some suspecting a [[transformer]] explosion, agents and bomb technicians from the ATF, FBI, and the NYPD quickly responded to the scene. The magnitude of the explosion was far beyond that of a transformer explosion and the FBI Laboratory Division technician, David Williams, who took charge of the crime scene, claimed to know prior to scientific testing the nature and size of the bomb which other lab specialists such as Stephen Burmeister and [[Frederic Whitehurst]] contradicted and later challenged with embarrassing consequences for the [[FBI Laboratory]].<ref>Newton, Michael. (2003). The FBI encyclopedia. Jefferson, N.C. : McFarland & Co. p. 376. {{ISBN|9780786417186}}.</ref>
In the days after the bombing, investigators surveyed the damage and looked for clues. About 300 FBI agents were deployed under the codename TRADEBOM.<ref name=poveda1>{{cite book|last=Poveda|first=Tony|title=The FBI: A Comprehensive Reference Guide|publisher=Greenwood|isbn=978-0897749916|page=94|author2=Powers, Richard|author3= Rosenfeld, Susan|author4= Theoharis, Athan G.}}</ref> While combing through the rubble in the underground parking area, a bomb technician located some internal component fragments from the vehicle that delivered the bomb. A [[vehicle identification number]] (VIN), found on a piece from an axle, gave investigators crucial information that led them to a [[Ryder]] truck rental outlet in Jersey City. Investigators determined that the vehicle had been rented by [[Mohammed A. Salameh]], one of Yousef's co-conspirators.<ref>Reeve (1999), pp. 27–32.</ref> Salameh had reported the van stolen, and when he returned on March 4, 1993, to get his deposit back, authorities arrested him.<ref>Reeve (1999), pp. 32–26.</ref>

Salameh's arrest led police to the apartment of [[Abdul Rahman Yasin]] in [[Jersey City, New Jersey]], which Yasin was sharing with his mother, in the same building as Ramzi Yousef's apartment. Yasin was taken to the FBI's Newark field office in [[Newark, New Jersey]], and was then released. The next day, he flew back to [[Ba'athist Iraq|Iraq]], via [[Amman, Jordan]]. Yasin was later indicted for the attack, and in 2001 he was placed on the initial list of the [[FBI Most Wanted Terrorists]], on which he remains today. He disappeared before the U.S. coalition invasion, [[Operation Iraqi Freedom]], in 2003. In March 1994, Salameh, Nidal Ayyad, [[Mahmud Abouhalima]] and [[Ahmad Ajaj]] were each convicted in the World Trade Center bombing. In May 1994, they were sentenced to life imprisonment.<ref>{{Cite web |url=http://edition.cnn.com/2013/11/05/us/1993-world-trade-center-bombing-fast-facts/ |title=Archived copy |access-date=September 17, 2015 |archive-url=https://web.archive.org/web/20150419021013/http://edition.cnn.com/2013/11/05/us/1993-world-trade-center-bombing-fast-facts/ |archive-date=April 19, 2015 |dead-url=no |df=mdy-all }}</ref>

The capture of Salameh and Yasin led authorities to [[Ramzi Yousef]]'s apartment, where they found bomb-making materials and a business card from [[Mohammed Jamal Khalifa]]. Khalifa was arrested on December 14, 1994, and was deported to [[Jordan]] by the INS on May 5, 1995. He was acquitted by a [[Jordan]]ian court and lived as a free man in [[Saudi Arabia]] until he was killed in 2007.<ref>{{cite web|url=https://www.theguardian.com/world/2007/mar/02/alqaida.saudiarabia|title=Gems, al-Qaida and murder. Mystery over killing of Osama Bin Laden's friend|work=The Guardian|accessdate=September 25, 2014|archive-url=https://web.archive.org/web/20170208134614/https://www.theguardian.com/world/2007/mar/02/alqaida.saudiarabia|archive-date=February 8, 2017|dead-url=no|df=mdy-all}}</ref> In 2002, it was made public that Yasin, the only person involved in the bombing who was never convicted by US authorities,<ref name="cbsnews">{{cite web|date = May 31, 2002|url = http://www.cbsnews.com/stories/2002/05/31/60minutes/main510795.shtml|title = 60 Minutes: The Man Who Got Away|publisher = [[60 Minutes]]|accessdate = February 4, 2012|last = 60 Minutes|archive-url = https://web.archive.org/web/20071012121849/http://www.cbsnews.com/stories/2002/05/31/60minutes/main510795.shtml|archive-date = October 12, 2007|dead-url = no|df = mdy-all}}</ref> was being held as a prisoner on the outskirts of Baghdad, Iraq since 1994.<ref name="cbsnews" /> When journalist Leslie Stahl interviewed him there for a segment on ''60 Minutes'' on May 23, 2002 <ref name="cbsnews" /> Yasin appeared in prison pyjamas and handcuffs.<ref name="cbsnews" /> Yasin has not been seen or heard from since the interview. He was not located during the 2003 invasion of Iraq.

None of the U.S. government's indictments against former al-Qaeda leader [[Osama bin Laden]] suggested that he had any connection with this bombing.<ref>{{cite web|url=https://www.fbi.gov/wanted/wanted_terrorists/usama-bin-laden |title=FBI — USAMA BIN LADEN |work=FBI |accessdate=September 25, 2014 |deadurl=yes |archiveurl=https://web.archive.org/web/20160526174037/https://www.fbi.gov/wanted/wanted_terrorists/usama-bin-laden |archivedate=May 26, 2016 |df= }}</ref>

==Aftermath==

===Victims===
[[File:4.28.12Feb93BombingPanelN-73ByLuigiNovi2.jpg|thumb|The names of the six victims, and the mention of an unborn child, of the attack are inscribed in panel N-73 of the North Pool at the [[National September 11 Memorial & Museum|9/11 Memorial]], where the North Tower formerly stood.]]

The bombing claimed the following six victims:
*'''John DiGiovanni''', age 45, a dental products salesperson.
*'''Robert "Bob" Kirkpatrick''', age 61, Senior Structural Maintenance Supervisor.
*'''Stephen Knapp''', age 47, Chief Maintenance Supervisor, Mechanical Section.
*'''Bill Macko''', age 57, General Maintenance Supervisor, Mechanical Section.
*'''Wilfredo Mercado''', age 37, a receiving agent for [[Windows on the World]] restaurant.
*'''Monica Rodriguez Smith''' age 35, a secretary, who was seven months [[pregnant]].

At the time of the bombing, Smith was checking time sheets in her office on the B-2 level, Kirkpatrick, Knapp and Macko were eating lunch together in an employees' break room next to Smith's office, Mercado was checking in deliveries for the restaurant, and DiGiovanni was parking in the underground garage.<ref>{{Cite web |url=https://www.911memorial.org/1993-wtc-bombing-victims |title=Archived copy |access-date=January 30, 2018 |archive-url=https://web.archive.org/web/20180219235002/https://www.911memorial.org/1993-wtc-bombing-victims |archive-date=February 19, 2018 |dead-url=no |df=mdy-all }}</ref>

===Memorial Fountain===
A granite memorial fountain honoring the victims of the bombing was designed by [[Elyn Zimmerman]] and dedicated in 1995 on Austin J. Tobin Plaza, directly above the site of the explosion. It contained the names of the six adults who were killed in the attack as well as an inscription that read:

''"On February 26, 1993, a bomb set by terrorists exploded below this site. This horrible act of violence killed innocent people, injured thousands, and made victims of us all."''<ref>{{cite web|url=http://voicesofseptember11.org/dev/content.php?idtocitems=1,6,1261283132,142752208|title=9/11 Living Memorial - 1993 WTC Bombing - Memorials|publisher=Voices of September 11|accessdate=March 10, 2012|archive-url=https://web.archive.org/web/20120720120842/http://www.voicesofseptember11.org/dev/content.php?idtocitems=1,6,1261283132,142752208|archive-date=July 20, 2012|dead-url=yes|df=mdy-all}}</ref>

The fountain was destroyed with the rest of the World Trade Center during the [[September 11 attacks]]. A recovered fragment from the 1993 bombing memorial with the text "John D", from bombing victim John DiGiovanni, was later incorporated into a temporary memorial designed by Port Authority architect Jacqueline Hanley, and erected on the Liberty Street side of the site following the September 11 attacks. The memorial was visible across a fence barrier but was not open to the public.<ref>{{cite news |url=http://www.downtownexpress.com/de_95/wtcmemorial.html |title=WTC Memorial for '93 victims unveiled |year=2005 |publisher=Downtown Express |accessdate=September 9, 2009 |archive-url=https://web.archive.org/web/20090409120916/http://www.downtownexpress.com/de_95/wtcmemorial.html |archive-date=April 9, 2009 |dead-url=no |df=mdy-all }}</ref>

At the [[National September 11 Memorial & Museum|9/11 Memorial]], which opened on the tenth anniversary of the 2001 attacks, the six adult victims of the 1993 bombing are memorialized at the North Pool, on Panel N-73.<ref>{{cite web|url=http://names.911memorial.org/#lang=en_US&page=person&id=4959|title=North Pool: Panel N-73|publisher=[[National September 11 Memorial & Museum]]|accessdate=December 9, 2011|archive-url=https://web.archive.org/web/20130727095710/http://names.911memorial.org/#lang=en_US&page=person&id=4959|archive-date=July 27, 2013|dead-url=yes|df=mdy-all}}</ref> The recovered fragment of the memorial fountain is on display among other artifacts<ref>{{Cite web |url=https://www.911memorial.org/1993-world-trade-center-bombing-artifacts |title=Archived copy |access-date=January 30, 2018 |archive-url=https://web.archive.org/web/20180131080907/https://www.911memorial.org/1993-world-trade-center-bombing-artifacts |archive-date=January 31, 2018 |dead-url=no |df=mdy-all }}</ref> related to the bombing inside the museum's historical exhibition.

Stephen Knapp's name is on the [[Postcards (memorial)|Postcards]] memorial in Staten Island, as the sole victim from the borough involved in the bombing.

===FBI involvement===
{{Expand section|date= May 2013}}
In the course of the trial it was revealed that the FBI had an [[informant]], a former [[Egypt]]ian army officer named [[Emad Salem]]. Salem claims to have informed the FBI of the plot to build a bomb that would eventually be used in the World Trade Center towers as early as February 6, 1992. Salem's role as informant allowed the FBI to quickly pinpoint the conspirators out of hundreds of possible suspects.
The transcripts do not make clear the extent to which Federal Authorities knew that there was a plan to bomb the World Trade Center, merely that a bombing of some sort was being discussed.

Salem claimed that the FBI's plan was for Salem to supply the conspirators with a harmless powder instead of actual explosive to build their bomb, but that the FBI chose to use him for other purposes instead. He secretly recorded hundreds of hours of telephone conversations with his FBI handlers.<ref>{{cite news |author=Blumenthal, Ralph |url=https://query.nytimes.com/gst/fullpage.html?res=9F0CE6DF1138F93BA15753C1A965958260 |title=Tapes Depict Proposal to Thwart Bomb Used in Trade Center Blast |work=The New York Times |date=October 28, 1993 |page=Section A, Page 1, Column 4 |accessdate=October 26, 2008 |archive-url=https://web.archive.org/web/20090201195746/http://query.nytimes.com/gst/fullpage.html?res=9F0CE6DF1138F93BA15753C1A965958260 |archive-date=February 1, 2009 |dead-url=no |df=mdy-all }}</ref>

===U.S. Diplomatic Security Service involvement===
[[File:1993 Bombing Aftermath in WTC by DSS Agent.png|thumb|Aftermath of the bombing, photographed by DSS agents]]
Although the FBI received the credit, [[Diplomatic Security Service]] (DSS) special agents actually found and arrested Ramzi Ahmed Yousef, the architect of the 1993 World Trade Center bombing. Special Agents Bill Miller and Jeff Riner were given a tip by an associate of Ramzi Yousef about his location. In coordination with the Pakistani [[Inter-Services Intelligence]] (ISI), DSS arrested Yousef.<ref name="katz">Katz, Samuel M. "Relentless Pursuit: The DSS and the manhunt for the al-Qaeda terrorists", 2002.</ref> After his arrest, [[Ramzi Yousef]] is alleged to have said to investigators "this is only the beginning."

===Allegations of Iraqi involvement===
In October 2001 in a [[PBS]] interview, former CIA Director [[R. James Woolsey, Jr.|James Woolsey]] claimed that Ramzi Yousef worked for Iraqi intelligence.<ref>{{cite web |url=https://www.pbs.org/wgbh/pages/frontline/shows/gunning/interviews/woolsey.html |work=Frontline: Gunning for Saddam |publisher=PBS |title=Interviews: R. James Woolsey |accessdate=October 16, 2008| archiveurl= https://web.archive.org/web/20081029173805/http://www.pbs.org/wgbh/pages/frontline/shows/gunning/interviews/woolsey.html| archivedate=October 29, 2008 | deadurl= no}}</ref> He suggested the grand jury investigation turned up evidence pointing to Iraq that the Justice Department "brushed aside." But Neil Herman, who headed the FBI investigation, noted "The one glaring connection that can't be overlooked is Yasin. We pursued that on every level, traced him to a relative and a location, and we made overtures to get him back." However, Herman says that Yasin's presence in Baghdad does not mean Iraq sponsored the attack: "We looked at that rather extensively. There were no ties to the Iraqi government." CNN terrorism reporter Peter L. Bergen writes, "In sum, by the mid-'90s, the Joint Terrorism Task Force in New York, the F.B.I., the U.S. Attorney's office in the Southern District of New York, the C.I.A., the N.S.C., and the State Department had all found no evidence implicating the Iraqi government in the first Trade Center attack."<ref name="armchair">{{cite news|author=Bergen, Peter |url=http://www.washingtonmonthly.com/features/2003/0312.bergen.html |title=Armchair Provocateur |date=December 2003 |work=Washington Monthly |accessdate=October 26, 2008 |archiveurl=https://web.archive.org/web/20081101154056/http://www.washingtonmonthly.com/features/2003/0312.bergen.html |archivedate=November 1, 2008 |deadurl=yes |df= }}</ref>

Claims of direct Iraqi involvement come from Dr. [[Laurie Mylroie]] of the [[American Enterprise Institute]] and former associate professor of the [[U.S. Naval War College]], with the claims rejected by others. CNN reporter [[Peter Bergen]] has called her a "crackpot" who claimed that "Saddam was not only behind the '93 Trade Center attack, but also every anti-American terrorist incident of the past decade, from the [[1998 United States embassy bombings|bombings of U.S. embassies in Kenya and Tanzania]] to the [[Oklahoma City bombing|leveling of the federal building in Oklahoma City bombing]] to September 11 itself."<ref name="armchair"/> Daniel Benjamin, a senior fellow at the [[Center for Strategic and International Studies]], writes: "The most knowledgeable analysts and investigators at the CIA and at the FBI believe that their work conclusively disproves Mylroie's claims."<ref>{{cite book |author=Benjamin, Daniel and Steven Simon |title=The Next Attack |publisher=Times Books |year=2005 |pages=145 |isbn=0-8050-7941-6}}</ref> Dr. Robert Leiken of the Nixon Center comments on the lack of evidence in her work: "Laurie has discovered Saddam's hand in every major attack on US interests since the Persian Gulf War, including U.S. embassies in Kenya and Tanzania and even the federal building in Oklahoma City. These allegations have all been definitively refuted by the FBI, the National Transportation Safety Board (NTSB) and other investigatory bodies...."<ref>{{cite news |url=http://www.frontpagemag.com/articles/readarticle.asp?ID=16986&p=1 |title=Symposium: The Saddam-Osama Connection: Part II |last=Glazov |first=Jamie |date=February 11, 2005 |work=[[FrontPage Magazine]] |access-date=October 26, 2008 |archive-url=https://archive.is/20120730144043/http://archive.frontpagemag.com/readArticle.aspx?ARTID=9604 |archive-date=30 July 2012}}</ref>

In March 2008, the Pentagon released its study of some 600,000 documents captured in Iraq after the 2003 invasion (see [[Saddam Hussein and Al Qaeda#2008 Pentagon report|2008 Pentagon Report]]). The study "found no 'smoking gun' (i.e., direct connection) between Saddam's Iraq and al Qaeda."<ref>{{cite web |url=https://fas.org/irp/eprint/iraqi/v1.pdf |author=Woods, Kevin M. and James Lacey |title=Saddam and Terrorism: Emerging Insights from Captured Iraqi Documents – Executive Summary; Volume 1 |publisher=Institute for Defense Analysis / Federation of American Scientists |date=November 2007 |pages=16, 18, 51 |format=PDF |accessdate=October 26, 2008| archiveurl= https://web.archive.org/web/20081031075111/https://fas.org/irp/eprint/iraqi/v1.pdf| archivedate=October 31, 2008 | deadurl= no}}</ref> Among the documents released by the Pentagon was a captured audio file of Saddam Hussein speculating that the 1993 attack on the World Trade Center had been carried out by Israel or American intelligence, or perhaps a Saudi or Egyptian faction. Saddam said that he did not trust the bomber Yasin, who was in Iraqi custody, because his testimony was too "organized." The Pentagon study found that Yasin "was a prisoner, and not a guest, in Iraq."<ref>Eli Lake, [http://www.nysun.com/foreign/report-details-saddams-terrorist-ties/72906/ Report Details Saddam's Terrorist Ties] {{Webarchive|url=https://web.archive.org/web/20080618130630/http://www.nysun.com/foreign/report-details-saddams-terrorist-ties/72906/ |date=June 18, 2008 }}, ''New York Sun'', March 14, 2008.</ref> Mylroie denied that this was proof of Saddam's non-involvement, claiming that "one common purpose of such meetings was to develop cover stories for whatever Iraq sought to conceal."<ref>Laurie Mylroie, [http://www.nysun.com/opinion/more-to-uncover-on-saddam/73991/ More To Uncover on Saddam] {{Webarchive|url=https://web.archive.org/web/20081123103650/http://www.nysun.com/opinion/more-to-uncover-on-saddam/73991/ |date=November 23, 2008 }}, ''New York Sun'', April 2, 2008.</ref>

===Improved security===
In the wake of the bombing and the chaotic evacuation which followed, the World Trade Center and many of the firms inside of it revamped emergency procedures, particularly with regard to evacuation of the towers. The New York Port Authority was to govern as the main security for the World Trade buildings. All packages were scanned at various checkpoints then sent up to the proper addressee. These policies played a role in evacuating the building during the [[September 11 attacks]], which destroyed the towers.

Free access to the roofs, which had enabled evacuation by police helicopter in the 1993 bombing, was ended soon after.{{citation needed|date=February 2013}}

==Legal responsibility==
The victims of the 1993 World Trade Center bombings sued the [[Port Authority of New York and New Jersey]] for damages. A decision was handed down in 2005, assigning liability for the bombings to the Port Authority.<ref>{{Cite web |url=https://www.nytimes.com/2005/10/27/nyregion/port-authority-found-negligent-in-1993-bombing.html |title=Archived copy |access-date=August 23, 2017 |archive-url=https://web.archive.org/web/20170701135532/http://www.nytimes.com/2005/10/27/nyregion/port-authority-found-negligent-in-1993-bombing.html |archive-date=July 1, 2017 |dead-url=no |df=mdy-all }}</ref> The decision declared that the agency was 68 percent responsible for the bombing, and the terrorists bore only 32 percent of the responsibility. In January 2008, the Port Authority asked a five-judge panel of the Appellate Division of the New York State Supreme Court in Manhattan to throw out the decision, describing the jury's verdict as "bizarre".<ref>{{cite news |title=Blame for 1993 Attack at Center Is Still at Issue |url=https://www.nytimes.com/2008/01/10/nyregion/10wtc.html |work=The New York Times |date=January 14, 2008 |accessdate=October 26, 2008 | first=Anemona | last=Hartocollis| archiveurl= https://web.archive.org/web/20081209231438/http://www.nytimes.com/2008/01/10/nyregion/10wtc.html?fta=y| archivedate=December 9, 2008 | deadurl= no}}</ref> On April 29, 2008, a New York State Appeals Court unanimously upheld the jury's verdict. Under New York law once a defendant is more than 50 percent at fault, he/she/it can be held fully financially liable.<ref>{{cite news |url=https://www.nytimes.com/2008/04/30/nyregion/30bombing.html?ref=nyregion" |title=Port Authority Liable in 1993 Trade Center Attack |author=Hartocollis, Anemona |date=April 30, 2008 |work=The New York Times |accessdate=October 26, 2008 |archive-url=https://web.archive.org/web/20190118181037/https://www.nytimes.com/2008/04/30/nyregion/30bombing.html?ref=nyregion%22 |archive-date=January 18, 2019 |dead-url=no |df=mdy-all }}</ref>
On September 22, 2011, the New York Court of Appeals, in a four to three ruling, excluded the Port Authority from claims of negligence related to the 1993 bombing.<ref>{{cite web|url=http://www.cnn.com/2013/11/05/us/1993-world-trade-center-bombing-fast-facts/|title=1993 World Trade Center Bombing Fast Facts|date=November 5, 2013|publisher=CNN|accessdate=September 25, 2014|archive-url=https://web.archive.org/web/20140927105747/http://www.cnn.com/2013/11/05/us/1993-world-trade-center-bombing-fast-facts|archive-date=September 27, 2014|dead-url=no|df=mdy-all}}</ref>

It has been argued that the problem with the apportionment of responsibility in the case is not the jury's verdict, but rather New York's tort-reform-produced state apportionment law. Traditionally, courts do not compare intentional and negligent fault. The Restatement Third of Torts: Apportionment of Liability recommends a rule to prevent juries from having to make comparisons like the terrorist-Port Authority comparison in this case. However, if a jurisdiction does compare these intentional and negligent torts, courts' second-best position is to do what the NYS Appeals Court did—to uphold all jury apportionments, even those that assign greater, or perhaps far greater, responsibility to negligent than intentional parties.<ref>Ellen M. Bublick, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1128414 Upside Down? Terrorists, Proprietors and Responsibility for Criminal Harm in the Post-9/11 Tort-Reform World] {{Webarchive|url=https://web.archive.org/web/20090410012753/http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1128414 |date=April 10, 2009 }}.</ref>

==See also==
{{Portal bar|New York City|Terrorism|1990s}}
*[[September 11 attacks]]
*[[1993 shootings at CIA Headquarters]]
*[[Oklahoma City bombing]]
*[[Bojinka plot]]
*[[Richard A. Clarke]]

==References==
'''Notes'''
{{Reflist|30em}}

'''Sources'''
{{refbegin}}
* {{cite book |author=Lance, Peter |title=1000 Years for Revenge | year=2003 |publisher=HarperCollins}}
* {{cite book |author=Reeve, Simon |title=The New Jackals: Ramzi Yousef, Osama bin Laden and the Future of Terrorism |publisher=Northeastern University Press |year=1999}}
* {{cite book |author=Wright, Lawrence |title=The Looming Tower: Al Qaeda and the Road to 9/11 |year=2006 |publisher= [[Alfred A. Knopf]] |isbn=0-375-41486-X}}
* {{cite book | last1=Dwyer | first1=Jim | authorlink1=Jim Dwyer (journalist) | last2=Kocieniewski | first2=David | last3=Murphy | first3=Deidre | last4=Tyre | first4=Peg | title=Two Seconds Under the World: terror comes to America—the conspiracy behind the World Trade Center bombing | date=1994 | publisher=[[Crown Publishing Group|Crown Publishers]] | place=New York | isbn=0-517-59767-5 | oclc=30623523}}
{{refend}}

==External links==
{{Commons category|1993 World Trade Center bombing}}
*[https://web.archive.org/web/20050907175540/http://www.rewardsforjustice.net/english/acts_of_terror/index.cfm?page=wt Rewards for Justice World Trade Center Bombing page]
*[http://news.bbc.co.uk/onthisday/hi/dates/stories/february/26/newsid_2516000/2516469.stm "1993: World Trade Center bomb terrorises New York"], ''BBC: On This Day''
*[https://www.fbi.gov/page2/feb08/tradebom_022608.html FBI – 1993 World Trade Center Bombing – Press Room] [[FBI]] February 26, 2008
*[http://toksook.deviantart.com/gallery/ Images from the 1993 World Trade Center Bombing]
*[http://www.sorabji.com/a/airchecks/wtc_1993.01.ram WCBS 880 radio aircheck, February 26, 1993] at sorabji.com
*[http://www.FDNewYork.com/wtc.asp A fire dispatcher's perspective] from FDNewYork.com
{{World Trade Center}}

{{Use mdy dates|date=June 2017}}

{{DEFAULTSORT:World Trade Center Bombing, 1993}}

[[Category:Attacks in the United States in 1993]]
[[Category:Explosions in 1993]]
[[Category:1993 murders in the United States]]
[[Category:1993 in New York City]]
[[Category:Presidency of Bill Clinton]]
[[Category:History of the United States (1991–present)]]
[[Category:Murder in New York (state)]]
[[Category:Terrorist incidents in New York City]]
[[Category:World Trade Center|Bombing]]
[[Category:Explosions in the United States]]
[[Category:Car and truck bombings in the United States]]
[[Category:Mass murder in 1993]]
[[Category:Terrorist incidents in the United States in 1993]]
[[Category:February 1993 events]]
[[Category:Attacks on office buildings]]
[[Category:Attacks on buildings and structures in the United States]]

Hamming(7,4)

2019-05-05T13:37:15Z

Crasshopper: /* Goal */

{{infobox code
| name = Hamming(7,4)-Code
| image = [[File:Hamming(7,4).svg|160px]]
| image_caption =
| namesake = [[Richard W. Hamming]]
| type = [[Linear block code]]
| block_length = 7
| message_length = 4
| rate = 4/7 ~ 0.571
| distance = 3
| alphabet_size = 2
| notation = [7,4,3]2-code
| decoding =
| properties = [[perfect code]]
}}
[[Image:Hamming(7,4).svg|thumb|300px|Graphical depiction of the 4 data bits ''d''1 to ''d''4 and 3 parity bits ''p''1 to ''p''3 and which parity bits apply to which data bits]]
In [[coding theory]], '''Hamming(7,4)''' is a [[linear code|linear error-correcting code]] that encodes four [[bit]]s of data into seven bits by adding three [[parity bit]]s. It is a member of a larger family of [[Hamming code]]s, but the term ''Hamming code'' often refers to this specific code that [[Richard W. Hamming]] introduced in 1950. At the time, Hamming worked at [[Bell Telephone Laboratories]] and was frustrated with the error-prone [[punched card]] reader, which is why he started working on error-correcting codes.<ref>{{cite web | url = http://biobio.loc.edu/chu/web/Courses/Cosi460/hamming_codes.htm | title = History of Hamming Codes | accessdate = 2008-04-03}}</ref>

The Hamming code adds three additional check bits to every four data bits of the message. Hamming's (7,4) [[algorithm]] can correct any single-bit error, or detect all single-bit and two-bit errors. In other words, the minimal [[Hamming distance]] between any two correct codewords is 3, and received words can be correctly decoded if they are at a distance of at most one from the codeword that was transmitted by the sender. This means that for transmission medium situations where [[error burst|burst errors]] do not occur, Hamming's (7,4) code is effective (as the medium would have to be extremely noisy for two out of seven bits to be flipped).

In [[quantum information]], the Hamming (7,4) is used as the base for the [[Steane code]], a type of [[CSS code]] used for [[quantum error correction]].

== Goal ==
The goal of the Hamming codes is to create a set of [[parity bit]]s that overlap so that a single-bit error in a data bit ''or'' a parity bit can be detected and corrected. While multiple overlaps can be created, the general method is presented in [[Hamming code#Hamming codes|Hamming codes]].

:{| class="wikitable"
|-
!Bit # !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7
|-
!Transmitted bit !! <math>p_1</math> !! <math>p_2</math> !! <math>d_1</math> !! <math>p_3</math> !! <math>d_2</math> !! <math>d_3</math> !! <math>d_4</math>
|-
| <math>p_1</math>
| {{yes}}
| {{No}}
| {{Yes}}
| {{No}}
| {{Yes}}
| {{No}}
| {{Yes}}
|-
| <math>p_2</math>
| {{No}}
| {{Yes}}
| {{Yes}}
| {{No}}
| {{No}}
| {{Yes}}
| {{Yes}}
|-
| <math>p_3</math>
| {{No}}
| {{No}}
| {{No}}
| {{Yes}}
| {{Yes}}
| {{Yes}}
| {{Yes}}
|}

This table describes which parity bits cover which transmitted bits in the encoded word. For example, ''p''2 provides an even parity for bits 2, 3, 6, and 7. It also details which transmitted bit is covered by which parity bit by reading the column. For example, ''d''1 is covered by ''p''1 and ''p''2 but not ''p''3 This table will have a striking resemblance to the parity-check matrix ('''H''') in the next section.

Furthermore, if the parity columns in the above table were removed
:{| class="wikitable"
|-
! !! <math>d_1</math> !! <math>d_2</math> !! <math>d_3</math> !! <math>d_4</math>
|-
| <math>p_1</math>
| {{Yes}}
| {{Yes}}
| {{No}}
| {{Yes}}
|-
| <math>p_2</math>
| {{Yes}}
| {{No}}
| {{Yes}}
| {{Yes}}
|-
| <math>p_3</math>
| {{No}}
| {{Yes}}
| {{Yes}}
| {{Yes}}
|}
then resemblance to rows 1, 2, and 4 of the code generator matrix ('''G''') below will also be evident.

So, by picking the parity bit coverage correctly, all errors with a Hamming distance of 1 can be detected and corrected, which is the point of using a Hamming code.

== Hamming matrices ==
Hamming codes can be computed in [[linear algebra]] terms through [[matrix (mathematics)|matrices]] because Hamming codes are [[linear code]]s. For the purposes of Hamming codes, two '''Hamming matrices''' can be defined: the '''code [[generator matrix]]''' '''G''' and the '''[[parity-check matrix]]''' '''H''':

:<math>\mathbf{G} := \begin{pmatrix}
1 & 1 & 0 & 1 \\
1 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}, \qquad \mathbf{H} := \begin{pmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & 1 \\
\end{pmatrix}.</math>

[[Image:Hamming(7,4) as bits.svg|thumb|300px|Bit position of the data and parity bits]]
As mentioned above, rows 1, 2, and 4 of '''G''' should look familiar as they map the data bits to their parity bits:
* ''p''1 covers ''d''1, ''d''2, ''d''4
* ''p''2 covers ''d''1, ''d''3, ''d''4
* ''p''3 covers ''d''2, ''d''3, ''d''4
The remaining rows (3, 5, 6, 7) map the data to their position in encoded form and there is only 1 in that row so it is an identical copy. In fact, these four rows are [[linearly independent]] and form the [[identity matrix]] (by design, not coincidence).

Also as mentioned above, the three rows of '''H''' should be familiar. These rows are used to compute the '''syndrome vector''' at the receiving end and if the syndrome vector is the [[null vector (vector space)|null vector]] (all zeros) then the received word is error-free; if non-zero then the value indicates which bit has been flipped.

The four data bits — assembled as a vector '''p''' — is pre-multiplied by '''G''' (i.e., '''Gp''') and taken [[Modulo operation|modulo]] 2 to yield the encoded value that is transmitted. The original 4 data bits are converted to seven bits (hence the name "Hamming(7,4)") with three parity bits added to ensure even parity using the above data bit coverages. The first table above shows the mapping between each data and parity bit into its final bit position (1 through 7) but this can also be presented in a [[Venn diagram]]. The first diagram in this article shows three circles (one for each parity bit) and encloses data bits that each parity bit covers. The second diagram (shown to the right) is identical but, instead, the bit positions are marked.

For the remainder of this section, the following 4 bits (shown as a column vector) will be used as a running example:
: <math>\mathbf{p} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \\ d_4 \\ \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}</math>

== Channel coding ==
[[Image:Hamming(7,4) example 1011.svg|thumb|300px|Mapping in the example '''x'''. The parity of the red, green, and blue circles are even.]]

Suppose we want to transmit this data (<code>1011</code>) over a [[signal noise|noisy]] [[channel (communications)|communications channel]]. Specifically, a [[binary symmetric channel]] meaning that error corruption does not favor either zero or one (it is symmetric in causing errors). Furthermore, all source vectors are assumed to be equiprobable. We take the product of '''G''' and '''p''', with entries modulo 2, to determine the transmitted codeword '''x''':

: <math>\mathbf{x} = \mathbf{G} \mathbf{p} =
\begin{pmatrix}
1 & 1 & 0 & 1 \\
1 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix} =
\begin{pmatrix} 2 \\ 3 \\ 1 \\ 2 \\ 0 \\ 1 \\ 1 \end{pmatrix} =
\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} </math>

This means that <code>0110011</code> would be transmitted instead of transmitting <code>1011</code>.

Programmers concerned about multiplication should observe that each row of the result is the least significant bit of the [[Hamming weight|Population Count]] of set bits resulting from the row and column being [[Bitwise AND]]ed together rather than multiplied.

In the adjacent diagram, the seven bits of the encoded word are inserted into their respective locations; from inspection it is clear that the parity of the red, green, and blue circles are even:
* red circle has two 1's
* green circle has two 1's
* blue circle has four 1's

What will be shown shortly is that if, during transmission, a bit is flipped then the parity of two or all three circles will be incorrect and the errored bit can be determined (even if one of the parity bits) by knowing that the parity of all three of these circles should be even.

== Parity check ==
If no error occurs during transmission, then the received codeword '''r''' is identical to the transmitted codeword '''x''':

:<math>\mathbf{r} = \mathbf{x}</math>

The receiver multiplies '''H''' and '''r''' to obtain the '''syndrome''' vector '''z''', which indicates whether an error has occurred, and if so, for which codeword bit. Performing this multiplication (again, entries modulo 2):

: <math>\mathbf{z} = \mathbf{H}\mathbf{r} =
\begin{pmatrix}
1 & 0 & 1 & 0 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & 1 \\
\end{pmatrix}
\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} =
\begin{pmatrix} 2 \\ 4 \\ 2 \end{pmatrix} =
\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} </math>

Since the syndrome '''z''' is the [[null vector (vector space)|null vector]], the receiver can conclude that no error has occurred. This conclusion is based on the observation that when the data vector is multiplied by '''G''', a change of basis occurs into a vector subspace that is the [[kernel (matrix)|kernel]] of '''H'''. As long as nothing happens during transmission, '''r''' will remain in the kernel of '''H''' and the multiplication will yield the null vector.

== Error correction ==
Otherwise, suppose a ''single'' bit error has occurred. Mathematically, we can write

:<math>\mathbf{r} = \mathbf{x} +\mathbf{e}_i</math>

modulo 2, where '''e'''''i'' is the <math>i_{th}</math> [[unit vector]], that is, a zero vector with a 1 in the <math>i^{th}</math>, counting from 1.

: <math>\mathbf{e}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}</math>

Thus the above expression signifies a single bit error in the <math>i^{th}</math> place.

Now, if we multiply this vector by '''H''':

:<math>\mathbf{Hr} = \mathbf{H} \left( \mathbf{x}+\mathbf{e}_i \right) = \mathbf{Hx} + \mathbf{He}_i</math>

Since '''x''' is the transmitted data, it is without error, and as a result, the product of '''H''' and '''x''' is zero. Thus

: <math> \mathbf{Hx} + \mathbf{He}_i = \mathbf{0} + \mathbf{He}_i = \mathbf{He}_i</math>

Now, the product of '''H''' with the <math>i^{th}</math> standard basis vector picks out that column of '''H''', we know the error occurs in the place where this column of '''H''' occurs.

For example, suppose we have introduced a bit error on bit #5

: <math>\mathbf{r} = \mathbf{x}+\mathbf{e}_5 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 1 \\ 1 \\ 1 \end{pmatrix}</math>

[[Image:Hamming(7,4) example 1011 bit 5 error.svg|thumb|300px|A bit error on bit 5 causes bad parity in the red and green circles]]
The diagram to the right shows the bit error (shown in blue text) and the bad parity created (shown in red text) in the red and green circles. The bit error can be detected by computing the parity of the red, green, and blue circles. If a bad parity is detected then the data bit that overlaps ''only'' the bad parity circles is the bit with the error. In the above example, the red and green circles have bad parity so the bit corresponding to the intersection of red and green but not blue indicates the errored bit.

Now,

: <math>\mathbf{z} = \mathbf{Hr} = \begin{pmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ \end{pmatrix}
\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \\ 3 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} </math>

which corresponds to the fifth column of '''H'''. Furthermore, the general algorithm used (''see [[Hamming code#General algorithm]]'') was intentional in its construction so that the syndrome of 101 corresponds to the binary value of 5, which indicates the fifth bit was corrupted. Thus, an error has been detected in bit 5, and can be corrected (simply flip or negate its value):

:<math> \mathbf{r}_{\text{corrected}} = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ \overline{1} \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} </math>

This corrected received value indeed, now, matches the transmitted value '''x''' from above.

== Decoding ==
Once the received vector has been determined to be error-free or corrected if an error occurred (assuming only zero or one bit errors are possible) then the received data needs to be decoded back into the original four bits.

First, define a matrix '''R''':

:<math>\mathbf{R} = \begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix} </math>

Then the received value, '''pr''', is equal to '''Rr'''. Using the running example from above

:<math>\mathbf{p_r} = \begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \end{pmatrix} </math>

== Multiple bit errors ==
[[Image:Hamming(7,4) example 1011 bits 4 & 5 error.svg|thumb|300px|A bit error on bit 4 & 5 are introduced (shown in blue text) with a bad parity only in the green circle (shown in red text)]]

It is not difficult to show that only single bit errors can be corrected using this scheme. Alternatively, Hamming codes can be used to detect single and double bit errors, by merely noting that the product of '''H''' is nonzero whenever errors have occurred. In the adjacent diagram, bits 4 and 5 were flipped. This yields only one circle (green) with an invalid parity but the errors are not recoverable.

However, the Hamming (7,4) and similar Hamming codes cannot distinguish between single-bit errors and two-bit errors. That is, two-bit errors appear the same as one-bit errors. If error correction is performed on a two-bit error the result will be incorrect.

Similarly, Hamming codes cannot detect or recover from an arbitrary three-bit error; Consider the diagram: if the bit in the green circle (colored red) were 1, the parity checking would return the null vector, indicating that there is no error in the codeword.

{{clear}}

== All codewords ==
Since the source is only 4 bits then there are only 16 possible transmitted words. Included is the eight-bit value if an extra parity bit is used (''see [[Hamming code#.5B7.2C4.5D Hamming code with an additional parity bit|Hamming(7,4) code with an additional parity bit]]''). (The data bits are shown in blue; the parity bits are shown in red; and the extra parity bit shown in green.)

{| class="wikitable"
|-
!rowspan="2"| Data <math>({\color{blue}d_1}, {\color{blue}d_2}, {\color{blue}d_3}, {\color{blue}d_4})</math>
!colspan="2"| Hamming(7,4)
!colspan="2"| Hamming(7,4) with extra parity bit (Hamming(8,4))
|-
! Transmitted <math>({\color{red}p_1}, {\color{red}p_2}, {\color{blue}d_1}, {\color{red}p_3}, {\color{blue}d_2}, {\color{blue}d_3}, {\color{blue}d_4})</math>
! Diagram
! Transmitted <math>({\color{red}p_1}, {\color{red}p_2}, {\color{blue}d_1}, {\color{red}p_3}, {\color{blue}d_2}, {\color{blue}d_3}, {\color{blue}d_4}, {\color{green}p_4})</math>
! Diagram
|-
| 0000
| 0000000
| [[Image:Hamming(7,4) example 0000.svg|150px|Hamming code for 0000 becomes 0000000]]
| 00000000
| [[Image:Hamming(7,4) example 0000 with extra parity.svg|150px|Hamming code for 0000 becomes 0000000 with extra parity bit 0]]
|-
| 1000
| 1110000
| [[Image:Hamming(7,4) example 1000.svg|150px|Hamming code for 1000 becomes 1000011]]
| 11100001
| [[Image:Hamming(7,4) example 1000 with extra parity.svg|150px|Hamming code for 1000 becomes 1000011 with extra parity bit 1]]
|-
| 0100
| 1001100
| [[Image:Hamming(7,4) example 0100.svg|150px|Hamming code for 0100 becomes 0100101]]
| 10011001
| [[Image:Hamming(7,4) example 0100 with extra parity.svg|150px|Hamming code for 0100 becomes 0100101 with extra parity bit 1]]
|-
| 1100
| 0111100
| [[Image:Hamming(7,4) example 1100.svg|150px|Hamming code for 1100 becomes 1100110]]
| 01111000
| [[Image:Hamming(7,4) example 1100 with extra parity.svg|150px|Hamming code for 1100 becomes 1100110 with extra parity bit 0]]
|-
| 0010
| 0101010
| [[Image:Hamming(7,4) example 0010.svg|150px|Hamming code for 0010 becomes 0010110]]
| 01010101
| [[Image:Hamming(7,4) example 0010 with extra parity.svg|150px|Hamming code for 0010 becomes 0010110 with extra parity bit 1]]
|-
| 1010
| 1011010
| [[Image:Hamming(7,4) example 1010.svg|150px|Hamming code for 1010 becomes 1010101]]
| 10110100
| [[Image:Hamming(7,4) example 1010 with extra parity.svg|150px|Hamming code for 1010 becomes 1010101 with extra parity bit 0]]
|-
| 0110
| 1100110
| [[Image:Hamming(7,4) example 0110.svg|150px|Hamming code for 0110 becomes 0110011]]
| 11001100
| [[Image:Hamming(7,4) example 0110 with extra parity.svg|150px|Hamming code for 0110 becomes 0110011 with extra parity bit 0]]
|-
| 1110
| 0010110
| [[Image:Hamming(7,4) example 1110.svg|150px|Hamming code for 1110 becomes 1110000]]
| 00101101
| [[Image:Hamming(7,4) example 1110 with extra parity.svg|150px|Hamming code for 1110 becomes 1110000 with extra parity bit 1]]
|-
| 0001
| 1101001
| [[Image:Hamming(7,4) example 0001.svg|150px|Hamming code for 0001 becomes 0001111]]
| 11010010
| [[Image:Hamming(7,4) example 0001 with extra parity.svg|150px|Hamming code for 0001 becomes 0001111 with extra parity bit 0]]
|-
| 1001
| 0011001
| [[Image:Hamming(7,4) example 1001.svg|150px|Hamming code for 1001 becomes 1001100]]
| 00110011
| [[Image:Hamming(7,4) example 1001 with extra parity.svg|150px|Hamming code for 1001 becomes 1001100 with extra parity bit 1]]
|-
| 0101
| 0100101
| [[Image:Hamming(7,4) example 0101.svg|150px|Hamming code for 0101 becomes 0101010]]
| 01001011
| [[Image:Hamming(7,4) example 0101 with extra parity.svg|150px|Hamming code for 0101 becomes 0101010 with extra parity bit 1]]
|-
| 1101
| 1010101
| [[Image:Hamming(7,4) example 1101.svg|150px|Hamming code for 1101 becomes 1101001]]
| 10101010
| [[Image:Hamming(7,4) example 1101 with extra parity.svg|150px|Hamming code for 1101 becomes 1101001 with extra parity bit 0]]
|-
| 0011
| 1000011
| [[Image:Hamming(7,4) example 0011.svg|150px|Hamming code for 0011 becomes 0011001]]
| 10000111
| [[Image:Hamming(7,4) example 0011 with extra parity.svg|150px|Hamming code for 0011 becomes 0011001 with extra parity bit 1]]
|-
| 1011
| 0110011
| [[Image:Hamming(7,4) example 1011.svg|150px|Hamming code for 1011 becomes 1011010]]
| 01100110
| [[Image:Hamming(7,4) example 1011 with extra parity.svg|150px|Hamming code for 1011 becomes 1011010 with extra parity bit 0]]
|-
| 0111
| 0001111
| [[Image:Hamming(7,4) example 0111.svg|150px|Hamming code for 0111 becomes 0111100]]
| 00011110
| [[Image:Hamming(7,4) example 0111 with extra parity.svg|150px|Hamming code for 0111 becomes 0111100 with extra parity bit 0]]
|-
| 1111
| 1111111
| [[Image:Hamming(7,4) example 1111.svg|150px|Hamming code for 1111 becomes 1111111]]
| 11111111
| [[Image:Hamming(7,4) example 1111 with extra parity.svg|150px|Hamming code for 1111 becomes 1111111 with extra parity bit 1]]
|}

== References ==
{{reflist}}

== External links ==
* [http://acm.timus.ru/problem.aspx?space=1&num=1792 A programming problem about the Hamming Code(7,4)]
* [http://toolmenow.com/34/Hamming(7,4)-Code-Calculator Hamming (7,4) Code Calculator]
* [http://www.toolmenow.com/31/Hamming(7,4)-Code-Checker Hamming (7,4) Code Checker]
[[Category:Coding theory]]
[[Category:Error detection and correction]]
[[Category:Computer arithmetic]]

Sphere packing

2019-05-04T05:11:40Z

Crasshopper:

{{Use dmy dates|date=November 2017}}
[[File:HCP Oranges.jpg|thumb|Sphere packing finds practical application in the stacking of [[orange (fruit)|oranges]].]]
In [[geometry]], a '''sphere packing''' is an arrangement of non-overlapping [[sphere]]s within a containing space. The spheres considered are usually all of identical size, and the space is usually three-[[dimension]]al [[Euclidean space]]. However, sphere [[packing problem]]s can be generalised to consider unequal spheres, to more dimensions, or to [[Non-Euclidean geometry|non-Euclidean]] spaces such as [[hyperbolic space]].

A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The amount of space filled by the spheres is called the density of the arrangement. (For spaces of infinite extent, a more careful definition of density is needed.)

For equal spheres in three dimensions the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%.

== Classification and terminology ==
A '''[[lattice (group)|lattice]]''' arrangement (commonly called a '''regular''' arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only ''n'' vectors to be uniquely defined (in ''n''-[[dimension]]al [[Euclidean space]]). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as '''irregular''') can still be periodic, but also '''[[aperiodic]]''' (properly speaking '''non-periodic''') or '''[[randomness|random]]'''. Lattice arrangements are easier to handle than irregular ones—their high degree of [[symmetry]] makes it easier to classify them and to measure their densities.

== Regular packing ==
[[File:Order and Chaos.tif|thumb|Regular arrangement of equal spheres in a plane changing to an irregular arrangement of unequal spheres (bubbles).]]
[[Image:close packing box.svg|thumb|right|160px| HCP lattice (left) and the FCC lattice (right) are the two most common highest density arrangements. Note that the two groups shown here are not ''[[Crystal structure|unit cells]]'' that are capable of [[Tessellation|tessellating]] in 3D space. These groups do, however, readily illustrate the difference between the two lattices.]]

[[Image:Closepacking.svg|thumb|right|160px|Two ways to stack three planes made of spheres]]

===Dense packing===
{{main|Close-packing of equal spheres}}
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called [[Close-packing of spheres|close-packed]] structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this "everywhere" in a second plane above the first, we create a new compact layer. A third layer can be placed directly above the first one, or the spheres can be offset, vertically above another set of hollows of the first layer. There are thus three types of planes, called A, B and C.

Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or [[face centred cubic]], "FCC")—where the layers are alternated in the ABCABC... sequence. The other is called hexagonal close packing ("HCP")—where the layers are alternated in the ABAB... sequence. But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere is surrounded by 12 other spheres, and the average density is
:<math>\frac{\pi}{3\sqrt{2}} \simeq 0.74048.</math>

[[Carl Friedrich Gauss]] proved in 1831 that these packings have the highest density amongst all possible lattice packings.<ref>{{cite journal|first=C. F.|last=Gauß|authorlink=Carl Friedrich Gauss|title=Besprechung des Buchs von L. A. Seeber: ''Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen'' usw|trans-title=Discussion of L. A. Seeber's book: ''Studies on the characteristics of positive ternary quadratic forms'' etc|journal=Göttingsche Gelehrte Anzeigen|year=1831}}</ref>

In 1611 [[Johannes Kepler]] had conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the [[Kepler conjecture]]. In 1998, [[Thomas Callister Hales]], following the approach suggested by [[László Fejes Tóth]] in 1953, announced a proof of the Kepler conjecture. Hales' proof is a [[proof by exhaustion]] involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014 Hales announced the completion of a formal proof using [[automated proof checking]], removing any doubt.<ref>{{cite web|url=https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion|website=Google Code Archive|title=Long-term storage for Google Code Project Hosting}}</ref>

=== Other common lattice packings ===

Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of <math> \frac{\pi}{6} \approx 0.5236</math>, the hexagonal lattice with a density of <math>\frac{\pi}{3\sqrt{3}}\approx 0.6046</math> and the tetrahedral lattice with a density of <math>\frac{\pi\sqrt{3}}{16}\approx 0.3401</math>, and loosest possible at a density of 0.0555.<ref>{{cite web | title=Wolfram Math World, Sphere packing | url=http://mathworld.wolfram.com/SpherePacking.html}}</ref>

===Jammed packings with a low density===
Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or [[Jamming (physics)|jammed]]. The strictly jammed sphere packing with the lowest density is a diluted ("tunneled") fcc crystal with a density of only 0.49365.<ref>{{Cite journal | last1 = Torquato | first1 = S. | authorlink1 = Salvatore Torquato | last2 = Stillinger | first2 = F. H. | title = Toward the jamming threshold of sphere packings: Tunneled crystals | journal = Journal of Applied Physics | volume = 102 | year = 2007 | pages = 093511 | doi = 10.1063/1.2802184 | arxiv = 0707.4263 | bibcode = 2007JAP...102i3511T }}
</ref>

== Irregular packing ==
{{main|Random close pack}}
If we attempt to build a densely packed collection of spheres, we will be tempted to always place the next sphere in a hollow between three packed spheres. If five spheres are assembled in this way, they will be consistent with one of the regularly packed arrangements described above. However, the sixth sphere placed in this way will render the structure inconsistent with any regular arrangement. This results in the possibility of a ''random close packing'' of spheres which is stable against compression.<ref>{{cite journal|title=Random thoughts|first=Paul|last=Chaikin|date=June 2007|journal=Physics Today|page=8|issn=0031-9228|publisher=American Institute of Physics|volume=60|issue=6|doi=10.1063/1.2754580|bibcode = 2007PhT....60f...8C }}</ref>

When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%. Recent research predicts analytically that it cannot exceed a density limit of 63.4%<ref name="nature">{{cite journal |last1=Song |first1=C. |last2=Wang |first2=P. |last3=Makse |first3=H. A. |date=29 May 2008 |title=A phase diagram for jammed matter |journal=[[Nature (journal)|Nature]] |volume=453 |pages=629–632 |doi=10.1038/nature06981 |url=http://www.nature.com/nature/journal/v453/n7195/full/nature06981.html |pmid=18509438 |issue=7195 |bibcode = 2008Natur.453..629S |arxiv = 0808.2196 }}</ref> This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres (that is, line segments or circles) will yield a regular packing.

==Hypersphere packing==
The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is [[circle packing|packing circles]] on a plane. In one dimension it is packing line segments into a linear universe.<ref>{{cite journal | last1 = Griffith | first1 = J.S. | year = 1962 | title = Packing of equal 0-spheres | url = | journal = Nature | volume = 196 | issue = | pages = 764–765 | doi = 10.1038/196764a0 | bibcode = 1962Natur.196..764G }}</ref>

In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions.<ref>{{MathWorld |title=Hypersphere Packing|urlname=HyperspherePacking}}</ref> Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing.<ref>{{cite journal | last=Sloane |first=N. J. A. | title=The Sphere-Packing Problem | year=1998 | pages=387–396 | journal=Documenta Mathematica|volume=3 | arxiv=math/0207256|bibcode = 2002math......7256S }}</ref>

In 2016, [[Maryna Viazovska]] announced a proof that the [[E8 lattice|E8 lattice]] provides the optimal packing (regardless of regularity) in eight-dimensional space,<ref>{{Cite journal|last=Viazovska|first=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 8|url=http://annals.math.princeton.edu/2017/185-3/p07|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=991–1015|arxiv=1603.04246|doi=10.4007/annals.2017.185.3.7|issn=0003-486X}}</ref> and soon afterwards she and a group of collaborators announced a similar proof that the [[Leech lattice]] is optimal in 24 dimensions.<ref>{{Cite journal|last=Cohn|first=Henry|last2=Kumar|first2=Abhinav|last3=Miller|first3=Stephen|last4=Radchenko|first4=Danylo|last5=Viazovska|first5=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 24|url=http://annals.math.princeton.edu/2017/185-3/p08|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=1017–1033|arxiv=1603.06518|doi=10.4007/annals.2017.185.3.8|issn=0003-486X}}</ref> This result built on and improved previous methods which showed that these two lattices are very close to optimal.<ref>{{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=Optimality and uniqueness of the Leech lattice among lattices | doi=10.4007/annals.2009.170.1003 | mr=2600869 | zbl=1213.11144 | year=2009 | journal=Annals of Mathematics | issn=1939-8980 | volume=170 | issue=3 | pages=1003–1050 | arxiv=math.MG/0403263 }} {{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=The densest lattice in twenty-four dimensions | doi=10.1090/S1079-6762-04-00130-1 | mr=2075897 | year=2004 | journal=Electronic Research Announcements of the American Mathematical Society | issn=1079-6762 | volume=10 | issue=07 | pages=58–67 |arxiv=math.MG/0408174 }}</ref>
The new proofs involve using the [[Laplace transform]] of a carefully chosen [[modular function]] to construct a [[Rotational symmetry|radially symmetric]] function {{mvar|f}} such that {{mvar|f}} and its [[Fourier transform]] {{mvar|f̂}} both equal one at the [[origin (mathematics)|origin]], and both vanish at all other points of the optimal lattice, with {{mvar|f}} negative outside the central sphere of the packing and {{mvar|f̂}} positive. Then, the [[Poisson summation formula]] for {{mvar|f}} is used to compare the density of the optimal lattice with that of any other packing.<ref>{{citation|url=https://www.youtube.com/watch?v=8qlZjarkS_g|first=Stephen D.|last=Miller|date=4 April 2016|title=The solution to the sphere packing problem in 24 dimensions via modular forms|publisher=[[Institute for Advanced Study]]}}. Video of an hour-long talk by one of Viazovska's co-authors explaining the new proofs.</ref> Before the proof had been [[Scholarly peer review|formally refereed]] and published, mathematician [[Peter Sarnak]] called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct."<ref>{{citation|last1=Klarreich|first1=Erica|authorlink1=Erica Klarreich|title=Sphere Packing Solved in Higher Dimensions|url=https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions|magazine=Quanta Magazine|date=30 March 2016}}</ref>

Another line of research in high dimensions is trying to find [[asymptotic]] bounds for the density of the densest packings. Currently the best known result is that there exists a lattice in dimension {{mvar|n}} with density bigger or equal to {{math|2−''n''''cn''}} for some number {{mvar|c}}.<ref>{{cite journal|last=Rogers|first=C. A.|title=Existence Theorems in the Geometry of Numbers|journal=Annals of Mathematics |series=Second Series|volume=48|issue=4|year=1947|pages=994–1002|jstor=1969390|doi=10.2307/1969390}}</ref>

==Unequal sphere packing==
[[File:Binary sphere packing LS3.png|thumb|A dense packing of spheres with a radius ratio of 0.64799 and a density of 0.74786<ref name="doi10.1021/jp206115p">{{Cite journal | last1 = O'Toole | first1 = P. I. | last2 = Hudson | first2 = T. S. | doi = 10.1021/jp206115p | title = New High-Density Packings of Similarly Sized Binary Spheres | journal = The Journal of Physical Chemistry C | volume = 115 | issue = 39 | pages = 19037 | year = 2011 | pmid = | pmc = }}</ref>]]
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or [[interstitial compound|interstitial]] packing. When many sizes of spheres (or a [[particle size distribution|distribution]]) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are available.

When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a close-packed arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends sensitively on the radius ratio, but in the limit of extreme size ratios, the smaller spheres can fill the gaps with the same density as the larger spheres filled space.<ref>{{Cite journal | last1 = Hudson | first1 = D. R. | title = Density and Packing in an Aggregate of Mixed Spheres | doi = 10.1063/1.1698327 | journal = Journal of Applied Physics | volume = 20 | issue = 2 | pages = 154–162| year = 1949 | pmid = | pmc = |bibcode = 1949JAP....20..154H }}</ref> Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere.<ref>{{Cite journal | last1 = Zong | first1 = C. | title = From deep holes to free planes | doi = 10.1090/S0273-0979-02-00950-3 | journal = Bulletin of the American Mathematical Society | volume = 39 | issue = 4 | pages = 533–555 | year = 2002 | pmid = | pmc = }}</ref>

When the smaller sphere has a radius greater than 0.41421 of the radius of the larger sphere, it is no longer possible to fit into even the octahedral holes of the close-packed structure. Thus, beyond this point, either the host structure must expand to accommodate the interstitials (which compromises the overall density), or rearrange into a more complex crystalline compound structure. Structures are known which exceed the close packing density for radius ratios up to 0.659786.<ref name="doi10.1021/jp206115p"/><ref>{{cite journal|first1=G. W.|last1=Marshall|first2=T. S.|last2=Hudson|journal=Contributions to Algebra and Geometry|title=Dense binary sphere packings|volume=51|issue=2|pages=337–344|year=2010|url=http://www.emis.de/journals/BAG/vol.51/no.2/3.html}}</ref>

Upper bounds for the density that can be obtained in such binary packings have also been obtained.<ref>{{cite journal |last1=de Laat |first1=David |last2=de Oliveira Filho |first2=Fernando Mário |last3=Vallentin |first3=Frank |title=Upper bounds for packings of spheres of several radii|arxiv=1206.2608|date=12 June 2012 |doi=10.1017/fms.2014.24 |volume=2 |journal=Forum of Mathematics, Sigma}}</ref>

In many chemical situations such as [[ionic crystal]]s, the [[stoichiometry]] is constrained by the charges of the constituent ions. This additional constraint on the packing, together with the need to minimize the [[Coulomb energy]] of interacting charges leads to a diversity of optimal packing arrangements.

==Hyperbolic space==
Although the concept of circles and spheres can be extended to [[hyperbolic space]], finding the densest packing becomes much more difficult. In a hyperbolic space there is no limit to the number of spheres that can surround another sphere (for example, [[Ford circle]]s can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an [[Infinity|infinite]] number of other circles). The concept of average density also becomes much more difficult to define accurately. The densest packings in any hyperbolic space are almost always irregular.<ref>{{Cite journal | last1 = Bowen | first1 = L. | last2 = Radin | first2 = C. | doi = 10.1007/s00454-002-2791-7 | title = Densest Packing of Equal Spheres in Hyperbolic Space | journal = Discrete and Computational Geometry | volume = 29 | pages = 23–39 | year = 2002 | pmid = | pmc = }}</ref>

Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic ''n''-space where ''n'' ≥ 2.<ref>{{Cite journal | last1 = Böröczky | first1 = K. | title = Packing of spheres in spaces of constant curvature | doi = 10.1007/BF01902361 | journal = Acta Mathematica Academiae Scientiarum Hungaricae | volume = 32 | issue = 3–4 | pages = 243–261 | year = 1978 | pmid = | pmc = }}</ref> In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the [[horosphere]] packing of the [[order-6 tetrahedral honeycomb]] with [[Schläfli symbol]] {3,3,6}.<ref>{{Cite journal | last1 = Böröczky | first1 = K. | last2 = Florian | first2 = A. | doi = 10.1007/BF01897041 | title = Über die dichteste Kugelpackung im hyperbolischen Raum | journal = Acta Mathematica Academiae Scientiarum Hungaricae | volume = 15 | pages = 237 | year = 1964 | pmid = | pmc = }}</ref> In addition to this configuration at least three other [[horosphere]] packings are known to exist in hyperbolic 3-space that realize the density upper bound.<ref>{{Cite journal | last1 = Kozma | first1 = R. T. | last2 = Szirmai | first2 = J. | doi = 10.1007/s00605-012-0393-x | title = Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types | journal = Monatshefte für Mathematik | volume = 168 | pages = 27 | year = 2012 | pmid = | pmc = | arxiv = 1007.0722 }}</ref>

==Touching pairs, triplets, and quadruples==
The [[contact graph]] of an arbitrary finite packing of unit balls is the graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. The cardinality of the edge set of the contact graph gives the number of touching pairs, the number of 3-cycles in the contact graph gives the number of touching triplets, and the number of tetrahedrons in the contact graph gives the number of touching quadruples (in general for a contact graph associated with a sphere packing in ''n'' dimensions that the cardinality of the set of ''n''-simplices in the contact graph gives the number of touching (''n'' + 1)-tuples in the sphere packing). In the case of 3-dimensional Euclidean space, non-trivial upper bounds on the number of touching pairs, triplets, and quadruples<ref>{{cite journal| last1 = Bezdek | first1 = Karoly | last2 = Reid | first2 = Samuel | title=Contact Graphs of Sphere Packings Revisited | year = 2013 |arxiv=1210.5756 |journal=Journal of Geometry |volume = 104 |issue = 1 | pages = 57–83 |doi = 10.1007/s00022-013-0156-4}}</ref> were proved by [[Karoly Bezdek]] and Samuel Reid at the University of Calgary.

==Other spaces==
Sphere packing on the corners of a hypercube (with the spheres defined by [[Hamming distance]]) corresponds to designing [[error-correcting codes]]: if the spheres have radius ''t'', then their centers are codewords of a (2''t'' + 1)-error-correcting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and error-correcting codes. For example, the [[binary Golay code]] is closely related to the 24-dimensional Leech lattice.

For further details on these connections, see the book ''Sphere Packings, Lattices and Groups'' by [[John Horton Conway|Conway]] and [[Neil Sloane|Sloane]].<ref>{{Cite book|url=https://books.google.com/books/about/Sphere_Packings_Lattices_and_Groups.html?id=ITDvBwAAQBAJ&printsec=frontcover&source=kp_read_button&hl=en#v=onepage&q&f=false|title=Sphere Packings, Lattices and Groups|last=Conway|first=John H.|authorlink=John Horton Conway|last2=Sloane|first2=Neil J. A.|authorlink2=Neil Sloane|publisher=Springer Science & Business Media|year=1998|isbn=0-387-98585-9|edition=3rd}}</ref>

==See also==
* [[Close-packing of equal spheres]]
* [[Apollonian sphere packing]]
* [[Hermite constant]]
* [[Kissing number problem]]
* [[Sphere-packing bound]]
* [[Random close pack]]

==References==
{{Reflist|2}}

==Bibliography==
* {{cite book |last1=Aste |first1=T. |last2=Weaire |first2=D. |title=The Pursuit of Perfect Packing |publisher=Institute of Physics Publishing |location=London |year=2000 |isbn=0-7503-0648-3}}
* {{cite book |last1=Conway |first1=J. H. |authorlink=John Horton Conway |last2=Sloane |first2=N. J. H. |authorlink2=Neil Sloane |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}
* {{cite journal | last1 = Sloane | first1 = N. J. A. | authorlink1=Neil Sloane | year = 1984 | title = The Packing of Spheres | url = | journal = Scientific American | volume = 250 | issue = | pages = 116–125 |bibcode = 1984SciAm.250e.116G |doi = 10.1038/scientificamerican0584-116 }}

==External links==
* Dana Mackenzie (May 2002) [http://www.ma.utexas.edu/users/radin/reviews/newscientist2.html "''A fine mess''"] (New Scientist)
:A non-technical overview of packing in hyperbolic space.
* {{MathWorld|urlname=CirclePacking |title=Circle Packing }}
* [http://www.3doro.de/e-kp.htm "Kugelpackungen (Sphere Packing)"] (T. E. Dorozinski)
*[http://alecjacobson.com/graphics/hw10b/ "3D Sphere Packing Applet"] Sphere Packing java applet
*[http://www.randomwalk.de/sphere/insphr/spheresinsphr.html "Densest Packing of spheres into a sphere"] java applet
*[http://codes.se/packings/ "Database of sphere packings"] (Erik Agrell)

{{Packing problem}}

[[Category:Discrete geometry]]
[[Category:Crystallography]]
[[Category:Packing problems]]
[[Category:Spheres]]

Sphere packing

2019-05-04T05:09:59Z

Crasshopper:

{{Use dmy dates|date=November 2017}}
[[File:HCP Oranges.jpg|thumb|Sphere packing finds practical application in the stacking of [[orange (fruit)|oranges]].]]
In [[geometry]], a '''sphere packing''' is an arrangement of non-overlapping [[sphere]]s within a containing space. The spheres considered are usually all of identical size, and the space is usually three-[[dimension]]al [[Euclidean space]]. However, sphere [[packing problem]]s can be generalised to consider unequal spheres, ''n''-dimensional Euclidean space (where the problem becomes [[circle packing]] in two dimensions, or [[hypersphere]] packing in higher dimensions) or to [[Non-Euclidean geometry|non-Euclidean]] spaces such as [[hyperbolic space]].

A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The amount of space filled by the spheres is called the density of the arrangement. (For spaces of infinite extent, a more careful definition of density is needed.)

For equal spheres in three dimensions the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 64%.

== Classification and terminology ==
A '''[[lattice (group)|lattice]]''' arrangement (commonly called a '''regular''' arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only ''n'' vectors to be uniquely defined (in ''n''-[[dimension]]al [[Euclidean space]]). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as '''irregular''') can still be periodic, but also '''[[aperiodic]]''' (properly speaking '''non-periodic''') or '''[[randomness|random]]'''. Lattice arrangements are easier to handle than irregular ones—their high degree of [[symmetry]] makes it easier to classify them and to measure their densities.

== Regular packing ==
[[File:Order and Chaos.tif|thumb|Regular arrangement of equal spheres in a plane changing to an irregular arrangement of unequal spheres (bubbles).]]
[[Image:close packing box.svg|thumb|right|160px| HCP lattice (left) and the FCC lattice (right) are the two most common highest density arrangements. Note that the two groups shown here are not ''[[Crystal structure|unit cells]]'' that are capable of [[Tessellation|tessellating]] in 3D space. These groups do, however, readily illustrate the difference between the two lattices.]]

[[Image:Closepacking.svg|thumb|right|160px|Two ways to stack three planes made of spheres]]

===Dense packing===
{{main|Close-packing of equal spheres}}
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called [[Close-packing of spheres|close-packed]] structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this "everywhere" in a second plane above the first, we create a new compact layer. A third layer can be placed directly above the first one, or the spheres can be offset, vertically above another set of hollows of the first layer. There are thus three types of planes, called A, B and C.

Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or [[face centred cubic]], "FCC")—where the layers are alternated in the ABCABC... sequence. The other is called hexagonal close packing ("HCP")—where the layers are alternated in the ABAB... sequence. But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere is surrounded by 12 other spheres, and the average density is
:<math>\frac{\pi}{3\sqrt{2}} \simeq 0.74048.</math>

[[Carl Friedrich Gauss]] proved in 1831 that these packings have the highest density amongst all possible lattice packings.<ref>{{cite journal|first=C. F.|last=Gauß|authorlink=Carl Friedrich Gauss|title=Besprechung des Buchs von L. A. Seeber: ''Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen'' usw|trans-title=Discussion of L. A. Seeber's book: ''Studies on the characteristics of positive ternary quadratic forms'' etc|journal=Göttingsche Gelehrte Anzeigen|year=1831}}</ref>

In 1611 [[Johannes Kepler]] had conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the [[Kepler conjecture]]. In 1998, [[Thomas Callister Hales]], following the approach suggested by [[László Fejes Tóth]] in 1953, announced a proof of the Kepler conjecture. Hales' proof is a [[proof by exhaustion]] involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014 Hales announced the completion of a formal proof using [[automated proof checking]], removing any doubt.<ref>{{cite web|url=https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion|website=Google Code Archive|title=Long-term storage for Google Code Project Hosting}}</ref>

=== Other common lattice packings ===

Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of <math> \frac{\pi}{6} \approx 0.5236</math>, the hexagonal lattice with a density of <math>\frac{\pi}{3\sqrt{3}}\approx 0.6046</math> and the tetrahedral lattice with a density of <math>\frac{\pi\sqrt{3}}{16}\approx 0.3401</math>, and loosest possible at a density of 0.0555.<ref>{{cite web | title=Wolfram Math World, Sphere packing | url=http://mathworld.wolfram.com/SpherePacking.html}}</ref>

===Jammed packings with a low density===
Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or [[Jamming (physics)|jammed]]. The strictly jammed sphere packing with the lowest density is a diluted ("tunneled") fcc crystal with a density of only 0.49365.<ref>{{Cite journal | last1 = Torquato | first1 = S. | authorlink1 = Salvatore Torquato | last2 = Stillinger | first2 = F. H. | title = Toward the jamming threshold of sphere packings: Tunneled crystals | journal = Journal of Applied Physics | volume = 102 | year = 2007 | pages = 093511 | doi = 10.1063/1.2802184 | arxiv = 0707.4263 | bibcode = 2007JAP...102i3511T }}
</ref>

== Irregular packing ==
{{main|Random close pack}}
If we attempt to build a densely packed collection of spheres, we will be tempted to always place the next sphere in a hollow between three packed spheres. If five spheres are assembled in this way, they will be consistent with one of the regularly packed arrangements described above. However, the sixth sphere placed in this way will render the structure inconsistent with any regular arrangement. This results in the possibility of a ''random close packing'' of spheres which is stable against compression.<ref>{{cite journal|title=Random thoughts|first=Paul|last=Chaikin|date=June 2007|journal=Physics Today|page=8|issn=0031-9228|publisher=American Institute of Physics|volume=60|issue=6|doi=10.1063/1.2754580|bibcode = 2007PhT....60f...8C }}</ref>

When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%. Recent research predicts analytically that it cannot exceed a density limit of 63.4%<ref name="nature">{{cite journal |last1=Song |first1=C. |last2=Wang |first2=P. |last3=Makse |first3=H. A. |date=29 May 2008 |title=A phase diagram for jammed matter |journal=[[Nature (journal)|Nature]] |volume=453 |pages=629–632 |doi=10.1038/nature06981 |url=http://www.nature.com/nature/journal/v453/n7195/full/nature06981.html |pmid=18509438 |issue=7195 |bibcode = 2008Natur.453..629S |arxiv = 0808.2196 }}</ref> This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres (that is, line segments or circles) will yield a regular packing.

==Hypersphere packing==
The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is [[circle packing|packing circles]] on a plane. In one dimension it is packing line segments into a linear universe.<ref>{{cite journal | last1 = Griffith | first1 = J.S. | year = 1962 | title = Packing of equal 0-spheres | url = | journal = Nature | volume = 196 | issue = | pages = 764–765 | doi = 10.1038/196764a0 | bibcode = 1962Natur.196..764G }}</ref>

In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions.<ref>{{MathWorld |title=Hypersphere Packing|urlname=HyperspherePacking}}</ref> Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing.<ref>{{cite journal | last=Sloane |first=N. J. A. | title=The Sphere-Packing Problem | year=1998 | pages=387–396 | journal=Documenta Mathematica|volume=3 | arxiv=math/0207256|bibcode = 2002math......7256S }}</ref>

In 2016, [[Maryna Viazovska]] announced a proof that the [[E8 lattice|E8 lattice]] provides the optimal packing (regardless of regularity) in eight-dimensional space,<ref>{{Cite journal|last=Viazovska|first=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 8|url=http://annals.math.princeton.edu/2017/185-3/p07|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=991–1015|arxiv=1603.04246|doi=10.4007/annals.2017.185.3.7|issn=0003-486X}}</ref> and soon afterwards she and a group of collaborators announced a similar proof that the [[Leech lattice]] is optimal in 24 dimensions.<ref>{{Cite journal|last=Cohn|first=Henry|last2=Kumar|first2=Abhinav|last3=Miller|first3=Stephen|last4=Radchenko|first4=Danylo|last5=Viazovska|first5=Maryna|date=1 January 2017|title=The sphere packing problem in dimension 24|url=http://annals.math.princeton.edu/2017/185-3/p08|journal=Annals of Mathematics|language=en-US|volume=185|issue=3|pages=1017–1033|arxiv=1603.06518|doi=10.4007/annals.2017.185.3.8|issn=0003-486X}}</ref> This result built on and improved previous methods which showed that these two lattices are very close to optimal.<ref>{{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=Optimality and uniqueness of the Leech lattice among lattices | doi=10.4007/annals.2009.170.1003 | mr=2600869 | zbl=1213.11144 | year=2009 | journal=Annals of Mathematics | issn=1939-8980 | volume=170 | issue=3 | pages=1003–1050 | arxiv=math.MG/0403263 }} {{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=The densest lattice in twenty-four dimensions | doi=10.1090/S1079-6762-04-00130-1 | mr=2075897 | year=2004 | journal=Electronic Research Announcements of the American Mathematical Society | issn=1079-6762 | volume=10 | issue=07 | pages=58–67 |arxiv=math.MG/0408174 }}</ref>
The new proofs involve using the [[Laplace transform]] of a carefully chosen [[modular function]] to construct a [[Rotational symmetry|radially symmetric]] function {{mvar|f}} such that {{mvar|f}} and its [[Fourier transform]] {{mvar|f̂}} both equal one at the [[origin (mathematics)|origin]], and both vanish at all other points of the optimal lattice, with {{mvar|f}} negative outside the central sphere of the packing and {{mvar|f̂}} positive. Then, the [[Poisson summation formula]] for {{mvar|f}} is used to compare the density of the optimal lattice with that of any other packing.<ref>{{citation|url=https://www.youtube.com/watch?v=8qlZjarkS_g|first=Stephen D.|last=Miller|date=4 April 2016|title=The solution to the sphere packing problem in 24 dimensions via modular forms|publisher=[[Institute for Advanced Study]]}}. Video of an hour-long talk by one of Viazovska's co-authors explaining the new proofs.</ref> Before the proof had been [[Scholarly peer review|formally refereed]] and published, mathematician [[Peter Sarnak]] called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct."<ref>{{citation|last1=Klarreich|first1=Erica|authorlink1=Erica Klarreich|title=Sphere Packing Solved in Higher Dimensions|url=https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions|magazine=Quanta Magazine|date=30 March 2016}}</ref>

Another line of research in high dimensions is trying to find [[asymptotic]] bounds for the density of the densest packings. Currently the best known result is that there exists a lattice in dimension {{mvar|n}} with density bigger or equal to {{math|2−''n''''cn''}} for some number {{mvar|c}}.<ref>{{cite journal|last=Rogers|first=C. A.|title=Existence Theorems in the Geometry of Numbers|journal=Annals of Mathematics |series=Second Series|volume=48|issue=4|year=1947|pages=994–1002|jstor=1969390|doi=10.2307/1969390}}</ref>

==Unequal sphere packing==
[[File:Binary sphere packing LS3.png|thumb|A dense packing of spheres with a radius ratio of 0.64799 and a density of 0.74786<ref name="doi10.1021/jp206115p">{{Cite journal | last1 = O'Toole | first1 = P. I. | last2 = Hudson | first2 = T. S. | doi = 10.1021/jp206115p | title = New High-Density Packings of Similarly Sized Binary Spheres | journal = The Journal of Physical Chemistry C | volume = 115 | issue = 39 | pages = 19037 | year = 2011 | pmid = | pmc = }}</ref>]]
Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or [[interstitial compound|interstitial]] packing. When many sizes of spheres (or a [[particle size distribution|distribution]]) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are available.

When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a close-packed arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends sensitively on the radius ratio, but in the limit of extreme size ratios, the smaller spheres can fill the gaps with the same density as the larger spheres filled space.<ref>{{Cite journal | last1 = Hudson | first1 = D. R. | title = Density and Packing in an Aggregate of Mixed Spheres | doi = 10.1063/1.1698327 | journal = Journal of Applied Physics | volume = 20 | issue = 2 | pages = 154–162| year = 1949 | pmid = | pmc = |bibcode = 1949JAP....20..154H }}</ref> Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere.<ref>{{Cite journal | last1 = Zong | first1 = C. | title = From deep holes to free planes | doi = 10.1090/S0273-0979-02-00950-3 | journal = Bulletin of the American Mathematical Society | volume = 39 | issue = 4 | pages = 533–555 | year = 2002 | pmid = | pmc = }}</ref>

When the smaller sphere has a radius greater than 0.41421 of the radius of the larger sphere, it is no longer possible to fit into even the octahedral holes of the close-packed structure. Thus, beyond this point, either the host structure must expand to accommodate the interstitials (which compromises the overall density), or rearrange into a more complex crystalline compound structure. Structures are known which exceed the close packing density for radius ratios up to 0.659786.<ref name="doi10.1021/jp206115p"/><ref>{{cite journal|first1=G. W.|last1=Marshall|first2=T. S.|last2=Hudson|journal=Contributions to Algebra and Geometry|title=Dense binary sphere packings|volume=51|issue=2|pages=337–344|year=2010|url=http://www.emis.de/journals/BAG/vol.51/no.2/3.html}}</ref>

Upper bounds for the density that can be obtained in such binary packings have also been obtained.<ref>{{cite journal |last1=de Laat |first1=David |last2=de Oliveira Filho |first2=Fernando Mário |last3=Vallentin |first3=Frank |title=Upper bounds for packings of spheres of several radii|arxiv=1206.2608|date=12 June 2012 |doi=10.1017/fms.2014.24 |volume=2 |journal=Forum of Mathematics, Sigma}}</ref>

In many chemical situations such as [[ionic crystal]]s, the [[stoichiometry]] is constrained by the charges of the constituent ions. This additional constraint on the packing, together with the need to minimize the [[Coulomb energy]] of interacting charges leads to a diversity of optimal packing arrangements.

==Hyperbolic space==
Although the concept of circles and spheres can be extended to [[hyperbolic space]], finding the densest packing becomes much more difficult. In a hyperbolic space there is no limit to the number of spheres that can surround another sphere (for example, [[Ford circle]]s can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an [[Infinity|infinite]] number of other circles). The concept of average density also becomes much more difficult to define accurately. The densest packings in any hyperbolic space are almost always irregular.<ref>{{Cite journal | last1 = Bowen | first1 = L. | last2 = Radin | first2 = C. | doi = 10.1007/s00454-002-2791-7 | title = Densest Packing of Equal Spheres in Hyperbolic Space | journal = Discrete and Computational Geometry | volume = 29 | pages = 23–39 | year = 2002 | pmid = | pmc = }}</ref>

Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic ''n''-space where ''n'' ≥ 2.<ref>{{Cite journal | last1 = Böröczky | first1 = K. | title = Packing of spheres in spaces of constant curvature | doi = 10.1007/BF01902361 | journal = Acta Mathematica Academiae Scientiarum Hungaricae | volume = 32 | issue = 3–4 | pages = 243–261 | year = 1978 | pmid = | pmc = }}</ref> In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the [[horosphere]] packing of the [[order-6 tetrahedral honeycomb]] with [[Schläfli symbol]] {3,3,6}.<ref>{{Cite journal | last1 = Böröczky | first1 = K. | last2 = Florian | first2 = A. | doi = 10.1007/BF01897041 | title = Über die dichteste Kugelpackung im hyperbolischen Raum | journal = Acta Mathematica Academiae Scientiarum Hungaricae | volume = 15 | pages = 237 | year = 1964 | pmid = | pmc = }}</ref> In addition to this configuration at least three other [[horosphere]] packings are known to exist in hyperbolic 3-space that realize the density upper bound.<ref>{{Cite journal | last1 = Kozma | first1 = R. T. | last2 = Szirmai | first2 = J. | doi = 10.1007/s00605-012-0393-x | title = Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types | journal = Monatshefte für Mathematik | volume = 168 | pages = 27 | year = 2012 | pmid = | pmc = | arxiv = 1007.0722 }}</ref>

==Touching pairs, triplets, and quadruples==
The [[contact graph]] of an arbitrary finite packing of unit balls is the graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. The cardinality of the edge set of the contact graph gives the number of touching pairs, the number of 3-cycles in the contact graph gives the number of touching triplets, and the number of tetrahedrons in the contact graph gives the number of touching quadruples (in general for a contact graph associated with a sphere packing in ''n'' dimensions that the cardinality of the set of ''n''-simplices in the contact graph gives the number of touching (''n'' + 1)-tuples in the sphere packing). In the case of 3-dimensional Euclidean space, non-trivial upper bounds on the number of touching pairs, triplets, and quadruples<ref>{{cite journal| last1 = Bezdek | first1 = Karoly | last2 = Reid | first2 = Samuel | title=Contact Graphs of Sphere Packings Revisited | year = 2013 |arxiv=1210.5756 |journal=Journal of Geometry |volume = 104 |issue = 1 | pages = 57–83 |doi = 10.1007/s00022-013-0156-4}}</ref> were proved by [[Karoly Bezdek]] and Samuel Reid at the University of Calgary.

==Other spaces==
Sphere packing on the corners of a hypercube (with the spheres defined by [[Hamming distance]]) corresponds to designing [[error-correcting codes]]: if the spheres have radius ''t'', then their centers are codewords of a (2''t'' + 1)-error-correcting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and error-correcting codes. For example, the [[binary Golay code]] is closely related to the 24-dimensional Leech lattice.

For further details on these connections, see the book ''Sphere Packings, Lattices and Groups'' by [[John Horton Conway|Conway]] and [[Neil Sloane|Sloane]].<ref>{{Cite book|url=https://books.google.com/books/about/Sphere_Packings_Lattices_and_Groups.html?id=ITDvBwAAQBAJ&printsec=frontcover&source=kp_read_button&hl=en#v=onepage&q&f=false|title=Sphere Packings, Lattices and Groups|last=Conway|first=John H.|authorlink=John Horton Conway|last2=Sloane|first2=Neil J. A.|authorlink2=Neil Sloane|publisher=Springer Science & Business Media|year=1998|isbn=0-387-98585-9|edition=3rd}}</ref>

==See also==
* [[Close-packing of equal spheres]]
* [[Apollonian sphere packing]]
* [[Hermite constant]]
* [[Kissing number problem]]
* [[Sphere-packing bound]]
* [[Random close pack]]

==References==
{{Reflist|2}}

==Bibliography==
* {{cite book |last1=Aste |first1=T. |last2=Weaire |first2=D. |title=The Pursuit of Perfect Packing |publisher=Institute of Physics Publishing |location=London |year=2000 |isbn=0-7503-0648-3}}
* {{cite book |last1=Conway |first1=J. H. |authorlink=John Horton Conway |last2=Sloane |first2=N. J. H. |authorlink2=Neil Sloane |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}
* {{cite journal | last1 = Sloane | first1 = N. J. A. | authorlink1=Neil Sloane | year = 1984 | title = The Packing of Spheres | url = | journal = Scientific American | volume = 250 | issue = | pages = 116–125 |bibcode = 1984SciAm.250e.116G |doi = 10.1038/scientificamerican0584-116 }}

==External links==
* Dana Mackenzie (May 2002) [http://www.ma.utexas.edu/users/radin/reviews/newscientist2.html "''A fine mess''"] (New Scientist)
:A non-technical overview of packing in hyperbolic space.
* {{MathWorld|urlname=CirclePacking |title=Circle Packing }}
* [http://www.3doro.de/e-kp.htm "Kugelpackungen (Sphere Packing)"] (T. E. Dorozinski)
*[http://alecjacobson.com/graphics/hw10b/ "3D Sphere Packing Applet"] Sphere Packing java applet
*[http://www.randomwalk.de/sphere/insphr/spheresinsphr.html "Densest Packing of spheres into a sphere"] java applet
*[http://codes.se/packings/ "Database of sphere packings"] (Erik Agrell)

{{Packing problem}}

[[Category:Discrete geometry]]
[[Category:Crystallography]]
[[Category:Packing problems]]
[[Category:Spheres]]

3-j symbol

2019-04-27T07:21:26Z

Crasshopper:

In [[quantum mechanics]], the '''Wigner 3-j symbols''', also called 3''-jm'' symbols, are an alternative to [[Clebsch–Gordan coefficients]] for the purpose of adding angular momenta.<ref name="Wigner1951">{{cite book |last=Wigner |first=E. P. |editor1-last=Wightman |editor1-first=Arthur S. |date=1951 |chapter=On the Matrices Which Reduce the Kronecker Products of Representations of S. R. Groups |title=The Collected Works of Eugene Paul Wigner |volume=3 |pages=608–654 |doi=10.1007/978-3-662-02781-3_42 |isbn=978-3-642-08154-5 }}</ref> While the two approaches address exactly the same physical problem, the 3-''j'' symbols do so more symmetrically.

== Mathematical relation to Clebsch-Gordan coefficients ==

The 3-''j'' symbols are given in terms of the Clebsch-Gordan coefficients by
:<math>
\begin{pmatrix}
j_1 & j_2 & j_3 \\
m_1 & m_2 & m_3
\end{pmatrix}
\equiv
\frac{(-1)^{j_1 - j_2 - m_3}}{\sqrt{2 j_3 + 1}}
\langle j_1 \, m_1 \, j_2 \, m_2 | j_3 \, (-m_3) \rangle.
</math>
The ''j'' 's and ''m'' 's are angular momentum quantum numbers, i.e., every {{math|''j''}} (and every corresponding {{math|''m''}}) is either a nonnegative integer or half-odd-integer. The exponent of the sign factor is always an integer, so it remains the same when transposed to the left hand side, and the inverse relation follows upon making the substitution {{math|''m''3 → −''m''3}}:
:<math>
\langle j_1 \, m_1 \, j_2 \, m_2 | j_3 \, m_3 \rangle
= (-1)^{-j_1 + j_2 - m_3} \sqrt{2 j_3 + 1}
\begin{pmatrix}
j_1 & j_2 & j_3 \\
m_1 & m_2 & -m_3
\end{pmatrix}
</math>.

== Definitional relation to Clebsch-Gordan coefficients ==

The C-G coefficients are defined so as to express the addition of two angular momenta in terms of a third:
:<math>
|j_3\, m_3\rangle
= \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2}
\langle j_1 \, m_1 \, j_2 \, m_2 | j_3 \, m_3 \rangle
|j_1 \, m_1 \, j_2 \, m_2 \rangle.
</math>
The 3-''j'' symbols, on the other hand, are the coefficients with which three angular momenta must be added so that the resultant is zero:
:<math>
\sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \sum_{m_3=-j_3}^{j_3}
|j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
= |0\,0\rangle.
</math>
Here, <math>|0\,0\rangle</math> is the zero angular momentum state (<math> j = m = 0</math>). It is apparent that the 3-''j'' symbol treats all three angular momenta involved in the addition problem on an equal footing, and is therefore more symmetrical than the C-G coefficient.

Since the state <math>|0\,0\rangle</math> is unchanged by rotation, one also says that the contraction of the product of three rotational states with a 3-''j'' symbol is invariant under rotations.

== Selection rules ==

The Wigner 3-''j'' symbol is zero unless all these conditions are satisfied:

:<math>\begin{align}
&m_i \in \{-j_i, -j_i + 1, -j_i + 2, \ldots, j_i\}, \quad (i = 1, 2, 3).\\
&m_1 + m_2 + m_3 = 0 \\
&|j_1 - j_2| \le j_3 \le j_1 + j_2 \\
&(j_1 + j_2 + j_3) \text{ is an integer (and, moreover, an even integer if } m_1 = m_2 = m_3 = 0 \text{)} \\
\end{align}</math>

== Symmetry properties ==
A 3-''j'' symbol is invariant under an even permutation of its columns:
:<math>
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
=
\begin{pmatrix}
j_2 & j_3 & j_1\\
m_2 & m_3 & m_1
\end{pmatrix}
=
\begin{pmatrix}
j_3 & j_1 & j_2\\
m_3 & m_1 & m_2
\end{pmatrix}.
</math>
An odd permutation of the columns gives a phase factor:
:<math>
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
j_2 & j_1 & j_3\\
m_2 & m_1 & m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
j_1 & j_3 & j_2\\
m_1 & m_3 & m_2
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
j_3 & j_2 & j_1\\
m_3 & m_2 & m_1
\end{pmatrix}.
</math>
Changing the sign of the <math>m</math> quantum numbers (time-reversal) also gives a phase:
:<math>
\begin{pmatrix}
j_1 & j_2 & j_3\\
-m_1 & -m_2 & -m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}.
</math>
The 3-''j'' symbols also have so-called Regge symmetries, which are not due to permutations or time-reversal.<ref>{{cite journal |first1=T. |last1=Regge|title=Symmetry Properties of Clebsch-Gordan Coefficients |journal=Nuovo Cimento |year=1958|volume=10 |issue=3|page=544 |doi=10.1007/BF02859841|bibcode=1958NCim...10..544R}}</ref> These symmetries are,
:<math>
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
=
\begin{pmatrix}
j_1 & \frac{j_2+j_3-m_1}{2} & \frac{j_2+j_3+m_1}{2}\\
j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3+m_1}{2}+m_3
\end{pmatrix}.
</math>
:<math>
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
=
(-1)^{j_1+j_2+j_3}
\begin{pmatrix}
\frac{j_2+j_3+m_1}{2} & \frac{j_1+j_3+m_2}{2} & \frac{j_1+j_2+m_3}{2}\\
j_1 - \frac{j_2+j_3-m_1}{2} & j_2 - \frac{j_1+j_3-m_2}{2} & j_3-\frac{j_1+j_2-m_3}{2}
\end{pmatrix}.
</math>
With the Regge symmetries, the 3-''j'' symbol has a total of 72 symmetries. These are best displayed by the definition of a Regge symbol which is a one-to-one correspondence between it and a 3-''j'' symbol and assumes the properties of a semi-magic square<ref>{{Cite journal |first1=J. |last1=Rasch
|first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932
}}</ref>
:<math>
R=
\begin{array}{|ccc|}
\hline
-j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\
j_1-m_1 & j_2-m_2 & j_3-m_3\\
j_1+m_1 & j_2+m_2 & j_3+m_3\\
\hline
\end{array}
</math>
whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. These facts can be used to devise an effective storage scheme.<ref>{{Cite journal |first1=J. |last1=Rasch |first2=A. C. H. |last2=Yu |title=Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients |journal=SIAM J. Sci. Comput. |volume=25 |issue=4 |year=2003 |pages=1416–1428 |doi=10.1137/s1064827503422932}}</ref>

== Orthogonality relations ==
A system of two angular momenta with magnitudes {{math|''j''1}} and {{math|''j''2}}, say, can be described either in terms of the uncoupled basis states (labeled by the quantum numbers {{math|''m''1}} and {{math|''m''2}}), or the coupled basis states (labeled by {{math|''j''3}} and {{math|''m''3}}). The 3-''j'' symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality relations,
:<math>
(2 j_3 + 1)\sum_{m_1 m_2}
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
\begin{pmatrix}
j_1 & j_2 & j'_3\\
m_1 & m_2 & m'_3
\end{pmatrix}
=\delta_{j_3, j'_3} \delta_{m_3, m'_3} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix}.
</math>
:<math>
\sum_{j_3 m_3} (2 j_3 + 1)
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1' & m_2' & m_3
\end{pmatrix}
=\delta_{m_1, m_1'} \delta_{m_2, m_2'}.
</math>
The ''triangular delta'' {{math|{''j''1  ''j''2  ''j''3}}} is equal to 1 when the triad (''j''1, ''j''2, ''j''3) satisfies the triangle conditions, and zero otherwise. The triangular delta itself is sometimes confusingly called<ref name="WormerPaldus2006">{{cite article|title=Angular Momentum Diagrams|author1=P.E.S. Wormer |author2=J. Paldus |journal=Advances in Quantum Chemistry|publisher=Elsevier|volume=51|pages=59–124|year=2006|issn=0065-3276|doi=10.1016/S0065-3276(06)51002-0|url=http://www.sciencedirect.com/science/article/pii/S0065327606510020|bibcode = 2006AdQC...51...59W }}</ref> a “3-j symbol” (without the “m”) in analogy to [[6-j symbol|6-j]] and [[9-j symbol|9-j]] symbols, all of which are irreducible summations of 3-jm symbols where no {{math|''m''}} variables remain.

==Relation to spherical harmonics==
The 3-''jm'' symbols give the integral of the products of three [[spherical harmonics]]
:<math>
\begin{align}
& {} \quad \int Y_{l_1m_1}(\theta,\varphi)Y_{l_2m_2}(\theta,\varphi)Y_{l_3m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\
& =
\sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}
\begin{pmatrix}
l_1 & l_2 & l_3 \\
0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
l_1 & l_2 & l_3\\
m_1 & m_2 & m_3
\end{pmatrix}
\end{align}
</math>
with <math>l_1</math>, <math>l_2</math> and <math>l_3</math> integers.

=== Relation to integrals of spin-weighted spherical harmonics ===

Similar relations exist for the [[spin-weighted spherical harmonics]] if <math>s_1+s_2+s_3=0</math>:
:<math>
\begin{align}
& {} \quad \int d{\mathbf{\hat n}}\,{}_{s_1} Y_{j_1 m_1}({\mathbf{\hat n}})
\,{}_{s_2} Y_{j_2m_2}({\mathbf{\hat n}})\, {}_{s_3} Y_{j_3m_3}({\mathbf{\hat
n}}) \\[8pt]
& = \sqrt{\frac{(2j_1+1)(2j_2+1)(2j_3+1)}{4\pi}}
\begin{pmatrix}
j_1 & j_2 & j_3\\
m_1 & m_2 & m_3
\end{pmatrix}
\begin{pmatrix}
j_1 & j_2 & j_3\\
-s_1 & -s_2 & -s_3
\end{pmatrix}
\end{align}
</math>

== Recursion relations ==
:<math>
\begin{align}
& {} \quad -\sqrt{(l_3\mp s_3)(l_3\pm s_3+1)}
\begin{pmatrix}
l_1 & l_2 & l_3\\
s_1 & s_2 & s_3\pm 1
\end{pmatrix}
\\
& = \sqrt{(l_1\mp s_1)(l_1\pm s_1+1)}
\begin{pmatrix}
l_1 & l_2 & l_3\\
s_1 \pm 1 & s_2 & s_3
\end{pmatrix}
+\sqrt{(l_2\mp s_2)(l_2\pm s_2+1)}
\begin{pmatrix}
l_1 & l_2 & l_3\\
s_1 & s_2 \pm 1 & s_3
\end{pmatrix}
\end{align}
</math>

== Asymptotic expressions ==
For <math>l_1\ll l_2,l_3</math> a non-zero 3-''j'' symbol has
:<math>
\begin{pmatrix}
l_1 & l_2 & l_3\\
m_1 & m_2 & m_3
\end{pmatrix}
\approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{2l_3+1}}
</math>
where <math>\cos(\theta) = -2m_3/(2l_3+1)</math> and <math>d^l_{mn}</math> is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by
:<math>
\begin{pmatrix}
l_1 & l_2 & l_3\\
m_1 & m_2 & m_3
\end{pmatrix}
\approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2+l_3+1}}
</math>
where <math>\cos(\theta) = (m_2-m_3)/(l_2+l_3+1)</math>.

== Metric tensor ==

The following quantity acts as a [[metric tensor]] in angular momentum theory and is also known as a ''Wigner 1-jm symbol'',<ref name="Wigner1951"/>
:<math>\begin{pmatrix}
j \\
m \quad m'
\end{pmatrix}
:= \sqrt{2 j + 1}
\begin{pmatrix}
j & 0 & j \\
m & 0 & m'
\end{pmatrix}
= (-1)^{j - m'} \delta_{m, -m'}
</math>
It can be used to perform time-reversal on angular momenta.

== Other properties ==
:<math>\sum_m (-1)^{j - m}
\begin{pmatrix}
j & j & J\\
m & -m & 0
\end{pmatrix} = \sqrt{2 j + 1} ~ \delta_{J, 0}
</math>

:<math>
\frac{1}{2} \int_{-1}^1 P_{l_1}(x)P_{l_2}(x)P_{l}(x) \, dx =
\begin{pmatrix}
l & l_1 & l_2 \\
0 & 0 & 0
\end{pmatrix} ^2
</math>

== Relation to Racah {{math|''V''}}-coefficients ==

Wigner 3-j symbols are related to [[Giulio Racah|Racah]] {{math|''V''}}-coefficients<ref>{{Cite journal |first=G. |last=Racah |title=Theory of Complex Spectra II |journal=[[Physical Review]] |volume=62 |issue=9–10 |pages=438–462 |year=1942 |doi=10.1103/PhysRev.62.438 |bibcode = 1942PhRv...62..438R }}</ref> by a simple phase:

:<math>
V(j_1 j_2 j_3; m_1 m_2 m_3) = (-1)^{j_1 - j_2 - j_3} \begin{pmatrix}
j_1 & j_2 & j_3 \\
m_1 & m_2 & m_3
\end{pmatrix}
</math>

==See also==
*[[Clebsch–Gordan coefficients]]
*[[Spherical harmonics]]
*[[6-j symbol]]
*[[9-j symbol]]

==References==
<references />

<div class="references">
*[[Lawrence Biedenharn|L. C. Biedenharn]] and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981.
* D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993.
* A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Princeton, 1960.
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</div>

==External links==
* {{cite web
|first1=Anthony
|last1=Stone
|url=http://www-stone.ch.cam.ac.uk/wigner.shtml
|title=Wigner coefficient calculator
}}
* {{cite web
|first1 = A.
|last1 = Volya
|url = http://www.volya.net/vc/vc.php
|archive-url = https://web.archive.org/web/20070929011249/http://www.volya.net/vc/vc.php
|dead-url = yes
|archive-date = 2007-09-29
|title = Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator
}} (Numerical)
* {{cite journal
|first1=Paul
|last1=Stevenson
|url=http://personal.ph.surrey.ac.uk/~phs3ps/cleb.html
|title=Clebsch-O-Matic
|journal=Computer Physics Communications
|volume=147
|issue=3
|pages=853–858
|doi=10.1016/S0010-4655(02)00462-9
|bibcode = 2002CoPhC.147..853S |year=2002
}}
* [http://plasma-gate.weizmann.ac.il/369j.html 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science] (Numerical)
* [http://geoweb.princeton.edu/people/simons/software.html Frederik J Simons: Matlab software archive, the code THREEJ.M]
* [http://www.sagemath.org/ Sage (mathematics software)] Gives exact answer for any value of j, m
* {{cite web
|first1=H.T.
|last1=Johansson
|first2=C.
|last2=Forssén
|title=(WIGXJPF)
|url=http://fy.chalmers.se/subatom/wigxjpf/
}} (accurate; C, fortran, python)
* {{cite web
|first1=H.T.
|last1=Johansson
|title=(FASTWIGXJ)
|url=http://fy.chalmers.se/subatom/fastwigxj/
}} (fast lookup, accurate; C, fortran)

[[Category:Rotational symmetry]]
[[Category:Representation theory of Lie groups]]
[[Category:Quantum mechanics]]

Whidbey Island

2019-03-08T07:03:30Z

Crasshopper: /* Other */

{{About|the island||Whidbey (disambiguation){{!}}Whidbey}}
{{Use mdy dates|date=October 2012}}
{{Infobox islands
| name = Whidbey Island
| image_name = WhidbeyIsland04.jpg
| image_caption = Map of Whidbey Island
| image_size = 220
| map = USA Washington
| coordinates = {{coord|48.136389|-122.5825|scale:500000_region:US-WA|display=it}}
| map_caption = Whidbey Island (Washington)
| native_name =
| native_name_link = Captain Joseph Whidbey
| nickname = "The Rock"<ref>{{cite web | url=http://www.whidbeynewstimes.com/news/22086709.html | title=Whidbey Island has a terrain that's set in stone | newspaper=[[Whidbey News-Times]] | date=July 3, 2008 | accessdate=February 10, 2012 | archive-url=https://web.archive.org/web/20150919193429/http://www.whidbeynewstimes.com/news/22086709.html | archive-date=September 19, 2015 | dead-url=yes | df=mdy-all }}</ref>
| location = [[Puget Sound]]
| archipelago =
| total_islands =
| major_islands =
| area_sqmi = 168.67
| length_mi = 37| width_mi = 10
| highest_mount =
| elevation_ft =
| country = United States
| country_admin_divisions_title = State
| country_admin_divisions = [[Washington (state)|Washington]]
| country_admin_divisions_title_1 = County
| country_admin_divisions_1 = [[Island County, Washington|Island County]]
| country_largest_city = [[Oak Harbor, Washington|Oak Harbor]]
| country_largest_city_population = 23,204 <ref>United States Census Bureau</ref>
| population = 80,022
| population_as_of =
| density_km2 = 133.25
| ethnic_groups =
| additional_info =
}}

[[File:Whidbey 2.JPG|thumb|Cultus Bay at Low Tide]] [[File:Whidbey Island.jpg|thumb|Double Bluff, with Useless Bay to the South (right) and Mutiny Bay to the North (left)]]
'''Whidbey Island''' (historical spellings '''Whidby''', '''Whitbey''',<ref>{{cite gnis|id=1509451|name=Whidbey Island}}</ref> or '''Whitby''') is the largest of the islands composing [[Island County, Washington|Island County]], [[Washington (state)|Washington]], in the United States. (The other large island is [[Camano Island]], east of Whidbey.) Whidbey is about {{convert|30|mi|km}} north of [[Seattle]], and lies between the [[Olympic Peninsula]] and the [[Interstate 5 (Washington)|I-5]] corridor of western Washington. The island forms the northern boundary of [[Puget Sound]]. It is home to [[Naval Air Station Whidbey Island]].

Whidbey Island is home to 80,022 residents (according to the [[United States Census, 2000|2000 census]]).<ref>[http://factfinder.census.gov/servlet/DTTable?_bm=y&-show_geoid=Y&-tree_id=4001&-_caller=geoselect&-context=dt&-errMsg=&-all_geo_types=N&-mt_name=DEC_2000_SF1_U_P001&-redoLog=true&-transpose=N&-search_map_config=|b=50|l=en|t=4001|zf=0.0|ms=sel_00dec|dw=0.14742116997381507|dh=0.0756441033284385|dt=gov.census.aff.domain.map.EnglishMapExtent|if=gif|cx=-90.71827740294499|cy=46.747841207516664|zl=4|pz=4|bo=318:317:316:314:313:323:319|bl=362:393:358:357:356:355:354|ft=350:349:335:389:388:332:331|fl=381:403:204:380:369:379:368|g=01000US&-PANEL_ID=p_dt_geo_map&-_lang=en&-geo_id=05000US53029&-geo_id=06000US5302990400&-geo_id=06000US5302990528&-geo_id=06000US5302992256&-geo_id=06000US5302993168&-CONTEXT=dt&-format=&-search_results=ALL&-ds_name=DEC_2000_SF1_U Island County, Washington and its subdivisions] United States Census Bureau</ref> An estimated 29,000 of Whidbey Island residents live in rural locations.

Whidbey Island is approximately {{convert|55|mi|km}} long (if measured along roads traveled from the extreme north to extreme south), or about {{convert|37|mi|km}} when measured along a straight line from north to south, and {{convert|1.5|to|10|mi|km}} wide, with a total land area of {{convert|168.67|sqmi|km2}},<ref>{{cite web|url=http://www.islandcounty.net/Assessor/|title=Island County Assessor|work=Island County Assessor|accessdate=February 16, 2015}}</ref> making it the [[List of islands of the United States by area|40th largest island in the United States]]. It is ranked as the fourth longest and fourth largest island in the contiguous United States, behind [[Long Island]], [[New York (state)|New York]];<ref name="peninsula">{{cite news |url=https://query.nytimes.com/gst/fullpage.html?res=9A07E3DB133FF932A15752C1A9629C8B63&sec=&spon=&pagewanted=all |title= Long Island at its Best; Who's the Longest of Them All? |accessdate=2008-10-16 |author= John Burbidge|date=November 21, 2004 |work=The New York Times}}</ref><ref>{{cite web|url=http://www.peakbagger.com/PBGeog/longisl.aspx|title=The Longest Islands in the United States - Peakbagger.com|website=www.peakbagger.com}}</ref> [[Padre Island]], Texas (the world's longest barrier island);<ref>{{cite web|url=https://www.nps.gov/pais/index.htm|title=Padre Island National Seashore (U.S. National Park Service)|website=www.nps.gov}}</ref> and [[Isle Royale]], [[Michigan]].<ref>{{cite web|url=https://www.nationalgeographic.com/travel/national-parks/isle-royale-national-park/|title=Isle Royale National Park - National Geographic|date=November 5, 2009|website=nationalgeographic.com}}</ref> In the state of Washington, it is the largest island, followed by [[Orcas Island]].

== History ==
Whidbey Island was inhabited by members of the [[Lower Skagit (tribe)|Lower Skagit]], [[Swinomish (tribe)|Swinomish]], [[Suquamish]], [[Snohomish (tribe)|Snohomish]] and other Native American tribes. The [[Salishan languages|Salishan]] name for the island was Tscha-kole-chy.<ref name="historylink.org3">{{cite web|url=http://www.historylink.org/File/7523|title=Island County -- Thumbnail History|website=historylink.org}}</ref> These were peaceful groups who lived off the sea and land, with fishing, harvesting nuts, berries and roots, which they preserved over the winter.<ref>{{cite web|url=http://whidbeyhistory.historywiz.org/tschakolecy.htm|title=Tschakolecy - Whidbey Island History: Tschakolecy|website=whidbeyhistory.historywiz.org}}</ref>

The first known European sighting of Whidbey Island was during the 1790 Spanish expedition of [[Manuel Quimper]] and [[Gonzalo López de Haro]] on the ''[[Princesa Real (sloop)|Princesa Real]]''.<ref>{{cite book |last= Hayes |first= Derek |title= Historical Atlas of the Pacific Northwest: Maps of exploration and Discovery |year= 1999 |publisher= Sasquatch Books |isbn= 1-57061-215-3 |pages= 70–71}}</ref>

Captain [[George Vancouver]] fully explored the island in 1792. In May of that year, [[Royal Navy]] officers and members of Vancouver's expedition, [[Joseph Whidbey]] (master of H. M. S. Discovery) and [[Peter Puget]] (a lieutenant on the ship), began to map and explore the areas of what would later be named [[Puget Sound]]. After Whidbey circumnavigated the island in June 1792, Vancouver named the island in his honor. By that time, Vancouver had claimed the area for Britain.<ref>{{Cite web |url=http://www.visitwhidbey.com/information/History-Images.html |title=Archived copy |access-date=August 23, 2017 |archive-url=https://web.archive.org/web/20170818030053/http://www.visitwhidbey.com/information/History-Images.html |archive-date=August 18, 2017 |dead-url=yes |df=mdy-all }}</ref>
<ref>{{cite web|url=http://www.historylink.org/File/5060|title=Joseph Whidbey circumnavigates Whidbey Island in June 1792. - HistoryLink.org|website=www.historylink.org}}</ref> On 4 June 1792, the King’s Birthday, near Possession Point at the southern end of Whidbey Island, Vancouver took formal possession of all the coast and hinterland contiguous to the Strait of Juan de Fuca, including Puget Sound, under the name of [[New Georgia]].<ref>W. Kaye Lamb (ed.), ''The Voyage of George Vancouver, 1791-1795,'' London, Hakluyt Society, 1984, Vol.1, p.569; also Freeman M. Tovell, ‘The Other Side of the Coin: the Viceroy, Bodega y Quadra, Vancouver, and the Nootka Crisis’, ''BC Studies,'' no.93, 1992, p.19.</ref>

The first known overnight stay by a non-Native American was made on May 26, 1840 by a Catholic missionary, Father [[François Norbert Blanchet]], during travel across Puget Sound. He had been invited by Chief Tslalakum.<ref name="historylink.org1">{{cite web|url=http://www.historylink.org/File/5258|title=Father Francis (or Francois) N. Blanchet visits Whidbey Island on May 26, 1840. - HistoryLink.org|website=www.historylink.org}}</ref> Blanchet remained on the island for nearly a year and guided the inhabitants in building a new log church.<ref>{{cite web|url=https://www.sos.wa.gov/legacy/timeline/detail.aspx|title=Legacy Washington - WA Secretary of State|first1=Contact Us Washington Secretary of StateLegacy WashingtonLegislative Building · PO Box|last1=40220Olympia|first2=WA 98504-0220Phone Numbers Privacy|last2=Policy|website=wa.gov}}</ref><ref name="historylink.org1"/>

Lieutenant [[Charles Wilkes]], commander of the [[United States Exploring Expedition]] of 1838–1842, sailed the [[USS Vincennes (1826)|USS ''Vincennes'']] into Penn Cove in 1841. By that time, the log church was already being built by the Native Americans beside a huge wooden cross (24 feet long) that they had erected. Wilkes ordered his men to use no force except in self-defense when dealing with the "savage and treacherous inhabitants". In fact, he encountered few problems with the indigenous people who had already been poorly treated by visitors and suffered from diseases they had introduced.<ref name="historylink.org2">{{cite web|url=http://www.historylink.org/File/5226|title=Wilkes, Charles (1798-1877)|website=historylink.org}}</ref>

Wilkes named the lower cove Holmes Harbor, after his assistant surgeon, Silas Holmes. During this time he charted Puget Sound.<ref name="historylink.org2"/> Other sites in the area that were given names by Wilkes included Maury Island (Vashon), Hammersley Inlet, Totten and Budd Inlets, Agate Passage between the Kitsap Peninsula, Hale Passage and Dana Passage.<ref name="historylink.org2"/>

Thomas W. Glasgow filed the first land claim on Whidbey Island in 1848, attempting to become the first settler. He built a small cabin near Penn Cove, planted some crops and married a local lady, Julia Pat-Ke-Nim.<ref name="hometownchronicles.com"/> Glasgow left in August of that year however, having been forced out by the local inhabitants.<ref name="historylink.org3" />
Colonel [[Isaac N. Ebey]] arrived from Columbus, Ohio, in 1850 and became the first permanent white settler, claiming a square mile (2.6 km²) of prairie with a southern shoreline on [[Admiralty Inlet]]. He took advantage of the 640 acres offered free of charge to each married couple, the first to do so, on October 15, 1850. In the fall of 1851, his children, his wife, three of her brothers and the Samuel Crockett family arrived to join Ebey.<ref name="hometownchronicles.com">{{Cite web |url=https://hometownchronicles.com/wa/island/cohist.html |title=Archived copy |access-date=August 23, 2017 |archive-url=https://web.archive.org/web/20170425233906/http://hometownchronicles.com/wa/island/cohist.html |archive-date=April 25, 2017 |dead-url=yes |df=mdy-all }}</ref>
In addition to farming potatoes and wheat, Eby was also the postmaster for [[Port Townsend, Washington]] and rowed a boat daily across the inlet in order to work at the post office there. Colonel Ebey also served as a representative in the Oregon Territory Legislative Assembly, as Island County's first Justice of the Peace, as a probate judge and as Collector of Customs for the Puget Sound District.<ref name="historylink.org3"/>

On August 11, 1857, at age 39, Colonel Ebey was murdered and beheaded by Native Americans, said to be [[Haida people|Haida]] who had traveled to this area from the [[Queen Charlotte Islands]]. Some sources however, refer to his killers as "Russian Indians called Kakes or Kikans, [from] Kufrinoff Island, near the head of Prince Frederick's Sound.<ref name="hometownchronicles.com"/> Ebey was slain in proxy-retaliation for the killing of a Haida chief or Tyee and 27 other indigenous people at [[Port Gamble]]. [[Fort Ebey]], named for the Colonel, was established in 1942 on the west side of the central part of the island, just northwest of [[Coupeville, Washington|Coupeville]].<ref name="historylink.org3"/>
[[File:Ebey's Landing N.H. Reserve.jpg|thumb|
On the Bluff Trail in Ebey's Landing National Historical Reserve
]]
[[Admiralty Head Lighthouse]] is located in this area, on the grounds of [[Fort Casey State Park]]. The area around Coupeville is the federally protected [[Ebey's Landing National Historical Reserve]], named in honor of Isaac Ebey.

In December 1984, the island was the site of a violent encounter between law enforcement and [[white nationalist]] and [[organized crime]] leader [[Robert Jay Mathews]] of the group The Order. A large shootout occurred between Mathews and FBI agents in which Mathews was killed during a house fire. Mathews' followers have since gathered on the island at the location where he was killed by FBI agents on the anniversary of his death to commemorate it.<ref>{{cite web|url=http://www.historylink.org/File/7921|title=Robert Jay Mathews, founder of the white-supremacist group The Order, is killed during an FBI siege on Whidbey Island on December 8, 1984.|website=historylink.org}}</ref>

== Government ==
Whidbey Island, along with [[Camano Island]], Ben Ure Island and six uninhabited islands, comprises [[Island County, Washington]]. The county seat is located in the town of [[Coupeville, Washington|Coupeville]] on Whidbey Island.
[[File:Swantown.jpg|thumb|
Looking east over Swantown Lake
]]
Population centers of Whidbey Island include the [[Oak Harbor, Washington|City of Oak Harbor]], the [[Coupeville, Washington|Town of Coupeville]], the [[Langley, Washington|City of Langley]], the [[Freeland, Washington|Village of Freeland]], the [[Greenbank, Washington|Community of Greenbank]], the [[Clinton, Washington|Village of Clinton]] and the [[Bayview, Island County, Washington|Community of Bayview]]. Only Oak Harbor, Coupeville and Langley are incorporated, the others (with the exception of Greenbank and Bayview) are all [[Census-designated place]]s, and all but Bayview have their own post offices and ZIP codes.

== Economy ==
[[File:DeceptionPass Bridge.jpg|thumb|
Deception Pass Bridge
]]
Whidbey Island is divided economically into two different regions: the northern end of the island (encompassing Oak Harbor and [[Naval Air Station Whidbey Island|Whidbey Island Naval Air Station]]), and the remainder of the island (encompassing Coupeville, Greenbank, Freeland, Langley, Clinton and the smaller communities in-between).

The economy of the northern end of Whidbey Island is strongly influenced by the presence of Whidbey Island Naval Air Station near [[Oak Harbor, Washington|Oak Harbor]] (N.A.S. Whidbey). N.A.S. Whidbey is Oak Harbor's largest employer; thus, Oak Harbor has a predominantly service-based economy and several national chain stores have been attracted to the Oak Harbor area.

The economy of Whidbey Island south of Oak Harbor relies heavily on tourism, small-scale agriculture, and the arts.

Tourism is especially important for both Whidbey and Camano Islands. On Whidbey, tourists find a wide range of amenities in the towns of Oak Harbor, Coupeville, Freeland and Langley. Coupeville's Penn Cove Mussel Farm exports large quantities of its highly renowned Penn Cove [[Mussel]]s. This aquaculture facility, along with a number of small farms, reflects the rural agricultural nature of most of central Whidbey Island. Many of these small farms host farm stands onsite, where customers may buy produce, flowers, meat, eggs and other locally raised products directly from the farmers.<ref>{{Cite web |url=http://goosefoot.org/pdf/farmstands.pdf |title=Archived copy |access-date=January 19, 2016 |archive-url=https://web.archive.org/web/20150701204046/http://www.goosefoot.org/pdf/farmstands.pdf |archive-date=July 1, 2015 |dead-url=yes |df=mdy-all }}</ref>

Often referred to as Puget Sound's Largest Artist's Colony, Whidbey is home to numerous working artists, writers, and performers. These include many well-known painters, sculptors, glass artists, wood workers, metal workers, mixed media artists, photographers, authors, poets, actors, and musicians.

In addition to being a haven for artists, the southern end of Whidbey Island also serves as a minor bedroom community for the nearby cities of [[Everett, Washington|Everett]], where the [[Boeing Everett Factory]] is located, and [[Seattle]]. Commuters to and from those areas use the [[Washington State Ferries]] system's run between [[Clinton, Washington|Clinton]] and [[Mukilteo, Washington|Mukilteo]].
[[File:ClintonSign.jpg|thumb|Carved sign welcoming visitors as they arrive by ferry at Clinton.]]

== Geography ==
Whidbey Island is often claimed to be the longest island in the continental United States (or another similar claim), but according to the [[Seattle Times]] it cannot be correctly considered so.<ref name ="DontStretchST">{{cite news|title=Whidbey is long, but let's not stretch it|url=http://community.seattletimes.nwsource.com/archive/?date=20000105&slug=A20000106010141|date=January 5, 2000|accessdate=August 25, 2011|first1=Steve|last1=Johnston|work=The Seattle Times}}</ref> Whidbey Island has four lakes that are part of its interior hydrology: Cranberry Lake (inside Deception Pass State Park), Deer Lake, Goss Lake and Lone Lake (both near the town of Langley).<ref>{{cite web|title=Lakes Monitored by Ecology's Lake Water Quality Monitoring Program From 1989 through 1997|url=http://www.ecy.wa.gov/programs/eap/lakes/wq/lake_assessments.html|accessdate=August 25, 2011|publisher=Department of Ecology, State of Washington}}</ref>

==Parks and reserve areas==
Whidbey Island contains [[Ebey's Landing National Historical Reserve]], the first national historic reserve in the US created by the [[National Park Service]] to preserve the rural history and culture of the island and to protect the area's rare and sensitive plants.

[[Washington State Park System|Washington State Parks]] located on the island include [[Deception Pass State Park]] (the most visited state park in Washington), [[Joseph Whidbey State Park]], [[Fort Ebey State Park]], [[Fort Casey State Park]], [[Possession Point State Park]], and [[South Whidbey Island State Park]]. There is also a series of county operated parks throughout the Island.

[[Earth Sanctuary]] is a nature reserve, sculpture garden and retreat center on Whidbey Island. The ponds and bog fen complex have been designated as a "habitat of local importance" by the Whidbey Audubon Society and Island County Critical Areas program.<ref>{{cite web|url=http://www.southwhidbeyrecord.com/lifestyle/35014484.html|title=ISLAND BIRDING - Islanders should speak up now to protect important bird habitat areas|work=South Whidbey Record|accessdate=February 16, 2015}}</ref><ref>{{cite web|url=http://www.southwhidbeyrecord.com/news/21571814.html|title=Newman Ponds to become 'Earth Sanctuary'|work=South Whidbey Record|accessdate=February 16, 2015}}</ref>

==Festivals==
Whidbey Island hosts many festivals and celebrations throughout the year.
*Whidbey Island Area Fair ("Island County Fair" until 2012<ref>{{cite web|url=http://www.southwhidbeyrecord.com/news/149304495.html|title=Welcome to the new Whidbey Island Fair - South Whidbey Record|date=April 27, 2012|website=southwhidbeyrecord.com}}</ref>), on the third weekend of July, includes rides, food, and animal shows.
*Wag'n'Walk, which takes place towards the end of August, is Western Washington's premier celebration of dogs and things dog-related. It includes vendors, games, competition, demonstrations and the Wag'n'Walk itself.
*Whidbey Island Kite Festival, in September
* Langley's Mystery Weekend in March or February. For the weekend the Town of Langley turns into the setting of a fictional murder mystery.
* Penn Cove Mussel Festival, in March, celebrates the bounty of the sea, especially the mussel.
*Loganberry Festival at the Greenbank Farm in July (This was discontinued after the 2016 festival.)
*Maxwelton Beach Fourth of July Parade and fireworks show, which takes place at the southern end of Maxwelton Road at Dave Mackie Park. After the parade, there are events for all ages, including three-legged races, divided into age groups, and the most popular event, the [[egg toss]].
*Choochokam is the annual street fair and arts festival, held in downtown Langley during the second weekend of July, detailed schedules and other information is generally available on the festival website. (This was discontinued after the 2016 festival.)
*Tour de Whidbey, in September, is a bike race spanning the length of Whidbey Island.
*The Whidbey Island Marathon and Half Marathon, in April since 2002.<ref>{{Cite web |url=http://www.whidbeyislandmarathon.com/ |title=Archived copy |access-date=February 25, 2008 |archive-url=https://web.archive.org/web/20021123192533/http://www.whidbeyislandmarathon.com/ |archive-date=November 23, 2002 |dead-url=yes |df=mdy-all }}</ref>
*Whidbey Island Race Week: a week-long sailing regatta every summer based out of Oak Harbor with daily racing in Penn Cove and/or Saratoga Passage (depending on wind conditions). Usually held third week of July, varies slightly due to tidal conditions.
*Whidbey Island Highland Games - 2nd Saturday in August. Competitions in Scottish Heavy Athletics, Highland Dancing, Pipe and drum bands.
*Whidbey Island Zucchini Festival - An annual festival hosted by residents of Whidbey island brought about by an excess of home-grown zucchini. The festival includes zucchini based musical performances, various types of zucchini based or themed visual art, all types of foods that feature zucchini, outdoor games and competitions using zucchini, and a giant zucchini slingshot.
*Oak Harbor Music Festival - An annual music festival held in the biggest city on the island, Oak Harbor. It is held over Labor Day Weekend, and consists of a wide variety of musical acts.
*DjangoFestNW - An annual 5-day music festival held in mid-September that celebrates the music of [[Django Reinhardt]] at Whidbey Island Center for the Arts<ref>{{cite web|url=https://www.djangofestnw.com/|title=Welcome|website=DjangofestNW.com}}</ref>

==Climate==
[[File:Fort Casey Cliff.jpg|right|thumb|A cliff on Whidbey Island near [[Fort Casey]]]]
Whidbey Island lies partially in the [[rain shadow]] of the [[Olympic Mountains|Olympic Mountain Range]] to the west, and has a variety of climate zones. This can be observed by rainfall amounts – wettest in the south with average rainfall of {{convert|36|in|mm}}, driest in the central district of [[Coupeville, Washington|Coupeville]] with average rainfall of {{convert|20|to|22|in|mm}}, and turning moister again farther north with average rainfall of {{convert|32|in|mm}}. [[Microclimate]]s abound, determined by proximity to water, elevation and prevailing winds.

{{Weather box
|location = [[Naval Air Station Whidbey Island|Whidbey Island NAS]] (1981−2010 normals)
|single line = Y
|Jan high F = 46.8
|Feb high F = 48.9
|Mar high F = 52.2
|Apr high F = 55.6
|May high F = 59.5
|Jun high F = 63.6
|Jul high F = 66.5
|Aug high F = 67.3
|Sep high F = 64.0
|Oct high F = 57.2
|Nov high F = 50.3
|Dec high F = 45.5
|year high F= 56.5
|Jan low F = 36.2
|Feb low F = 35.4
|Mar low F = 38.4
|Apr low F = 41.5
|May low F = 46.1
|Jun low F = 50.0
|Jul low F = 52.1
|Aug low F = 51.8
|Sep low F = 48.0
|Oct low F = 43.2
|Nov low F = 39.2
|Dec low F = 35.1
|year low F= 43.1
|precipitation colour = green
|Jan precipitation inch = 2.23
|Feb precipitation inch = 1.47
|Mar precipitation inch = 1.67
|Apr precipitation inch = 1.65
|May precipitation inch = 1.56
|Jun precipitation inch = 1.28
|Jul precipitation inch = 0.74
|Aug precipitation inch = 0.96
|Sep precipitation inch = 1.15
|Oct precipitation inch = 2.07
|Nov precipitation inch = 3.40
|Dec precipitation inch = 2.11
|year precipitation inch=20.29
|snow colour = green
|Jan snow inch = 0.9
|Feb snow inch = 1.5
|Mar snow inch = 0.1
|Apr snow inch = 0.1
|May snow inch = 0.0
|Jun snow inch = 0.0
|Jul snow inch = 0.0
|Aug snow inch = 0.0
|Sep snow inch = 0.0
|Oct snow inch = 0.0
|Nov snow inch = 0.9
|Dec snow inch = 1.7
|year snow inch =5.2
|unit precipitation days = 0.01 in
|Jan precipitation days = 16.4
|Feb precipitation days = 10.7
|Mar precipitation days = 11.5
|Apr precipitation days = 11.9
|May precipitation days = 10.0
|Jun precipitation days = 5.9
|Jul precipitation days = 3.7
|Aug precipitation days = 3.8
|Sep precipitation days = 4.1
|Oct precipitation days = 12.6
|Nov precipitation days = 20.7
|Dec precipitation days = 17.3
|year precipitation days=144.7
|unit snow days = 0.1 in
|Jan snow days = 1.0
|Feb snow days = 0.5
|Mar snow days = 0.1
|Apr snow days = 0.0
|May snow days = 0.0
|Jun snow days = 0.0
|Jul snow days = 0.0
|Aug snow days = 0.0
|Sep snow days = 0.0
|Oct snow days = 0.0
|Nov snow days = 0.4
|Dec snow days = 0.9
|year snow days =2.9
|source 1 = NOAA<ref name="NOAA txt">
{{cite web
| url = ftp://ftp.ncdc.noaa.gov/pub/data/normals/1981-2010/products/station/USW00024255.normals.txt
| publisher = National Oceanic and Atmospheric Administration
| title = WA Whidbey Island NAS
| accessdate = 28 May 2014}}</ref>
|date=May 2014
}}

==Ecology==

===Flora===
Vegetation varies greatly from one end of the island to the other. Vegetation in the south is more similar to that of mainland Washington. The principal trees are [[Douglas fir]], [[red alder]], [[bigleaf maple]], [[western red cedar]], and [[tsuga|western hemlock]].<ref name="McCreary1975">{{cite book|author=Fred R. McCreary|title=Soil Survey of Jefferson County Area, Washington|url=https://books.google.com/books?id=EC5xchy3HEMC&pg=PA4|year=1975|publisher=U.S. Department of Agriculture, Soil Conservation Service|pages=4–}}</ref> Compared to the rest of western Washington state, [[vine maple]] is notably absent, except where they have been planted. Other under-story plants include the evergreen huckleberry, lower longleaf [[Oregon grape]], [[elderberry]], [[salal]], [[oceanspray]], and varieties of [[Urtica|nettle]]. Non-native introduced plants such as [[foxglove]], [[ivy]] and [[holly]] are also evident.<ref>{{cite web |title=All About Whidbey Island |url=https://whidbeyisland.us/all-about-whidbey-island/ |website=Whidbey Island |accessdate=14 January 2019}}</ref>

Farther up the island, however, the shorter Oregon-Grape and the blue Evergreen Huckleberry is seen less, while tall Oregon-grape and Red Huckleberry predominate. The native [[Pacific rhododendron]] is much more visible. Amongst the [[deciduous]] varieties, [[Garry oak]] (from which Oak Harbor takes its name) are seen more frequently in the northern portion of the island and [[arbutus|Pacific madrone]] is also notably present.<ref>{{cite web |title=Preservation |url=https://ohgarryoaksociety.org/preservation/ |website=Oak Harbor Garry Oak Society |accessdate=14 January 2019}}</ref> In the [[conifer]] classification, [[grand fir]] is found more in the northern part of Whidbey Island along with [[Sitka spruce]] and [[Lodgepole Pine|shore pine]]. There are three open prairie areas on Whidbey Island – Smith Prairie, Crockett Prairie and Ebey Prairie.<ref>{{cite book|title=Ebey's Landing National Historical Reserve, General Management Plan: Environmental Impact Statement|url=https://books.google.com/books?id=5TM3AQAAMAAJ&pg=PA89|year=2006|pages=89–90}}</ref> Some patches of [[Opuntia|prickly pear cactus]] are found along the slopes near Partridge Point.<ref name="Mass2015">{{cite book|author=Clifford Mass|title=The Weather of the Pacific Northwest|url=https://books.google.com/books?id=xPiACgAAQBAJ&pg=PA194|date=1 September 2015|publisher=University of Washington Press|isbn=978-0-295-99836-7|pages=194–}}</ref>

===Fauna===
[[Gray whale]]s migrate between Whidbey and Camano Islands during March and April and can be seen from both ship and shore.<ref>{{Cite news|url=https://whidbeycamanoislands.com/things-to-do/beaches-and-waterways/whale-watching/|title=Whale Watching around Whidbey and Camano Islands.|work=Whidbey and Camano Islands|access-date=2018-03-08|language=en-US}}</ref> [[Orca]] also make use of the waters surrounding Whidbey Island.

[[Clams]] and [[oysters]] are abundant locally and may be harvested from some public beaches.<ref>{{cite web |url=http://wdfw.wa.gov/fishing/shellfish/beaches/MapArea/08/ |title=Map data|website=wdfw.wa.gov}}</ref> The Washington State Department of Health provides an online guide to assist in identifying shellfish varieties as well as providing guidance about where to find specific varieties.<ref>{{cite web|url=http://www.doh.wa.gov/Portals/1/Documents/4400/332-087-Shellfish-ID.pdf|title=Shellfish Identification :: Washington State Department of Health|website=www.doh.wa.gov}}</ref>

According to the Whidbey Audubon Society, Approximately 230 [[bird]] species are reported to take advantage of the diverse habitats on the island. <ref>{{cite web |url=http://www.whidbeyaudubon.org/birdlist/birdlist.htm |title=Birds of Whidbey Island|website=www.whidbeyaudubon.org}}</ref>

== Education ==

=== Public school districts ===
Whidbey Island is served by three public [[school district]]s.

[[Oak Harbor School District]] operates in [[Oak Harbor, Washington|Oak Harbor]]. Within the district, there is one [[Oak Harbor High School (Washington)|high school]], one alternative high school, two middle schools, and five elementary schools. Within the [[Washington Interscholastic Activities Association]], Oak Harbor High is listed as a 3-A school.

[[Coupeville School District]] operates in [[Coupeville, Washington]] and [[Greenbank, Washington]]. Within the district, there is one high school, one middle school, and one elementary school. Within the [[Washington Interscholastic Activities Association]], Coupeville High is listed as a 1-A school.

[[South Whidbey School District]] serves the southern end of the island, including [[Freeland, Washington|Freeland]], [[Langley, Washington|Langley]], and [[Clinton, Washington|Clinton]]. Within the district, there is one high school (grades 9–12), one alternative school (grades K–12), one middle school (grades 5-8) split between 2 campuses, and one elementary school (grades K–4). Within the Washington Interscholastic Activities Association, South Whidbey High is listed as a 1-A school.

=== Colleges ===

[[Skagit Valley College]] has a campus located in Oak Harbor, and a limited service campus in South Whidbey.

[[Seattle Pacific University]] owns Camp Casey, a retreat center near [[Coupeville, Washington|Coupeville]], which was once the barracks for the adjacent [[Fort Casey State Park|Fort Casey]].

==Notable people==
===Actors===
*[[Lana Condor]] known for her role in To All The Boys I've Loved Before
===Politicians===
*[[Jack Metcalf]] (1927-2007), [[United States House of Representatives]], grew up on Whidbey Island in the 1930s.<ref>[http://seattletimes.nwsource.com/html/localnews/2003620723_metcalf16m.html Jack Metcalf obituary] {{webarchive|url=https://web.archive.org/web/20070319204602/http://seattletimes.nwsource.com/html/localnews/2003620723_metcalf16m.html |date=March 19, 2007 }}, ''Seattle Times'', March 16, 2007</ref>

===Writers and artists===

*[[Shayla Beesley]], actress, grew up in Oak Harbor
*[[Juliet Winters Carpenter]], prize-winning translator of Japanese literature and author
*[[Aleah Chapin]], painter, grew up on Whidbey Island
*[[Pete Dexter]], National Book Award fiction writer
*[[Elizabeth George]], mystery author.
*[[Aaron Parks]], jazz pianist.
*[[David Ossman]], founder of [[Firesign Theater]]
*[[Nancy Horan]], best selling author
*[[Jeff Alexander]], conductor and arranger
*Eli Moore and Ashley Eriksson of [[Lake (American band)|Lake]]
*[[David Whyte (poet)|David Whyte]], poet

===Other===
*[[Robert Jay Mathews]], American neo-Nazi terrorist and the leader of [[The Order (white supremacist group)]], an American white supremacist militant group, died on Whidbey Island during a shoot-out with federal law enforcement agents.
*[[Bruce Bochte]], American baseball player. Bochte lived on Whidbey Island for over three years after his baseball playing days were over.<ref>{{cite web|url=http://www.seattlepi.com/allstar/30690_bochte09.shtml|title=Bochte has moved a long way from baseball|website=seattlepi.com}}</ref>
*[[Marti Malloy]], London 2012 Olympic Bronze medalist, Judo women's 57 kg
*Mark Sargent, flat farther

==Infrastructure==

===Transportation===
[[File:ClintonFerry.jpg|thumb|right|Ferry at Clinton]] The only bridge that reaches Whidbey Island is the [[Deception Pass Bridge]], [[Washington State Route 20|State Route 20]], which connects the north end of Whidbey to the mainland via [[Fidalgo Island]]. Prior to the completion of the bridge in 1935, Whidbey Island was linked to [[Fidalgo Island]] by the [[Deception Pass ferry]], which ran from 1924 to 1935. Modern ferry service is available via State Route 20 on the [[Coupeville, Washington|Coupeville]] to [[Port Townsend, Washington|Port Townsend]] ferry, and via [[Washington State Route 525|State Route 525]] on the [[Clinton, Washington|Clinton]] to [[Mukilteo, Washington|Mukilteo]] ferry service on the southern east coast.

Travel on the island involves use of an extensive county road system, or city infrastructure depending on location, all of which act as feeders to the two state highways [[Washington State Route 525|State Route 525]] and [[Washington State Route 20|State Route 20]].

Whidbey Island's State Routes [[State Route 525 (Washington)|525]]/[[State Route 20 (Washington)|20]] is the only nationally designated Scenic Byway on an island. It is appropriately named the "Whidbey Island Scenic Isle Way."

[[Mass transit|Public transportation]] is provided by [[Island Transit (Washington)|Island Transit]], which provides a [[zero-fare]] bus service paid for by a 6/10th of 1% sales tax within the county. There are currently 11 bus routes serving Whidbey Island. No service is available on Sundays or major holidays.

Two public airports provide service to Whidbey Island. Whidbey Air Park is located {{convert|2|mi|km}} southwest of [[Langley, Washington|Langley]] with a {{convert|2470|ft|m}} long runway. [[Wes Lupien Airport]] is located {{convert|3|mi|km}} southwest of [[Oak Harbor, Washington|Oak Harbor]] with a {{convert|3265|ft|m|abbr=on}} long runway. In addition, there are approximately half dozen private dirt strips on the island{{citation needed|date=February 2017}}. [[Kenmore Air Express]] ran a scheduled airline service to Whidbey Island serving the Oak Harbor airport from 2006 to 2009.<ref>{{Cite web |url=http://www.kenmoreair.com/content.php?content_id=261 |title=Archived copy |access-date=August 4, 2009 |archive-url=https://web.archive.org/web/20081007031605/http://www.kenmoreair.com/content.php?content_id=261 |archive-date=October 7, 2008 |dead-url=yes |df=mdy-all }}</ref>

The [[United States Navy]] operates two airports on Whidbey Island. The largest is a two-runway airport located at [[Whidbey Island Naval Air Station]] north of [[Oak Harbor, Washington|Oak Harbor]]. In addition, the Navy also operates a flight training facility named [[Coupeville Naval Outlying Field|Coupeville Outlying Landing Field (Coupeville OLF)]] located just southeast of [[Coupeville, Washington|Coupeville]]. The Navy named [[USS Whidbey Island (LSD-41)]] in honor of the island.

=== Health systems ===

[[Whidbey Health]] is the regional, county-run hospital. Located in Coupeville, the hospital has an extension clinic in Oak Harbor. The Naval Air Station in Oak Harbor has a limited service hospital for military personnel, veteran retirees, and their dependents.

==Communities ==
North to south:
*[[Deception Pass]]
*[[Oak Harbor, Washington|Oak Harbor]] - Largest city
*[[West Beach, Washington|West Beach]]
*[[San De Fuca, Washington|San De Fuca]]
*[[Coupeville, Washington|Coupeville]] – County Seat
*[[Keystone, Island County, Washington|Keystone]]
*[[Admiral's Cove, Washington|Admiral's Cove]]
*[[Lagoon Point, Washington|Lagoon Point]]
*[[Greenbank, Washington|Greenbank]]
*[[Langley, Washington|Langley]]
*[[Freeland, Washington|Freeland]]
*[[Bayview, Island County, Washington|Bayview]]
*[[Clinton, Washington|Clinton]]
*[[Maxwelton, Washington|Maxwelton]]
*[[Glendale, Washington|Glendale]]

==See also==
*[[Admiralty Head Lighthouse]]
*[[Meerkerk Rhododendron Gardens]]

==References==
{{reflist|30em}}

==External links==
{{commons category}}
{{wikivoyage|Whidbey Island}}
*[http://www.WhidbeyCamanoIslands.com Whidbey Island & Camano Island Official Tourism Website]
*[http://content.lib.washington.edu/vanolindaweb/index.html University of Washington Libraries Digital Collections – Oliver S. Van Olinda Photographs] A collection of 420 photographs depicting life on Vashon Island, Whidbey Island, Seattle and other communities of Washington State's Puget Sound from the 1880s to the 1930s.
{{Washington}}

[[Category:Landforms of Island County, Washington]]
[[Category:Islands of Washington (state)]]
[[Category:Islands of Puget Sound]]

Whidbey Island

2019-03-08T07:02:42Z

Crasshopper: /* Other */

{{About|the island||Whidbey (disambiguation){{!}}Whidbey}}
{{Use mdy dates|date=October 2012}}
{{Infobox islands
| name = Whidbey Island
| image_name = WhidbeyIsland04.jpg
| image_caption = Map of Whidbey Island
| image_size = 220
| map = USA Washington
| coordinates = {{coord|48.136389|-122.5825|scale:500000_region:US-WA|display=it}}
| map_caption = Whidbey Island (Washington)
| native_name =
| native_name_link = Captain Joseph Whidbey
| nickname = "The Rock"<ref>{{cite web | url=http://www.whidbeynewstimes.com/news/22086709.html | title=Whidbey Island has a terrain that's set in stone | newspaper=[[Whidbey News-Times]] | date=July 3, 2008 | accessdate=February 10, 2012 | archive-url=https://web.archive.org/web/20150919193429/http://www.whidbeynewstimes.com/news/22086709.html | archive-date=September 19, 2015 | dead-url=yes | df=mdy-all }}</ref>
| location = [[Puget Sound]]
| archipelago =
| total_islands =
| major_islands =
| area_sqmi = 168.67
| length_mi = 37| width_mi = 10
| highest_mount =
| elevation_ft =
| country = United States
| country_admin_divisions_title = State
| country_admin_divisions = [[Washington (state)|Washington]]
| country_admin_divisions_title_1 = County
| country_admin_divisions_1 = [[Island County, Washington|Island County]]
| country_largest_city = [[Oak Harbor, Washington|Oak Harbor]]
| country_largest_city_population = 23,204 <ref>United States Census Bureau</ref>
| population = 80,022
| population_as_of =
| density_km2 = 133.25
| ethnic_groups =
| additional_info =
}}

[[File:Whidbey 2.JPG|thumb|Cultus Bay at Low Tide]] [[File:Whidbey Island.jpg|thumb|Double Bluff, with Useless Bay to the South (right) and Mutiny Bay to the North (left)]]
'''Whidbey Island''' (historical spellings '''Whidby''', '''Whitbey''',<ref>{{cite gnis|id=1509451|name=Whidbey Island}}</ref> or '''Whitby''') is the largest of the islands composing [[Island County, Washington|Island County]], [[Washington (state)|Washington]], in the United States. (The other large island is [[Camano Island]], east of Whidbey.) Whidbey is about {{convert|30|mi|km}} north of [[Seattle]], and lies between the [[Olympic Peninsula]] and the [[Interstate 5 (Washington)|I-5]] corridor of western Washington. The island forms the northern boundary of [[Puget Sound]]. It is home to [[Naval Air Station Whidbey Island]].

Whidbey Island is home to 80,022 residents (according to the [[United States Census, 2000|2000 census]]).<ref>[http://factfinder.census.gov/servlet/DTTable?_bm=y&-show_geoid=Y&-tree_id=4001&-_caller=geoselect&-context=dt&-errMsg=&-all_geo_types=N&-mt_name=DEC_2000_SF1_U_P001&-redoLog=true&-transpose=N&-search_map_config=|b=50|l=en|t=4001|zf=0.0|ms=sel_00dec|dw=0.14742116997381507|dh=0.0756441033284385|dt=gov.census.aff.domain.map.EnglishMapExtent|if=gif|cx=-90.71827740294499|cy=46.747841207516664|zl=4|pz=4|bo=318:317:316:314:313:323:319|bl=362:393:358:357:356:355:354|ft=350:349:335:389:388:332:331|fl=381:403:204:380:369:379:368|g=01000US&-PANEL_ID=p_dt_geo_map&-_lang=en&-geo_id=05000US53029&-geo_id=06000US5302990400&-geo_id=06000US5302990528&-geo_id=06000US5302992256&-geo_id=06000US5302993168&-CONTEXT=dt&-format=&-search_results=ALL&-ds_name=DEC_2000_SF1_U Island County, Washington and its subdivisions] United States Census Bureau</ref> An estimated 29,000 of Whidbey Island residents live in rural locations.

Whidbey Island is approximately {{convert|55|mi|km}} long (if measured along roads traveled from the extreme north to extreme south), or about {{convert|37|mi|km}} when measured along a straight line from north to south, and {{convert|1.5|to|10|mi|km}} wide, with a total land area of {{convert|168.67|sqmi|km2}},<ref>{{cite web|url=http://www.islandcounty.net/Assessor/|title=Island County Assessor|work=Island County Assessor|accessdate=February 16, 2015}}</ref> making it the [[List of islands of the United States by area|40th largest island in the United States]]. It is ranked as the fourth longest and fourth largest island in the contiguous United States, behind [[Long Island]], [[New York (state)|New York]];<ref name="peninsula">{{cite news |url=https://query.nytimes.com/gst/fullpage.html?res=9A07E3DB133FF932A15752C1A9629C8B63&sec=&spon=&pagewanted=all |title= Long Island at its Best; Who's the Longest of Them All? |accessdate=2008-10-16 |author= John Burbidge|date=November 21, 2004 |work=The New York Times}}</ref><ref>{{cite web|url=http://www.peakbagger.com/PBGeog/longisl.aspx|title=The Longest Islands in the United States - Peakbagger.com|website=www.peakbagger.com}}</ref> [[Padre Island]], Texas (the world's longest barrier island);<ref>{{cite web|url=https://www.nps.gov/pais/index.htm|title=Padre Island National Seashore (U.S. National Park Service)|website=www.nps.gov}}</ref> and [[Isle Royale]], [[Michigan]].<ref>{{cite web|url=https://www.nationalgeographic.com/travel/national-parks/isle-royale-national-park/|title=Isle Royale National Park - National Geographic|date=November 5, 2009|website=nationalgeographic.com}}</ref> In the state of Washington, it is the largest island, followed by [[Orcas Island]].

== History ==
Whidbey Island was inhabited by members of the [[Lower Skagit (tribe)|Lower Skagit]], [[Swinomish (tribe)|Swinomish]], [[Suquamish]], [[Snohomish (tribe)|Snohomish]] and other Native American tribes. The [[Salishan languages|Salishan]] name for the island was Tscha-kole-chy.<ref name="historylink.org3">{{cite web|url=http://www.historylink.org/File/7523|title=Island County -- Thumbnail History|website=historylink.org}}</ref> These were peaceful groups who lived off the sea and land, with fishing, harvesting nuts, berries and roots, which they preserved over the winter.<ref>{{cite web|url=http://whidbeyhistory.historywiz.org/tschakolecy.htm|title=Tschakolecy - Whidbey Island History: Tschakolecy|website=whidbeyhistory.historywiz.org}}</ref>

The first known European sighting of Whidbey Island was during the 1790 Spanish expedition of [[Manuel Quimper]] and [[Gonzalo López de Haro]] on the ''[[Princesa Real (sloop)|Princesa Real]]''.<ref>{{cite book |last= Hayes |first= Derek |title= Historical Atlas of the Pacific Northwest: Maps of exploration and Discovery |year= 1999 |publisher= Sasquatch Books |isbn= 1-57061-215-3 |pages= 70–71}}</ref>

Captain [[George Vancouver]] fully explored the island in 1792. In May of that year, [[Royal Navy]] officers and members of Vancouver's expedition, [[Joseph Whidbey]] (master of H. M. S. Discovery) and [[Peter Puget]] (a lieutenant on the ship), began to map and explore the areas of what would later be named [[Puget Sound]]. After Whidbey circumnavigated the island in June 1792, Vancouver named the island in his honor. By that time, Vancouver had claimed the area for Britain.<ref>{{Cite web |url=http://www.visitwhidbey.com/information/History-Images.html |title=Archived copy |access-date=August 23, 2017 |archive-url=https://web.archive.org/web/20170818030053/http://www.visitwhidbey.com/information/History-Images.html |archive-date=August 18, 2017 |dead-url=yes |df=mdy-all }}</ref>
<ref>{{cite web|url=http://www.historylink.org/File/5060|title=Joseph Whidbey circumnavigates Whidbey Island in June 1792. - HistoryLink.org|website=www.historylink.org}}</ref> On 4 June 1792, the King’s Birthday, near Possession Point at the southern end of Whidbey Island, Vancouver took formal possession of all the coast and hinterland contiguous to the Strait of Juan de Fuca, including Puget Sound, under the name of [[New Georgia]].<ref>W. Kaye Lamb (ed.), ''The Voyage of George Vancouver, 1791-1795,'' London, Hakluyt Society, 1984, Vol.1, p.569; also Freeman M. Tovell, ‘The Other Side of the Coin: the Viceroy, Bodega y Quadra, Vancouver, and the Nootka Crisis’, ''BC Studies,'' no.93, 1992, p.19.</ref>

The first known overnight stay by a non-Native American was made on May 26, 1840 by a Catholic missionary, Father [[François Norbert Blanchet]], during travel across Puget Sound. He had been invited by Chief Tslalakum.<ref name="historylink.org1">{{cite web|url=http://www.historylink.org/File/5258|title=Father Francis (or Francois) N. Blanchet visits Whidbey Island on May 26, 1840. - HistoryLink.org|website=www.historylink.org}}</ref> Blanchet remained on the island for nearly a year and guided the inhabitants in building a new log church.<ref>{{cite web|url=https://www.sos.wa.gov/legacy/timeline/detail.aspx|title=Legacy Washington - WA Secretary of State|first1=Contact Us Washington Secretary of StateLegacy WashingtonLegislative Building · PO Box|last1=40220Olympia|first2=WA 98504-0220Phone Numbers Privacy|last2=Policy|website=wa.gov}}</ref><ref name="historylink.org1"/>

Lieutenant [[Charles Wilkes]], commander of the [[United States Exploring Expedition]] of 1838–1842, sailed the [[USS Vincennes (1826)|USS ''Vincennes'']] into Penn Cove in 1841. By that time, the log church was already being built by the Native Americans beside a huge wooden cross (24 feet long) that they had erected. Wilkes ordered his men to use no force except in self-defense when dealing with the "savage and treacherous inhabitants". In fact, he encountered few problems with the indigenous people who had already been poorly treated by visitors and suffered from diseases they had introduced.<ref name="historylink.org2">{{cite web|url=http://www.historylink.org/File/5226|title=Wilkes, Charles (1798-1877)|website=historylink.org}}</ref>

Wilkes named the lower cove Holmes Harbor, after his assistant surgeon, Silas Holmes. During this time he charted Puget Sound.<ref name="historylink.org2"/> Other sites in the area that were given names by Wilkes included Maury Island (Vashon), Hammersley Inlet, Totten and Budd Inlets, Agate Passage between the Kitsap Peninsula, Hale Passage and Dana Passage.<ref name="historylink.org2"/>

Thomas W. Glasgow filed the first land claim on Whidbey Island in 1848, attempting to become the first settler. He built a small cabin near Penn Cove, planted some crops and married a local lady, Julia Pat-Ke-Nim.<ref name="hometownchronicles.com"/> Glasgow left in August of that year however, having been forced out by the local inhabitants.<ref name="historylink.org3" />
Colonel [[Isaac N. Ebey]] arrived from Columbus, Ohio, in 1850 and became the first permanent white settler, claiming a square mile (2.6 km²) of prairie with a southern shoreline on [[Admiralty Inlet]]. He took advantage of the 640 acres offered free of charge to each married couple, the first to do so, on October 15, 1850. In the fall of 1851, his children, his wife, three of her brothers and the Samuel Crockett family arrived to join Ebey.<ref name="hometownchronicles.com">{{Cite web |url=https://hometownchronicles.com/wa/island/cohist.html |title=Archived copy |access-date=August 23, 2017 |archive-url=https://web.archive.org/web/20170425233906/http://hometownchronicles.com/wa/island/cohist.html |archive-date=April 25, 2017 |dead-url=yes |df=mdy-all }}</ref>
In addition to farming potatoes and wheat, Eby was also the postmaster for [[Port Townsend, Washington]] and rowed a boat daily across the inlet in order to work at the post office there. Colonel Ebey also served as a representative in the Oregon Territory Legislative Assembly, as Island County's first Justice of the Peace, as a probate judge and as Collector of Customs for the Puget Sound District.<ref name="historylink.org3"/>

On August 11, 1857, at age 39, Colonel Ebey was murdered and beheaded by Native Americans, said to be [[Haida people|Haida]] who had traveled to this area from the [[Queen Charlotte Islands]]. Some sources however, refer to his killers as "Russian Indians called Kakes or Kikans, [from] Kufrinoff Island, near the head of Prince Frederick's Sound.<ref name="hometownchronicles.com"/> Ebey was slain in proxy-retaliation for the killing of a Haida chief or Tyee and 27 other indigenous people at [[Port Gamble]]. [[Fort Ebey]], named for the Colonel, was established in 1942 on the west side of the central part of the island, just northwest of [[Coupeville, Washington|Coupeville]].<ref name="historylink.org3"/>
[[File:Ebey's Landing N.H. Reserve.jpg|thumb|
On the Bluff Trail in Ebey's Landing National Historical Reserve
]]
[[Admiralty Head Lighthouse]] is located in this area, on the grounds of [[Fort Casey State Park]]. The area around Coupeville is the federally protected [[Ebey's Landing National Historical Reserve]], named in honor of Isaac Ebey.

In December 1984, the island was the site of a violent encounter between law enforcement and [[white nationalist]] and [[organized crime]] leader [[Robert Jay Mathews]] of the group The Order. A large shootout occurred between Mathews and FBI agents in which Mathews was killed during a house fire. Mathews' followers have since gathered on the island at the location where he was killed by FBI agents on the anniversary of his death to commemorate it.<ref>{{cite web|url=http://www.historylink.org/File/7921|title=Robert Jay Mathews, founder of the white-supremacist group The Order, is killed during an FBI siege on Whidbey Island on December 8, 1984.|website=historylink.org}}</ref>

== Government ==
Whidbey Island, along with [[Camano Island]], Ben Ure Island and six uninhabited islands, comprises [[Island County, Washington]]. The county seat is located in the town of [[Coupeville, Washington|Coupeville]] on Whidbey Island.
[[File:Swantown.jpg|thumb|
Looking east over Swantown Lake
]]
Population centers of Whidbey Island include the [[Oak Harbor, Washington|City of Oak Harbor]], the [[Coupeville, Washington|Town of Coupeville]], the [[Langley, Washington|City of Langley]], the [[Freeland, Washington|Village of Freeland]], the [[Greenbank, Washington|Community of Greenbank]], the [[Clinton, Washington|Village of Clinton]] and the [[Bayview, Island County, Washington|Community of Bayview]]. Only Oak Harbor, Coupeville and Langley are incorporated, the others (with the exception of Greenbank and Bayview) are all [[Census-designated place]]s, and all but Bayview have their own post offices and ZIP codes.

== Economy ==
[[File:DeceptionPass Bridge.jpg|thumb|
Deception Pass Bridge
]]
Whidbey Island is divided economically into two different regions: the northern end of the island (encompassing Oak Harbor and [[Naval Air Station Whidbey Island|Whidbey Island Naval Air Station]]), and the remainder of the island (encompassing Coupeville, Greenbank, Freeland, Langley, Clinton and the smaller communities in-between).

The economy of the northern end of Whidbey Island is strongly influenced by the presence of Whidbey Island Naval Air Station near [[Oak Harbor, Washington|Oak Harbor]] (N.A.S. Whidbey). N.A.S. Whidbey is Oak Harbor's largest employer; thus, Oak Harbor has a predominantly service-based economy and several national chain stores have been attracted to the Oak Harbor area.

The economy of Whidbey Island south of Oak Harbor relies heavily on tourism, small-scale agriculture, and the arts.

Tourism is especially important for both Whidbey and Camano Islands. On Whidbey, tourists find a wide range of amenities in the towns of Oak Harbor, Coupeville, Freeland and Langley. Coupeville's Penn Cove Mussel Farm exports large quantities of its highly renowned Penn Cove [[Mussel]]s. This aquaculture facility, along with a number of small farms, reflects the rural agricultural nature of most of central Whidbey Island. Many of these small farms host farm stands onsite, where customers may buy produce, flowers, meat, eggs and other locally raised products directly from the farmers.<ref>{{Cite web |url=http://goosefoot.org/pdf/farmstands.pdf |title=Archived copy |access-date=January 19, 2016 |archive-url=https://web.archive.org/web/20150701204046/http://www.goosefoot.org/pdf/farmstands.pdf |archive-date=July 1, 2015 |dead-url=yes |df=mdy-all }}</ref>

Often referred to as Puget Sound's Largest Artist's Colony, Whidbey is home to numerous working artists, writers, and performers. These include many well-known painters, sculptors, glass artists, wood workers, metal workers, mixed media artists, photographers, authors, poets, actors, and musicians.

In addition to being a haven for artists, the southern end of Whidbey Island also serves as a minor bedroom community for the nearby cities of [[Everett, Washington|Everett]], where the [[Boeing Everett Factory]] is located, and [[Seattle]]. Commuters to and from those areas use the [[Washington State Ferries]] system's run between [[Clinton, Washington|Clinton]] and [[Mukilteo, Washington|Mukilteo]].
[[File:ClintonSign.jpg|thumb|Carved sign welcoming visitors as they arrive by ferry at Clinton.]]

== Geography ==
Whidbey Island is often claimed to be the longest island in the continental United States (or another similar claim), but according to the [[Seattle Times]] it cannot be correctly considered so.<ref name ="DontStretchST">{{cite news|title=Whidbey is long, but let's not stretch it|url=http://community.seattletimes.nwsource.com/archive/?date=20000105&slug=A20000106010141|date=January 5, 2000|accessdate=August 25, 2011|first1=Steve|last1=Johnston|work=The Seattle Times}}</ref> Whidbey Island has four lakes that are part of its interior hydrology: Cranberry Lake (inside Deception Pass State Park), Deer Lake, Goss Lake and Lone Lake (both near the town of Langley).<ref>{{cite web|title=Lakes Monitored by Ecology's Lake Water Quality Monitoring Program From 1989 through 1997|url=http://www.ecy.wa.gov/programs/eap/lakes/wq/lake_assessments.html|accessdate=August 25, 2011|publisher=Department of Ecology, State of Washington}}</ref>

==Parks and reserve areas==
Whidbey Island contains [[Ebey's Landing National Historical Reserve]], the first national historic reserve in the US created by the [[National Park Service]] to preserve the rural history and culture of the island and to protect the area's rare and sensitive plants.

[[Washington State Park System|Washington State Parks]] located on the island include [[Deception Pass State Park]] (the most visited state park in Washington), [[Joseph Whidbey State Park]], [[Fort Ebey State Park]], [[Fort Casey State Park]], [[Possession Point State Park]], and [[South Whidbey Island State Park]]. There is also a series of county operated parks throughout the Island.

[[Earth Sanctuary]] is a nature reserve, sculpture garden and retreat center on Whidbey Island. The ponds and bog fen complex have been designated as a "habitat of local importance" by the Whidbey Audubon Society and Island County Critical Areas program.<ref>{{cite web|url=http://www.southwhidbeyrecord.com/lifestyle/35014484.html|title=ISLAND BIRDING - Islanders should speak up now to protect important bird habitat areas|work=South Whidbey Record|accessdate=February 16, 2015}}</ref><ref>{{cite web|url=http://www.southwhidbeyrecord.com/news/21571814.html|title=Newman Ponds to become 'Earth Sanctuary'|work=South Whidbey Record|accessdate=February 16, 2015}}</ref>

==Festivals==
Whidbey Island hosts many festivals and celebrations throughout the year.
*Whidbey Island Area Fair ("Island County Fair" until 2012<ref>{{cite web|url=http://www.southwhidbeyrecord.com/news/149304495.html|title=Welcome to the new Whidbey Island Fair - South Whidbey Record|date=April 27, 2012|website=southwhidbeyrecord.com}}</ref>), on the third weekend of July, includes rides, food, and animal shows.
*Wag'n'Walk, which takes place towards the end of August, is Western Washington's premier celebration of dogs and things dog-related. It includes vendors, games, competition, demonstrations and the Wag'n'Walk itself.
*Whidbey Island Kite Festival, in September
* Langley's Mystery Weekend in March or February. For the weekend the Town of Langley turns into the setting of a fictional murder mystery.
* Penn Cove Mussel Festival, in March, celebrates the bounty of the sea, especially the mussel.
*Loganberry Festival at the Greenbank Farm in July (This was discontinued after the 2016 festival.)
*Maxwelton Beach Fourth of July Parade and fireworks show, which takes place at the southern end of Maxwelton Road at Dave Mackie Park. After the parade, there are events for all ages, including three-legged races, divided into age groups, and the most popular event, the [[egg toss]].
*Choochokam is the annual street fair and arts festival, held in downtown Langley during the second weekend of July, detailed schedules and other information is generally available on the festival website. (This was discontinued after the 2016 festival.)
*Tour de Whidbey, in September, is a bike race spanning the length of Whidbey Island.
*The Whidbey Island Marathon and Half Marathon, in April since 2002.<ref>{{Cite web |url=http://www.whidbeyislandmarathon.com/ |title=Archived copy |access-date=February 25, 2008 |archive-url=https://web.archive.org/web/20021123192533/http://www.whidbeyislandmarathon.com/ |archive-date=November 23, 2002 |dead-url=yes |df=mdy-all }}</ref>
*Whidbey Island Race Week: a week-long sailing regatta every summer based out of Oak Harbor with daily racing in Penn Cove and/or Saratoga Passage (depending on wind conditions). Usually held third week of July, varies slightly due to tidal conditions.
*Whidbey Island Highland Games - 2nd Saturday in August. Competitions in Scottish Heavy Athletics, Highland Dancing, Pipe and drum bands.
*Whidbey Island Zucchini Festival - An annual festival hosted by residents of Whidbey island brought about by an excess of home-grown zucchini. The festival includes zucchini based musical performances, various types of zucchini based or themed visual art, all types of foods that feature zucchini, outdoor games and competitions using zucchini, and a giant zucchini slingshot.
*Oak Harbor Music Festival - An annual music festival held in the biggest city on the island, Oak Harbor. It is held over Labor Day Weekend, and consists of a wide variety of musical acts.
*DjangoFestNW - An annual 5-day music festival held in mid-September that celebrates the music of [[Django Reinhardt]] at Whidbey Island Center for the Arts<ref>{{cite web|url=https://www.djangofestnw.com/|title=Welcome|website=DjangofestNW.com}}</ref>

==Climate==
[[File:Fort Casey Cliff.jpg|right|thumb|A cliff on Whidbey Island near [[Fort Casey]]]]
Whidbey Island lies partially in the [[rain shadow]] of the [[Olympic Mountains|Olympic Mountain Range]] to the west, and has a variety of climate zones. This can be observed by rainfall amounts – wettest in the south with average rainfall of {{convert|36|in|mm}}, driest in the central district of [[Coupeville, Washington|Coupeville]] with average rainfall of {{convert|20|to|22|in|mm}}, and turning moister again farther north with average rainfall of {{convert|32|in|mm}}. [[Microclimate]]s abound, determined by proximity to water, elevation and prevailing winds.

{{Weather box
|location = [[Naval Air Station Whidbey Island|Whidbey Island NAS]] (1981−2010 normals)
|single line = Y
|Jan high F = 46.8
|Feb high F = 48.9
|Mar high F = 52.2
|Apr high F = 55.6
|May high F = 59.5
|Jun high F = 63.6
|Jul high F = 66.5
|Aug high F = 67.3
|Sep high F = 64.0
|Oct high F = 57.2
|Nov high F = 50.3
|Dec high F = 45.5
|year high F= 56.5
|Jan low F = 36.2
|Feb low F = 35.4
|Mar low F = 38.4
|Apr low F = 41.5
|May low F = 46.1
|Jun low F = 50.0
|Jul low F = 52.1
|Aug low F = 51.8
|Sep low F = 48.0
|Oct low F = 43.2
|Nov low F = 39.2
|Dec low F = 35.1
|year low F= 43.1
|precipitation colour = green
|Jan precipitation inch = 2.23
|Feb precipitation inch = 1.47
|Mar precipitation inch = 1.67
|Apr precipitation inch = 1.65
|May precipitation inch = 1.56
|Jun precipitation inch = 1.28
|Jul precipitation inch = 0.74
|Aug precipitation inch = 0.96
|Sep precipitation inch = 1.15
|Oct precipitation inch = 2.07
|Nov precipitation inch = 3.40
|Dec precipitation inch = 2.11
|year precipitation inch=20.29
|snow colour = green
|Jan snow inch = 0.9
|Feb snow inch = 1.5
|Mar snow inch = 0.1
|Apr snow inch = 0.1
|May snow inch = 0.0
|Jun snow inch = 0.0
|Jul snow inch = 0.0
|Aug snow inch = 0.0
|Sep snow inch = 0.0
|Oct snow inch = 0.0
|Nov snow inch = 0.9
|Dec snow inch = 1.7
|year snow inch =5.2
|unit precipitation days = 0.01 in
|Jan precipitation days = 16.4
|Feb precipitation days = 10.7
|Mar precipitation days = 11.5
|Apr precipitation days = 11.9
|May precipitation days = 10.0
|Jun precipitation days = 5.9
|Jul precipitation days = 3.7
|Aug precipitation days = 3.8
|Sep precipitation days = 4.1
|Oct precipitation days = 12.6
|Nov precipitation days = 20.7
|Dec precipitation days = 17.3
|year precipitation days=144.7
|unit snow days = 0.1 in
|Jan snow days = 1.0
|Feb snow days = 0.5
|Mar snow days = 0.1
|Apr snow days = 0.0
|May snow days = 0.0
|Jun snow days = 0.0
|Jul snow days = 0.0
|Aug snow days = 0.0
|Sep snow days = 0.0
|Oct snow days = 0.0
|Nov snow days = 0.4
|Dec snow days = 0.9
|year snow days =2.9
|source 1 = NOAA<ref name="NOAA txt">
{{cite web
| url = ftp://ftp.ncdc.noaa.gov/pub/data/normals/1981-2010/products/station/USW00024255.normals.txt
| publisher = National Oceanic and Atmospheric Administration
| title = WA Whidbey Island NAS
| accessdate = 28 May 2014}}</ref>
|date=May 2014
}}

==Ecology==

===Flora===
Vegetation varies greatly from one end of the island to the other. Vegetation in the south is more similar to that of mainland Washington. The principal trees are [[Douglas fir]], [[red alder]], [[bigleaf maple]], [[western red cedar]], and [[tsuga|western hemlock]].<ref name="McCreary1975">{{cite book|author=Fred R. McCreary|title=Soil Survey of Jefferson County Area, Washington|url=https://books.google.com/books?id=EC5xchy3HEMC&pg=PA4|year=1975|publisher=U.S. Department of Agriculture, Soil Conservation Service|pages=4–}}</ref> Compared to the rest of western Washington state, [[vine maple]] is notably absent, except where they have been planted. Other under-story plants include the evergreen huckleberry, lower longleaf [[Oregon grape]], [[elderberry]], [[salal]], [[oceanspray]], and varieties of [[Urtica|nettle]]. Non-native introduced plants such as [[foxglove]], [[ivy]] and [[holly]] are also evident.<ref>{{cite web |title=All About Whidbey Island |url=https://whidbeyisland.us/all-about-whidbey-island/ |website=Whidbey Island |accessdate=14 January 2019}}</ref>

Farther up the island, however, the shorter Oregon-Grape and the blue Evergreen Huckleberry is seen less, while tall Oregon-grape and Red Huckleberry predominate. The native [[Pacific rhododendron]] is much more visible. Amongst the [[deciduous]] varieties, [[Garry oak]] (from which Oak Harbor takes its name) are seen more frequently in the northern portion of the island and [[arbutus|Pacific madrone]] is also notably present.<ref>{{cite web |title=Preservation |url=https://ohgarryoaksociety.org/preservation/ |website=Oak Harbor Garry Oak Society |accessdate=14 January 2019}}</ref> In the [[conifer]] classification, [[grand fir]] is found more in the northern part of Whidbey Island along with [[Sitka spruce]] and [[Lodgepole Pine|shore pine]]. There are three open prairie areas on Whidbey Island – Smith Prairie, Crockett Prairie and Ebey Prairie.<ref>{{cite book|title=Ebey's Landing National Historical Reserve, General Management Plan: Environmental Impact Statement|url=https://books.google.com/books?id=5TM3AQAAMAAJ&pg=PA89|year=2006|pages=89–90}}</ref> Some patches of [[Opuntia|prickly pear cactus]] are found along the slopes near Partridge Point.<ref name="Mass2015">{{cite book|author=Clifford Mass|title=The Weather of the Pacific Northwest|url=https://books.google.com/books?id=xPiACgAAQBAJ&pg=PA194|date=1 September 2015|publisher=University of Washington Press|isbn=978-0-295-99836-7|pages=194–}}</ref>

===Fauna===
[[Gray whale]]s migrate between Whidbey and Camano Islands during March and April and can be seen from both ship and shore.<ref>{{Cite news|url=https://whidbeycamanoislands.com/things-to-do/beaches-and-waterways/whale-watching/|title=Whale Watching around Whidbey and Camano Islands.|work=Whidbey and Camano Islands|access-date=2018-03-08|language=en-US}}</ref> [[Orca]] also make use of the waters surrounding Whidbey Island.

[[Clams]] and [[oysters]] are abundant locally and may be harvested from some public beaches.<ref>{{cite web |url=http://wdfw.wa.gov/fishing/shellfish/beaches/MapArea/08/ |title=Map data|website=wdfw.wa.gov}}</ref> The Washington State Department of Health provides an online guide to assist in identifying shellfish varieties as well as providing guidance about where to find specific varieties.<ref>{{cite web|url=http://www.doh.wa.gov/Portals/1/Documents/4400/332-087-Shellfish-ID.pdf|title=Shellfish Identification :: Washington State Department of Health|website=www.doh.wa.gov}}</ref>

According to the Whidbey Audubon Society, Approximately 230 [[bird]] species are reported to take advantage of the diverse habitats on the island. <ref>{{cite web |url=http://www.whidbeyaudubon.org/birdlist/birdlist.htm |title=Birds of Whidbey Island|website=www.whidbeyaudubon.org}}</ref>

== Education ==

=== Public school districts ===
Whidbey Island is served by three public [[school district]]s.

[[Oak Harbor School District]] operates in [[Oak Harbor, Washington|Oak Harbor]]. Within the district, there is one [[Oak Harbor High School (Washington)|high school]], one alternative high school, two middle schools, and five elementary schools. Within the [[Washington Interscholastic Activities Association]], Oak Harbor High is listed as a 3-A school.

[[Coupeville School District]] operates in [[Coupeville, Washington]] and [[Greenbank, Washington]]. Within the district, there is one high school, one middle school, and one elementary school. Within the [[Washington Interscholastic Activities Association]], Coupeville High is listed as a 1-A school.

[[South Whidbey School District]] serves the southern end of the island, including [[Freeland, Washington|Freeland]], [[Langley, Washington|Langley]], and [[Clinton, Washington|Clinton]]. Within the district, there is one high school (grades 9–12), one alternative school (grades K–12), one middle school (grades 5-8) split between 2 campuses, and one elementary school (grades K–4). Within the Washington Interscholastic Activities Association, South Whidbey High is listed as a 1-A school.

=== Colleges ===

[[Skagit Valley College]] has a campus located in Oak Harbor, and a limited service campus in South Whidbey.

[[Seattle Pacific University]] owns Camp Casey, a retreat center near [[Coupeville, Washington|Coupeville]], which was once the barracks for the adjacent [[Fort Casey State Park|Fort Casey]].

==Notable people==
===Actors===
*[[Lana Condor]] known for her role in To All The Boys I've Loved Before
===Politicians===
*[[Jack Metcalf]] (1927-2007), [[United States House of Representatives]], grew up on Whidbey Island in the 1930s.<ref>[http://seattletimes.nwsource.com/html/localnews/2003620723_metcalf16m.html Jack Metcalf obituary] {{webarchive|url=https://web.archive.org/web/20070319204602/http://seattletimes.nwsource.com/html/localnews/2003620723_metcalf16m.html |date=March 19, 2007 }}, ''Seattle Times'', March 16, 2007</ref>

===Writers and artists===

*[[Shayla Beesley]], actress, grew up in Oak Harbor
*[[Juliet Winters Carpenter]], prize-winning translator of Japanese literature and author
*[[Aleah Chapin]], painter, grew up on Whidbey Island
*[[Pete Dexter]], National Book Award fiction writer
*[[Elizabeth George]], mystery author.
*[[Aaron Parks]], jazz pianist.
*[[David Ossman]], founder of [[Firesign Theater]]
*[[Nancy Horan]], best selling author
*[[Jeff Alexander]], conductor and arranger
*Eli Moore and Ashley Eriksson of [[Lake (American band)|Lake]]
*[[David Whyte (poet)|David Whyte]], poet

===Other===
*[[Robert Jay Mathews]], American neo-Nazi terrorist and the leader of [[The Order (white supremacist group)]], an American white supremacist militant group, died on Whidbey Island during a shoot-out with federal law enforcement agents.
*[[Bruce Bochte]], American baseball player. Bochte lived on Whidbey Island for over three years after his baseball playing days were over.<ref>{{cite web|url=http://www.seattlepi.com/allstar/30690_bochte09.shtml|title=Bochte has moved a long way from baseball|website=seattlepi.com}}</ref>
*[[Marti Malloy]], London 2012 Olympic Bronze medalist, Judo women's 57 kg
*[[Mark Sargent]], flat farther

==Infrastructure==

===Transportation===
[[File:ClintonFerry.jpg|thumb|right|Ferry at Clinton]] The only bridge that reaches Whidbey Island is the [[Deception Pass Bridge]], [[Washington State Route 20|State Route 20]], which connects the north end of Whidbey to the mainland via [[Fidalgo Island]]. Prior to the completion of the bridge in 1935, Whidbey Island was linked to [[Fidalgo Island]] by the [[Deception Pass ferry]], which ran from 1924 to 1935. Modern ferry service is available via State Route 20 on the [[Coupeville, Washington|Coupeville]] to [[Port Townsend, Washington|Port Townsend]] ferry, and via [[Washington State Route 525|State Route 525]] on the [[Clinton, Washington|Clinton]] to [[Mukilteo, Washington|Mukilteo]] ferry service on the southern east coast.

Travel on the island involves use of an extensive county road system, or city infrastructure depending on location, all of which act as feeders to the two state highways [[Washington State Route 525|State Route 525]] and [[Washington State Route 20|State Route 20]].

Whidbey Island's State Routes [[State Route 525 (Washington)|525]]/[[State Route 20 (Washington)|20]] is the only nationally designated Scenic Byway on an island. It is appropriately named the "Whidbey Island Scenic Isle Way."

[[Mass transit|Public transportation]] is provided by [[Island Transit (Washington)|Island Transit]], which provides a [[zero-fare]] bus service paid for by a 6/10th of 1% sales tax within the county. There are currently 11 bus routes serving Whidbey Island. No service is available on Sundays or major holidays.

Two public airports provide service to Whidbey Island. Whidbey Air Park is located {{convert|2|mi|km}} southwest of [[Langley, Washington|Langley]] with a {{convert|2470|ft|m}} long runway. [[Wes Lupien Airport]] is located {{convert|3|mi|km}} southwest of [[Oak Harbor, Washington|Oak Harbor]] with a {{convert|3265|ft|m|abbr=on}} long runway. In addition, there are approximately half dozen private dirt strips on the island{{citation needed|date=February 2017}}. [[Kenmore Air Express]] ran a scheduled airline service to Whidbey Island serving the Oak Harbor airport from 2006 to 2009.<ref>{{Cite web |url=http://www.kenmoreair.com/content.php?content_id=261 |title=Archived copy |access-date=August 4, 2009 |archive-url=https://web.archive.org/web/20081007031605/http://www.kenmoreair.com/content.php?content_id=261 |archive-date=October 7, 2008 |dead-url=yes |df=mdy-all }}</ref>

The [[United States Navy]] operates two airports on Whidbey Island. The largest is a two-runway airport located at [[Whidbey Island Naval Air Station]] north of [[Oak Harbor, Washington|Oak Harbor]]. In addition, the Navy also operates a flight training facility named [[Coupeville Naval Outlying Field|Coupeville Outlying Landing Field (Coupeville OLF)]] located just southeast of [[Coupeville, Washington|Coupeville]]. The Navy named [[USS Whidbey Island (LSD-41)]] in honor of the island.

=== Health systems ===

[[Whidbey Health]] is the regional, county-run hospital. Located in Coupeville, the hospital has an extension clinic in Oak Harbor. The Naval Air Station in Oak Harbor has a limited service hospital for military personnel, veteran retirees, and their dependents.

==Communities ==
North to south:
*[[Deception Pass]]
*[[Oak Harbor, Washington|Oak Harbor]] - Largest city
*[[West Beach, Washington|West Beach]]
*[[San De Fuca, Washington|San De Fuca]]
*[[Coupeville, Washington|Coupeville]] – County Seat
*[[Keystone, Island County, Washington|Keystone]]
*[[Admiral's Cove, Washington|Admiral's Cove]]
*[[Lagoon Point, Washington|Lagoon Point]]
*[[Greenbank, Washington|Greenbank]]
*[[Langley, Washington|Langley]]
*[[Freeland, Washington|Freeland]]
*[[Bayview, Island County, Washington|Bayview]]
*[[Clinton, Washington|Clinton]]
*[[Maxwelton, Washington|Maxwelton]]
*[[Glendale, Washington|Glendale]]

==See also==
*[[Admiralty Head Lighthouse]]
*[[Meerkerk Rhododendron Gardens]]

==References==
{{reflist|30em}}

==External links==
{{commons category}}
{{wikivoyage|Whidbey Island}}
*[http://www.WhidbeyCamanoIslands.com Whidbey Island & Camano Island Official Tourism Website]
*[http://content.lib.washington.edu/vanolindaweb/index.html University of Washington Libraries Digital Collections – Oliver S. Van Olinda Photographs] A collection of 420 photographs depicting life on Vashon Island, Whidbey Island, Seattle and other communities of Washington State's Puget Sound from the 1880s to the 1930s.
{{Washington}}

[[Category:Landforms of Island County, Washington]]
[[Category:Islands of Washington (state)]]
[[Category:Islands of Puget Sound]]

Sheaf (mathematics)

2018-12-07T11:00:51Z

Crasshopper:

{{about|sheaves on [[topological space]]s|sheaves on a site|Grothendieck topology|and|Topos}}
In [[mathematics]], '''sheaves''' track locally defined data as it varies globally. Values are defined on the [[open set]]s of a [[topological space]], but also must match on overlapping open sets. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the [[ring (mathematics)|ring]]s of [[continuous function|continuous]] or [[smooth function|smooth]] [[real numbers|real]]-valued [[function (mathematics)|function]]s defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of [[set (mathematics)|sets]] or sheaves of rings, depending on the type of data assigned to open sets.

There are also [[map (mathematics)|map]]s (or [[morphism]]s) from one sheaf to another; sheaves (of a specific type, such as sheaves of [[abelian group]]s) with their [[morphism]]s on a fixed topological space form a [[category (mathematics)|category]]. On the other hand, to each [[continuous map]] there is associated both a [[direct image functor]], taking sheaves and their morphisms on the [[domain (mathematics)|domain]] to sheaves and morphisms on the [[codomain]], and an [[inverse image functor]] operating in the opposite direction. These [[functor]]s, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in topology and especially in [[algebraic geometry|algebraic]] and [[differential geometry]]. First, geometric structures such as that of a [[differentiable manifold]] or a [[scheme (mathematics)|scheme]] can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as [[vector bundles]] or [[divisor (algebraic geometry)|divisors]] are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general [[sheaf cohomology|cohomology theory]], which encompasses also the "usual" topological cohomology theories such as [[singular cohomology]]. Especially in algebraic geometry and the theory of [[complex manifold]]s, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of [[D-module]]s, which provide applications to the theory of [[differential equation]]s. In addition, generalisations of sheaves to more general settings than topological spaces, such as [[Grothendieck topology]], have provided applications to [[mathematical logic]] and [[number theory]].

== Overview ==
In [[topology]], [[differential geometry]], and [[algebraic geometry]], several structures defined on a [[topological space]] (e.g., a [[differentiable manifold]]) can be naturally ''localised'' or ''restricted'' to [[open set|open]] [[subset]]s of the space: typical examples include [[continuous function|continuous]] [[real numbers|real]] or [[complex number|complex]]-valued functions, ''n'' times [[differentiable function|differentiable]] (real or complex-valued) functions, [[bounded function|bounded]] real-valued functions, [[vector field]]s, and [[section (fiber bundle)|section]]s of any [[vector bundle]] on the space.

''Presheaves'' formalise the situation common to the examples above: a presheaf (of sets) on a topological space is a structure that associates to each open set ''U'' of the space a set ''F''(''U'') of ''sections'' on ''U'', and to each open set ''V'' included in ''U'' a map ''F''(''U'') → ''F''(''V'') giving ''restrictions'' of sections over ''U'' to ''V''. Each of the examples above defines a presheaf by taking the restriction maps to be the usual restriction of functions, vector fields and sections of a vector bundle. Moreover, in each of these examples the sets of sections have additional [[algebraic structure]]: pointwise operations make them [[abelian group]]s, and in the examples of real and complex-valued functions the sets of sections even have a [[ring (mathematics)|ring]] structure. In addition, in each example the restriction maps are [[homomorphism]]s of the corresponding algebraic structure. This observation leads to the natural definition of presheaves with additional algebraic structure such as presheaves of groups, of abelian groups, of rings: sets of sections are required to have the specified algebraic structure, and the restrictions are required to be homomorphisms. Thus for example continuous real-valued functions on a topological space form a presheaf of rings on the space.

Given a presheaf, a natural question to ask is to what extent its sections over an open set ''U'' are specified by their restrictions to smaller open sets ''V''''i'' of an [[open cover]] of ''U''. A presheaf is ''separated'' if its sections are "locally determined": whenever two sections over ''U'' coincide when restricted to each of ''V''''i'', the two sections are identical. All examples of presheaves discussed above are separated, since in each case the sections are specified by their values at the points of the underlying space. Finally, a separated presheaf is a ''sheaf'' if ''compatible sections can be glued together'', i.e., whenever there is a section of the presheaf over each of the covering sets ''V''''i'', chosen so that they match on the overlaps of the covering sets, these sections correspond to a (unique) section on ''U'', of which they are restrictions. It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is ''local'' in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.

On the other hand, a function can be bounded on each set of an (infinite) open cover of a space without being bounded on all of the space; thus bounded functions provide an example of a presheaf that in general fails to be a sheaf. Another example of a presheaf that fails to be a sheaf is the ''constant presheaf'' that associates the same fixed set (or abelian group, or ring,...) to each open set: it follows from the gluing property of sheaves that the set of sections on a disjoint union of two open sets is the [[Cartesian product]] of the sets of sections over the two open sets. The correct way to define the [[constant sheaf]] ''FA'' (associated to for instance a set ''A'') on a topological space is to require sections on an open set ''U'' to be continuous maps from ''U'' to ''A'' equipped with the [[discrete topology]]; then in particular ''FA''(''U'') = ''A'' for [[connected space|connected]] ''U''.

Maps between sheaves or presheaves (called [[morphism]]s), consist of maps between the sets of sections over each open set of the underlying space, compatible with restrictions of sections. If the presheaves or sheaves considered are provided with additional algebraic structure, these maps are assumed to be homomorphisms. Sheaves endowed with nontrivial endomorphisms, such as the action of an [[algebraic torus]] or a [[Galois group]], are of particular interest.

Presheaves and sheaves are typically denoted by capital letters, ''F'' being particularly common, presumably for the [[French language|French]] word for sheaves, ''faisceaux''. Use of calligraphic letters such as <math>\mathcal{F}</math> is also common.

== Formal definitions ==
The first step in defining a sheaf is to define a ''presheaf'', which captures the idea of associating data and restriction maps to the open sets of a topological space. The second step is to require the normalisation and gluing axioms. A presheaf that satisfies these axioms is a sheaf.

=== Presheaves ===
{{See also|Presheaf (category theory)}}

Let ''X'' be a topological space, and let '''C''' be a [[category (category theory)|category]]. Usually '''C''' is the [[category of sets]], the [[category of groups]], the [[category of abelian groups]], or the [[category of commutative rings]]. A '''presheaf''' ''F'' on ''X'' is a functor with values in '''C''' given by the following data:
*For each open set ''U'' of ''X'', there corresponds an object ''F''(''U'') in '''C'''
*For each inclusion of open sets ''V'' ⊆ ''U'', there corresponds a [[morphism]] <math>\operatorname{res}_{V, U} \colon F(U) \rightarrow F(V)</math> in the category '''C'''.
The morphisms res''V'',''U'' are called '''restriction morphisms'''. If {{nowrap|''s'' ∈ ''F''(''U'')}}, then its restriction {{nowrap|res''V'',''U''(s)}} is often denoted ''s''|''V'' by analogy with restriction of functions. The restriction morphisms are required to satisfy two properties:
*For every open set ''U'' of ''X'', the restriction morphism res''U'',''U'' : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
*If we have three open sets ''W'' ⊆ ''V'' ⊆ ''U'', then the [[function composition|composite]] {{nowrap|1=res''W'',''V'' o res''V'',''U'' equals res''W'',''U''.}}
Informally, the second axiom says it doesn't matter whether we restrict to ''W'' in one step or restrict first to ''V'', then to ''W''.

There is a compact way to express the notion of a presheaf in terms of [[category theory]]. First we define the category of open sets on ''X'' to be the [[posetal category]] ''O''(''X'') whose objects are the open sets of ''X'' and whose morphisms are inclusions. Then a '''C'''-valued presheaf on ''X'' is the same as a [[contravariant functor]] from ''O''(''X'') to '''C'''. This definition can be generalized to the case when the source category is not of the form ''O''(''X'') for any ''X''; see [[presheaf (category theory)]].

If ''F'' is a '''C'''-valued presheaf on ''X'', and ''U'' is an open subset of ''X'', then ''F''(''U'') is called the '''sections of ''F'' over ''U'''''. If '''C''' is a [[concrete category]], then each element of ''F''(''U'') is called a '''section'''. A section over ''X'' is called a '''global section'''. A common notation (used also below) for the restriction res''V'',''U''(''s'') of a section is ''s''|''V''. This terminology and notation is by analogy with sections of [[fiber bundle]]s or sections of the étalé space of a sheaf; see below. ''F''(''U'') is also often denoted Γ(''U'',''F''), especially in contexts such as [[sheaf cohomology]] where ''U'' tends to be fixed and ''F'' tends to be variable.

=== Sheaves ===
For simplicity, consider first the case where the sheaf takes values in the category of sets. In fact, this definition applies more generally to the situation where the category is a [[concrete category]] whose underlying set functor is [[Conservative functor|conservative]], meaning that if the underlying map of sets is a bijection, then the original morphism is an isomorphism.

A ''sheaf'' is a presheaf with values in the category of sets that satisfies the following two axioms:
# (Locality) If (''U''''i'') is an open [[cover (topology)|covering]] of an open set ''U'', and if ''s'',''t'' ∈ ''F''(''U'') are such that ''s''|''U''''i'' = ''t''|''U''''i'' for each set ''U''''i'' of the covering, then ''s'' = ''t''; and
# ([[Gluing axiom|Gluing]]) If (''U''''i'') is an open covering of an open set ''U'', and if for each ''i'' a section ''s''''i'' ∈ ''F''(''U''''i'') is given such that for each pair ''U''''i'',''U''''j'' of the covering sets the restrictions of ''s''''i'' and ''s''''j'' agree on the overlaps: ''s''''i''|''U''''i''∩''U''''j'' = ''s''''j''|''U''''i''∩''U''''j'', then there is a section ''s'' ∈ ''F''(''U'') such that ''s''|''U''''i'' = ''s''''i'' for each ''i''.

The section ''s'' whose existence is guaranteed by axiom 2 is called the '''gluing''', '''concatenation''', or '''collation''' of the sections ''s''''i''. By axiom 1 it is unique. Sections ''s''''i'' satisfying the condition of axiom 2 are often called ''compatible''; thus axioms 1 and 2 together state that ''compatible sections can be uniquely glued together''. A '''separated presheaf''', or '''monopresheaf''', is a presheaf satisfying axiom 1.<ref>{{Citation | last1=Tennison | first1=B. R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975}}</ref>

If '''C''' has [[product (category theory)|products]], the sheaf axioms are equivalent to the requirement that, for any open covering ''U''''i'', the first arrow in the following diagram is an [[equalizer (mathematics)|equalizer]]:

:<math>F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i, j} F(U_i \cap U_j).</math>

Here the first map is the product of the restriction maps

:<math>\operatorname{res}_{U_i, U} \colon F(U) \rightarrow F(U_i)</math>

and the pair of arrows the products of the two sets of restrictions

:<math>\operatorname{res}_{U_i \cap U_j, U_i} \colon F(U_i) \rightarrow F(U_i \cap U_j)</math>

and

:<math>\operatorname{res}_{U_i \cap U_j, U_j} \colon F(U_j) \rightarrow F(U_i \cap U_j).</math>

For a separated presheaf, the first arrow need only be injective.

In general, for an open set ''U'' and open covering (''U''''i''), construct a category ''J'' whose objects are the sets ''Ui'' and the intersections {{nowrap|''Ui'' ∩ ''Uj''}} and whose morphisms are the inclusions of {{nowrap|''Ui'' ∩ ''Uj''}} in ''Ui'' and ''Uj''. The sheaf axioms for ''U'' and (''U''''i'') are that the [[limit (category theory)|limit]] of the functor ''F'' restricted to the category ''J'' must be isomorphic to ''F''(''U'').

Notice that the empty subset of a topological space is covered by the empty family of sets. The product of an empty family or the limit of an empty family is a terminal object, and consequently the value of a sheaf on the empty set must be a terminal object. If sheaf values are in the category of sets, applying the local identity axiom to the empty family shows that over the empty set, there is at most one section, and applying the gluing axiom to the empty family shows that there is at least one section. This property is called the '''normalisation axiom'''.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a [[basis (topology)|basis]] for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis. (see [[gluing axiom#Sheaves on a basis of open sets]].)

=== Morphisms ===
A morphism of sheaves is a clean function between them. Because sheaves contain data relative to every open set of a topological space, a morphism of sheaves is defined as a collection of functions, one for each open set. The collection satisfies a compatibility condition.

Let ''F'' and ''G'' be two sheaves on ''X'' with values in the category '''C'''. A ''[[morphism]]'' <math>\varphi:G\to F</math> consists of a morphism <math>\varphi_U:G(U)\to F(U)</math> for each open set {{mvar|U}} of {{mvar|X}}, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset ''V'' of an open set ''U'', the following diagram is [[commutative diagram|commutative]].

:<math>\begin{array}{rcl}
G(U) & \xrightarrow{\quad\varphi_U\quad} & F(U)\\
r_{V,U}\Biggl\downarrow & & \Biggl\downarrow r_{V,U}\\
G(V) & \xrightarrow[{\quad\varphi_V\quad}]{} & F(V)
\end{array}</math>

Sheaves are themselves functors, from "data" to a space. In this language, a morphism of sheaves is a [[natural transformation]] of the corresponding functors. With this notion of morphism, there is a category of '''C'''-valued sheaves on ''X'' for any '''C'''. The objects are the '''C'''-valued sheaves, and the morphisms are morphisms of sheaves. An ''[[isomorphism]]'' of sheaves is an isomorphism in this category.

An isomorphism of sheaves is an isomorphism on each open set ''U''. In other words, φ is an isomorphism if and only if for each ''U'', φ(''U'') is an isomorphism. A morphism of sheaves φ is an isomorphism if and only if there exists an open cover <math>\{U_\alpha\}</math> such that <math>\varphi|_{U_\alpha}</math> are isomorphisms of sheaves for all <math>\alpha</math>. The same facts are true of [[monomorphism]]s. However, they are false for [[epimorphism]]s, and their failure is measured by [[sheaf cohomology]].

Notice that we did not use the gluing axiom in defining a morphism of sheaves. Consequently, the above definition makes sense for presheaves as well. The category of '''C'''-valued presheaves is then a [[functor category]], the category of contravariant functors from ''O''(''X'') to '''C'''.

== Examples ==
Because sheaves encode exactly the data needed to pass between local and global situations, they have been used across mathematical subfields—wherever local and global facts [[Hairy ball theorem|differ]]. Here are some additional examples of sheaves:

* Any continuous map of topological spaces determines a sheaf of sets. Let ''f'' : ''Y'' → ''X'' be a continuous map. We define a sheaf Γ(''Y''/''X'') on ''X'' by setting Γ(''Y''/''X'')(U) equal to the sections ''U'' → ''Y'', that is, Γ(''Y''/''X'')(U) is the set of all continuous functions ''s'' : ''U'' → ''Y'' such that ''f ∘ s'' = ''id''''U''. Restriction is given by restriction of functions. This sheaf is called the '''sheaf of sections''' of ''f'', and it is especially important when ''f'' is the projection of a [[fiber bundle]] onto its base space. Notice that if the image of ''f'' does not contain ''U'', then Γ(''Y''/''X'')(''U'') is empty. For a concrete example, take ''X'' = '''C''' \ {0}, ''Y'' = '''C''', and ''f''(''z'') = exp(''z''). Γ(''Y''/''X'')(''U'') is the set of branches of the logarithm on ''U''.
* Fix a point ''x'' in ''X'' and an object ''S'' in a category '''C'''. The ''skyscraper sheaf over ''x'' with stalk'' ''S'' is the sheaf ''S''''x'' defined as follows: If ''U'' is an open set containing ''x'', then ''S''''x''(''U'') = ''S''. If ''U'' does not contain ''x'', then ''S''''x''(''U'') is the terminal object of '''C'''. The restriction maps are either the identity on ''S'', if both open sets contain ''x'', or the unique map from ''S'' to the terminal object of '''C'''.

=== Sheaves on manifolds ===
In the following examples ''M'' is an ''n''-dimensional ''C''''k''-manifold. The table lists the values of certain sheaves over open subsets ''U'' of ''M'' and their restriction maps.

{| class="wikitable"
|-
! Sheaf !! Sections over an open set ''U'' !! Restriction maps !! Remarks
|-
! Sheaf of ''j''-times continuously differentiable functions <math>\mathcal{O}^j_M</math>, ''j'' ≤ ''k''
| ''C''''j''-functions ''U'' → '''R'''
| Restriction of functions.
| This is a sheaf of rings with addition and multiplication given by pointwise addition and multiplication. When ''j'' = ''k'', this sheaf is called the '''structure sheaf''' and is denoted <math>\mathcal{O}_M</math>.
|-
! Sheaf of nonzero ''k''-times continuously differentiable functions <math>\mathcal{O}_X^\times</math>
| Nowhere zero ''C''''k''-functions ''U'' → '''R'''
| Restriction of functions.
| A sheaf of groups under pointwise multiplication.
|-
! '''Cotangent sheaves''' Ω''p''''M''
| [[Differential form]]s of degree ''p'' on ''U''
| Restriction of differential forms.
| Ω1''M'' and Ωn''M'' are commonly denoted Ω''M'' and ω''M'', respectively.
|-
! '''Sheaf of distributions''' {{mathcal|DB}}
| [[Distribution (mathematics)|Distribution]]s on ''U''
| The dual map to extension of smooth compactly supported functions by zero.
| Here ''M'' is assumed to be smooth.
|-
! '''Sheaf of differential operators''' <math>\mathcal{D}_M</math>
| Finite-order [[differential operator]]s on ''U''
| Restriction of differential operators.
| Here ''M'' is assumed to be smooth.
|}

=== Presheaves that are not sheaves ===
Here are two examples of presheaves that are not sheaves:
* Let ''X'' be the [[discrete two-point space|two-point topological space]] {''x'', ''y''} with the discrete topology. Define a presheaf ''F'' as follows: ''F''(∅) = {∅}, ''F''({''x''}) = '''R''', ''F''({''y''}) = '''R''', ''F''({''x'', ''y''}) = '''R''' × '''R''' × '''R'''. The restriction map ''F''({''x'', ''y''}) → ''F''({''x''}) is the projection of '''R''' × '''R''' × '''R''' onto its first coordinate, and the restriction map ''F''({''x'', ''y''}) → ''F''({''y''}) is the projection of '''R''' × '''R''' × '''R''' onto its second coordinate. ''F'' is a presheaf that is not separated: A global section is determined by three numbers, but the values of that section over {''x''} and {''y''} determine only two of those numbers. So while we can glue any two sections over {''x''} and {''y''}, we cannot glue them uniquely.
* Let ''X'' be the [[real line]], and let ''F''(''U'') be the set of [[bounded function|bounded]] continuous functions on ''U''. This is not a sheaf because it is not always possible to glue. For example, let ''U''''i'' be the set of all ''x'' such that |''x''| < ''i''. The identity function ''f''(''x'') = ''x'' is bounded on each ''U''''i''. Consequently we get a section ''s''''i'' on ''U''''i''. However, these sections do not glue, because the function ''f'' is not bounded on the real line. Consequently ''F'' is a presheaf, but not a sheaf. In fact, ''F'' is separated because it is a sub-presheaf of the sheaf of continuous functions.

== Turning a presheaf into a sheaf ==
It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf ''F'' and produces a new sheaf ''aF'' called the '''sheaving''', '''sheafification''' or '''sheaf associated to the presheaf''' ''F''. The functor ''a'' is called the '''sheaving functor''', '''sheafification functor''', or '''associated sheaf functor'''. There is a natural morphism of presheaves <math>i\colon F\to aF</math> that has the universal property that for any sheaf ''G'' and any morphism of presheaves <math>f\colon F\to G</math>, there is a unique morphism of sheaves <math>\tilde f \colon aF \rightarrow G</math> such that <math>f = \tilde f i</math>. In fact ''a'' is the left [[adjoint functor]] to the inclusion functor (or [[forgetful functor]]) from the category of sheaves to the category of presheaves, and ''i'' is the [[adjoint functor#Unit and co-unit|unit]] of the adjunction. In this way, the category of sheaves turns into a [[Giraud subcategory]] of presheaves.

One concrete way of constructing the sheaf ''aF'' is to identify it with the sheaf of sections of an appropriate topological space. This space is analogous to the [[#The étalé space of a sheaf|étalé space]] of a sheaf. Briefly, the underlying set of the topological space is the disjoint union of the [[#Stalks of a sheaf|stalks]] of ''F'', denoted {{nowrap|Spé ''F''}}. There is a natural map {{nowrap|φ : Spé ''F'' → ''X''}} that sends each germ to the point of ''X'' over which it lies. For each open set ''U'' and each section ''s'' of ''F'' over ''U'', we define a section <math>\bar s</math> of φ that sends ''x'' to the germ ''s''''x''. Then {{nowrap|Spé ''F''}} is given the finest topology for which all sections <math>\bar s</math> are continuous, and ''aF'' is the sheaf of continuous sections of φ for this topology.

There are other constructions of the sheaf ''aF''. In particular, [[Alexander Grothendieck]] and [[Jean-Louis Verdier]] ([[Séminaire de Géométrie Algébrique du Bois Marie#SGA 4|SGA 4]] II 3.0.5) define a functor ''L'' from presheaves to presheaves which, when applied to a presheaf, yields a separated presheaf and, when applied to a separated presheaf, yields a sheaf. Applying the functor ''L'' twice therefore turns a presheaf into a sheaf, and in fact ''LLF'' is the associated sheaf ''aF''.

== Operations ==
If ''K'' is a [[Subobject|subsheaf]] of a sheaf ''F'' of abelian groups, then the '''quotient sheaf''' ''Q'' is the sheaf associated to the presheaf <math>U \mapsto F(U)/K(U)</math>; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;
:<math>0 \to K \to F \to Q \to 0.</math>
(this is also called a [[sheaf extension]].)

Let ''F'', ''G'' be sheaves of abelian groups. The set of morphisms of sheaves from ''F'' to ''G'' forms an abelian group (by the abelian group structure of ''G''). The '''sheaf hom''' of ''F'' and ''G'', denoted by,
:<math>\mathcal{Hom}(F, G)</math>
is the sheaf of abelian groups <math>U \mapsto \operatorname{\mathcal{Hom}}(F|_U, G|_U)</math> where <math>F|_U</math> is the sheaf on ''U'' given by <math>(F|_U)(V) = F(V)</math> (Note sheafification is not needed here). The direct sum of ''F'' and ''G'' is the sheaf given by <math>U \mapsto F(U) \oplus G(U) </math>, and the tensor product of ''F'' and ''G'' is the sheaf associated to the presheaf <math>U \mapsto F(U) \otimes G(U)</math>.

All of these operations extend to [[sheaf of modules|sheaves of modules]] over a [[sheaf of rings]] ''A''; the above is the special case when ''A'' is the [[constant sheaf]] <math>\underline{\mathbf{Z}}</math>.

== Images of sheaves ==
{{Images of sheaves}}
{{Main|Image functors for sheaves}}

The definition of a morphism on sheaves makes sense only for sheaves on the same space ''X''. This is because the data contained in a sheaf is indexed by the open sets of the space. If we have two sheaves on different spaces, then their data is indexed differently. There is no way to go directly from one set of data to the other.

However, it is possible to move a sheaf from one space to another using a continuous function. Let ''f'' : ''X'' → ''Y'' be a continuous function from a topological space ''X'' to a topological space ''Y''. If we have a sheaf on ''X'', we can move it to ''Y'', and vice versa. There are four ways in which sheaves can be moved.
* A sheaf <math>\mathcal{F}</math> on ''X'' can be moved to ''Y'' using the [[direct image functor]] <math>f_*</math> or the [[direct image with proper support functor]] <math>f_!</math>.
* A sheaf <math>\mathcal{G}</math> on ''Y'' can be moved to ''X'' using the [[inverse image functor]] <math>f^{-1}</math> or the [[twisted inverse image functor]] <math>f^!</math>.

The twisted inverse image functor <math>f^!</math> is, in general, only defined as a functor between [[derived category|derived categories]]. These functors come in adjoint pairs: <math>f^{-1}</math> and <math>f_*</math> are left and right adjoints of each other, and <math>Rf_!</math> and <math>f^!</math> are left and right adjoints of each other. The functors are intertwined with each other by [[Coherent duality|Grothendieck duality]] and [[Verdier duality]].

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted <math>f^*</math> and it is distinct from <math>f^{-1}</math>. See [[inverse image functor]].

== Stalks of a sheaf ==
{{Main|Stalk (sheaf)}}

The '''stalk''' <math>\mathcal{F}_x</math> of a sheaf <math>\mathcal{F}</math> captures the properties of a sheaf "around" a point ''x'' ∈ ''X''.
Here, "around" means that, conceptually speaking, one looks at smaller and smaller [[neighborhood (mathematics)|neighborhoods]] of the point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.

The stalk is defined by
:<math>\mathcal{F}_x = \varinjlim_{U\ni x} \mathcal{F}(U),</math>
the [[direct limit]] being over all open subsets of ''X'' containing the given point ''x''. In other words, an element of the stalk is given by a section over some open neighborhood of ''x'', and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism ''F''(''U'') → ''F''''x'' takes a section ''s'' in ''F''(''U'') to its ''germ''. This generalises the usual definition of a [[germ (mathematics)|germ]].

A different way of defining the stalk is
:<math>\mathcal{F}_x := i^{-1}\mathcal{F}(\{x\}),</math>
where ''i'' is the inclusion of the one-point space {''x''} into ''X''. The equivalence follows from the definition of the [[inverse image functor|inverse image]].

In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. They also find use in constructions such as [[Godement resolution]]s.

== Ringed spaces and locally ringed spaces ==
{{Main|Ringed space}}

A pair <math>(X, \mathcal{O}_X)</math> consisting of a topological space ''X'' and a sheaf of rings on ''X'' is called a '''[[ringed space]]'''. Many types of spaces can be defined as certain types of ringed spaces. The sheaf <math>\mathcal{O}_X</math> is called the '''structure sheaf''' of the space. A very common situation is when all the stalks of the structure sheaf are [[local ring]]s, in which case the pair is called a '''locally ringed space'''. Here are examples of definitions made in this way:
* An ''n''-dimensional ''C''''k'' manifold ''M'' is a locally ringed space whose structure sheaf is an <math>\underline{\mathbf{R}}</math>-algebra and is locally isomorphic to the sheaf of ''C''k real-valued functions on '''R'''''n''.
* A [[complex analytic space]] is a locally ringed space whose structure sheaf is a <math>\underline{\mathbf{C}}</math>-algebra and is locally isomorphic to the vanishing locus of a finite set of holomorphic functions together with the restriction (to the vanishing locus) of the sheaf of holomorphic functions on '''C'''''n'' for some ''n''.
* A [[scheme (mathematics)|scheme]] is a locally ringed space that is locally isomorphic to the [[spectrum of a ring]].
* A [[semialgebraic space]] is a locally ringed space that is locally isomorphic to a [[semialgebraic set]] in Euclidean space together with its sheaf of semialgebraic functions.

== Sheaves of modules ==
{{main|Sheaf of modules}}
Let <math>(X, \mathcal{O}_X)</math> be a ringed space. A '''sheaf of modules''' is a sheaf <math>\mathcal{M}</math> such that on every open set ''U'' of ''X'', <math>\mathcal{M}(U)</math> is an <math>\mathcal{O}_X(U)</math>-module and for every inclusion of open sets ''V'' ⊆ ''U'', the restriction map <math>\mathcal{M}(U) \to \mathcal{M}(V)</math> is a homomorphism of <math>\mathcal{O}_X(U)</math>-modules.

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between [[vector bundle]]s and [[locally free sheaf|locally free sheaves]] of <math>\mathcal{O}_X</math>-modules. Sheaves of solutions to differential equations are [[D-module]]s, that is, modules over the sheaf of differential operators.

A particularly important case are [[abelian sheaf|abelian sheaves]], which are modules over the constant sheaf <math>\underline{\mathbf{Z}}</math>. Every sheaf of modules is an abelian sheaf.

=== Finiteness conditions for sheaves of modules ===
{{Further|Coherent sheaf}}
The condition that a module is finitely generated or finitely presented can also be formulated for a sheaf of modules. <math>\mathcal{M}</math> is '''finitely generated''' if, for every point ''x'' of ''X'', there exists an open neighborhood ''U'' of ''x'', a natural number ''n'' (possibly depending on ''U''), and a surjective morphism of sheaves <math>\mathcal{O}_X^n|_U \to \mathcal{M}|_U</math>. Similarly, <math>\mathcal{M}</math> is '''finitely presented''' if in addition there exists a natural number ''m'' (again possibly depending on ''U'') and a morphism of sheaves <math>\mathcal{O}_X^m|_U \to \mathcal{O}_X^n|_U</math> such that the sequence of morphisms <math>\mathcal{O}_X^m|_U \to \mathcal{O}_X^n|_U \to \mathcal{M}</math> is exact. Equivalently, the kernel of the morphism <math>\mathcal{O}_X^n|_U \to \mathcal{M}</math> is itself a finitely generated sheaf.

These, however, are not the only possible finiteness conditions on a sheaf. The most important finiteness condition for a sheaf is coherence. <math>\mathcal{M}</math> is '''coherent''' if it is of finite type and if, for every open set ''U'' and every morphism of sheaves <math>\phi : \mathcal{O}_X^n \to \mathcal{M}</math> (not necessarily surjective), the kernel of φ is of finite type. <math>\mathcal{O}_X</math> is '''coherent''' if it is coherent as a module over itself. Note that coherence is a strictly stronger condition than finite presentation: <math>\mathcal{O}_X</math> is always finitely presented as a module over itself, but it is not always coherent. For example, let ''X'' be a point, let <math>\mathcal{O}_X</math> be the ring {{nowrap begin}}''R'' = '''C'''[''x''1, ''x''2, ...]{{nowrap end}} of complex polynomials in countably many indeterminates. Choose {{nowrap begin}}''n'' = 1{{nowrap end}}, and for the morphism φ, take the map that sends every variable to zero. The kernel of this map is not finitely generated, so <math>\mathcal{O}_X</math> is not coherent.

== The étalé space of a sheaf ==
{{anchor|Etale space}}
In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called the ''étalé space'', from the French word étalé {{IPA-fr|etale|}}, meaning roughly "spread out". If ''F'' is a sheaf over ''X'', then the '''étalé space''' of ''F'' is a topological space ''E'' together with a [[local homeomorphism]] ''π'' : ''E'' → ''X'' such that the sheaf of sections of ''π'' is ''F''. The space ''E'' is usually very strange, and even if the sheaf ''F'' arises from a natural topological situation, ''E'' may not have any clear topological interpretation. For example, if ''F'' is the sheaf of sections of a continuous function ''f'' : ''Y'' → ''X'', then ''E'' = ''Y'' if and only if ''f'' is a [[local homeomorphism]].

The étalé space ''E'' is constructed from the stalks of ''F'' over ''X''. As a set, it is their [[disjoint union]] and ''π'' is the obvious map that takes the value ''x'' on the stalk of ''F'' over ''x'' ∈ ''X''. The topology of ''E'' is defined as follows. For each element ''s'' of ''F''(''U'') and each ''x'' in ''U'', we get a germ of ''s'' at ''x''. These germs determine points of ''E''. For any ''U'' and ''s'' ∈ ''F''(''U''), the union of these points (for all ''x'' ∈ ''U'') is declared to be open in ''E''. Notice that each stalk has the [[discrete topology]] as subspace topology. Two morphisms between sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.

The construction above determines an [[equivalence of categories]] between the category of sheaves of sets on ''X'' and the category of étalé spaces over ''X''. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf.

This construction makes all sheaves into [[representable functor]]s on certain categories of topological spaces. As above, let ''F'' be a sheaf on ''X'', let ''E'' be its étalé space, and let ''π'' : ''E'' → ''X'' be the natural projection. Consider the category '''Top'''/''X'' of topological spaces over ''X'', that is, the category of topological spaces together with fixed continuous maps to ''X''. Every object of this space is a continuous map ''f'' : ''Y'' → ''X'', and a morphism from ''Y'' → ''X'' to ''Z'' → ''X'' is a continuous map ''Y'' → ''Z'' that commutes with the two maps to ''X''. There is a functor Γ from '''Top'''/''X'' to the category of sets that takes an object ''f'' : ''Y'' → ''X'' to (''f''−1''F'')(''Y''). For example, if ''i'' : ''U'' → ''X'' is the inclusion of an open subset, then Γ(''i'') = (''i''−1''F'')(''U'') agrees with the usual ''F''(''U''), and if ''i'' : {''x''} → ''X'' is the inclusion of a point, then Γ({''x''}) = (''i''−1''F'')({''x''}) is the stalk of ''F'' at ''x''. There is a natural isomorphism
:<math>(f^{-1}F)(Y) \cong \operatorname{Hom}_{\mathbf{Top}/X}(f, \pi)</math>,
which shows that ''E'' represents the functor Γ.

''E'' is constructed so that the projection map π is a covering map. In algebraic geometry, the natural analog of a covering map is called an [[étale morphism]]. Despite its similarity to "étalé", the word étale {{IPA-fr|etal|}} has a different meaning in French. It is possible to turn ''E'' into a [[scheme (mathematics)|scheme]] and π into a morphism of schemes in such a way that π retains the same universal property, but π is ''not'' in general an étale morphism because it is not quasi-finite. It is, however, formally étale.

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as [[mathematical analysis]].

== Sheaf cohomology ==
{{Main|Sheaf cohomology}}

It was noted above that the functor <math>\Gamma(U,-)</math> preserves isomorphisms and monomorphisms, but not epimorphisms. If ''F'' is a sheaf of abelian groups, or more generally a sheaf with values in an [[abelian category]], then <math>\Gamma(U,-)</math> is actually a [[left exact functor]]. This means that it is possible to construct [[derived functor]]s of <math>\Gamma(U,-)</math>. These derived functors are called the ''cohomology groups'' (or ''modules'') of ''F'' and are written <math>H^i(U,-)</math>. Grothendieck proved in his "[[Tohoku paper]]" ({{harvtxt|Grothendieck|1957}}) that every category of sheaves of abelian groups contains enough [[injective object]]s, so these derived functors always exist.

However, computing sheaf cohomology using injective resolutions is nearly impossible. In practice, it is much more common to find a different and more tractable resolution of ''F''. A general construction is provided by [[Godement resolution]]s, and particular resolutions may be constructed using [[soft sheaf|soft sheaves]], [[fine sheaf|fine sheaves]], and [[flabby sheaf|flabby sheaves]] (also known as ''flasque sheaves'' from the French ''flasque'' meaning flabby). As a consequence, it can become possible to compare sheaf cohomology with other cohomology theories. For example, the [[de Rham complex]] is a resolution of the constant sheaf <math>\underline{\mathbf{R}}</math> on any smooth manifold, so the sheaf cohomology of <math>\underline{\mathbf{R}}</math> is equal to its [[de Rham cohomology]]. In fact, comparing sheaf cohomology to de Rham cohomology and singular cohomology provides a proof of de Rham's theorem that the two cohomology theories are isomorphic.

A different approach is by [[Čech cohomology]]. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct <math>H^1</math> but incorrect higher cohomology groups. To get around this, [[Jean-Louis Verdier]] developed [[hypercover]]ings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of [[Pierre Deligne]]'s [[mixed Hodge structure]]s.

A much cleaner approach to the computation of some cohomology groups is the [[Borel–Bott–Weil theorem]], which identifies the cohomology groups of some [[line bundle]]s on [[flag manifold]]s with [[irreducible representation]]s of [[Lie group]]s. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space.

In many cases there is a duality theory for sheaves that generalizes [[Poincaré duality]]. See [[Coherent duality|Grothendieck duality]] and [[Verdier duality]].

== Sites and topoi ==
{{Main|Grothendieck topology|Topos}}

[[André Weil]]'s [[Weil conjectures]] stated that there was a [[Weil cohomology theory|cohomology theory]] for [[algebraic variety|algebraic varieties]] over [[finite field]]s that would give an analogue of the [[Riemann hypothesis]]. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf <math>\underline{\mathbf{C}}</math> in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the [[Zariski topology]], and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). [[Alexandre Grothendieck]] solved this problem by introducing [[Grothendieck topology|Grothendieck topologies]], which axiomatize the notion of ''covering''. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to define [[étale cohomology]] and [[l-adic cohomology]], which eventually were used to prove the Weil conjectures.

A category with a Grothendieck topology is called a ''site''. A category of sheaves on a site is called a ''topos'' or a ''Grothendieck topos''. The notion of a topos was later abstracted by [[William Lawvere]] and Miles Tierney to define an [[elementary topos]], which has connections to [[mathematical logic]].

== History ==

{{unreferenced section|date=January 2016}}

The first origins of '''sheaf theory''' are hard to pin down — they may be co-extensive with the idea of [[analytic continuation]]{{Clarify|date=July 2010}}. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on [[cohomology]].
* 1936 [[Eduard Čech]] introduces the ''[[Nerve of an open covering|nerve]]'' construction, for associating a [[simplicial complex]] to an open covering.
* 1938 [[Hassler Whitney]] gives a 'modern' definition of cohomology, summarizing the work since [[James Waddell Alexander II|J. W. Alexander]] and [[Kolmogorov]] first defined ''[[cochain]]s''.
* 1943 [[Norman Steenrod]] publishes on homology ''with [[local coefficients]]''.
* 1945 [[Jean Leray]] publishes work carried out as a [[prisoner of war]], motivated by proving [[Fixed point (mathematics)|fixed point]] theorems for application to [[Partial differential equation|PDE]] theory; it is the start of sheaf theory and [[spectral sequence]]s.
* 1947 [[Henri Cartan]] reproves the [[de Rham theorem]] by sheaf methods, in correspondence with [[André Weil]] (see [[De Rham–Weil theorem]]). Leray gives a sheaf definition in his courses via closed sets (the later ''carapaces'').
* 1948 The Cartan seminar writes up sheaf theory for the first time.
* 1950 The "second edition" sheaf theory from the Cartan seminar: the [[sheaf space]] (''espace étalé'') definition is used, with stalkwise structure. [[Support (mathematics)|Support]]s are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time [[Kiyoshi Oka]] introduces an idea (adjacent to that) of a sheaf of ideals, in [[several complex variables]].
* 1951 The Cartan seminar proves [[Theorems A and B]], based on Oka's work.
* 1953 The finiteness theorem for [[coherent sheaf|coherent sheaves]] in the analytic theory is proved by Cartan and [[Jean-Pierre Serre]], as is [[Serre duality]].
* 1954 Serre's paper ''[[#CITEREFSerre1955|Faisceaux algébriques cohérents]]'' (published in 1955) introduces sheaves into [[algebraic geometry]]. These ideas are immediately exploited by [[Friedrich Hirzebruch]], who writes a major 1956 book on topological methods.
* 1955 [[Alexander Grothendieck]] in lectures in [[Kansas]] defines [[abelian category]] and ''presheaf'', and by using [[injective resolution]]s allows direct use of sheaf cohomology on all topological spaces, as [[derived functor]]s.
* 1956 [[Oscar Zariski]]'s report ''[[#CITEREFMartinChernZariski1956|Algebraic sheaf theory]]''
* 1957 Grothendieck's [[#CITEREFGrothendieck1957|''Tohoku'' paper]] rewrites [[homological algebra]]; he proves [[Coherent duality|Grothendieck duality]] (i.e., Serre duality for possibly [[Mathematical singularity|singular]] algebraic varieties).
* 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: [[Scheme (mathematics)|scheme]]s and general sheaves on them, [[local cohomology]], [[derived category|derived categories]] (with Verdier), and [[Grothendieck topologies]]. There emerges also his influential schematic idea of 'six operations' in homological algebra.
* 1958 [[Roger Godement]]'s book on sheaf theory is published. At around this time [[Mikio Sato]] proposes his [[hyperfunction]]s, which will turn out to have sheaf-theoretic nature.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to [[algebraic topology]]. It was later discovered that the logic in categories of sheaves is [[intuitionistic logic]] (this observation is now often referred to as [[Kripke–Joyal semantics]], but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as [[Gottfried Wilhelm Leibniz|Leibniz]].

== See also ==
* [[Coherent sheaf]]
* [[Cosheaf]]
* [[Gerbe]]
* [[Stack (mathematics)]]
* [[Sheaf of spectra]]
* [[Presheaf of spaces]]
* [[Base change theorems]]
* [[Locally constant sheaf]]
* [[Constructible sheaf]]

== Notes ==
{{Reflist}}

== References ==
* {{Citation | last1=Bredon | first1=Glen E. | author1-link = Glen Bredon | title=Sheaf theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94905-5 | mr=1481706 | edition=2nd | year=1997 | volume=170}} (oriented towards conventional topological applications)
* {{Citation | last1=Godement | first1=Roger | author1-link = Roger Godement | title=Topologie algébrique et théorie des faisceaux | publisher=Hermann | location=Paris | mr=0345092 | year=1973}}
* {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Sur quelques points d'algèbre homologique | mr=0102537 | year=1957 | journal=The Tohoku Mathematical Journal. Second Series | issn=0040-8735 | volume=9 | pages=119–221 | doi=10.2748/tmj/1178244839}}
* {{Citation | last1=Hirzebruch | first1=Friedrich | author1-link = Friedrich Hirzebruch | title=Topological methods in algebraic geometry | publisher=Springer-Verlag | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58663-0 | mr=1335917 | year=1995}} (updated edition of a classic using enough sheaf theory to show its power)
* {{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | last2=Schapira | first2=Pierre | title=Sheaves on manifolds | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-51861-7 | mr=1299726 | year=1994 | volume=292}}(advanced techniques such as the [[derived category]] and [[vanishing cycle]]s on the most reasonable spaces)
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link = Saunders Mac Lane | last2=Moerdijk | first2=Ieke | author2-link = Ieke Moerdijk | title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-0-387-97710-2 | mr=1300636 | year=1994}} (category theory and toposes emphasised)
* {{Citation | last1=Martin | first1=William T. | last2=Chern | first2=Shiing-Shen | author2-link=Shiing-Shen Chern | last3=Zariski | first3=Oscar | author3-link=Oscar Zariski | title=Scientific report on the Second Summer Institute, several complex variables | mr=0077995 | year=1956 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=62 | pages=79–141 | doi=10.1090/S0002-9904-1956-10013-X | issue=2}}
* [[J. Arthur Seebach]], Linda A. Seebach & [[Lynn A. Steen]] (1970) "What is a Sheaf", [[American Mathematical Monthly]] 77:681–703 {{MR|id=0263073}}.
* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Faisceaux algébriques cohérents | url=http://www.mat.uniroma1.it/people/arbarello/FAC.pdf | mr=0068874 | year=1955 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=61 | pages=197–278 | doi=10.2307/1969915 | jstor=1969915 | issue=2 }}
* {{Citation | last1=Swan | first1=Richard G. | authorlink=Richard Swan| title=The Theory of Sheaves | publisher=[[University of Chicago Press]]| year=1964}} (concise lecture notes)
* {{Citation | last1=Tennison | first1=Barry R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975}} (pedagogic treatment)

== External links ==
* {{planetmath reference|id=5648|title=Sheaf}}

[[Category:Sheaf theory|*]]
[[Category:Topological methods of algebraic geometry]]
[[Category:Algebraic topology]]

Sheaf (mathematics)

2018-12-07T10:57:18Z

Crasshopper: /* Examples */

{{about|sheaves on [[topological space]]s|sheaves on a site|Grothendieck topology|and|Topos}}
In [[mathematics]], a '''sheaf''' is a tool for systematically tracking locally defined data attached to the [[open set]]s of a [[topological space]]. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the [[ring (mathematics)|ring]]s of [[continuous function|continuous]] or [[smooth function|smooth]] [[real numbers|real]]-valued [[function (mathematics)|function]]s defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of [[set (mathematics)|sets]] or sheaves of rings, depending on the type of data assigned to open sets.

There are also [[map (mathematics)|map]]s (or [[morphism]]s) from one sheaf to another; sheaves (of a specific type, such as sheaves of [[abelian group]]s) with their [[morphism]]s on a fixed topological space form a [[category (mathematics)|category]]. On the other hand, to each [[continuous map]] there is associated both a [[direct image functor]], taking sheaves and their morphisms on the [[domain (mathematics)|domain]] to sheaves and morphisms on the [[codomain]], and an [[inverse image functor]] operating in the opposite direction. These [[functor]]s, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in topology and especially in [[algebraic geometry|algebraic]] and [[differential geometry]]. First, geometric structures such as that of a [[differentiable manifold]] or a [[scheme (mathematics)|scheme]] can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as [[vector bundles]] or [[divisor (algebraic geometry)|divisors]] are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general [[sheaf cohomology|cohomology theory]], which encompasses also the "usual" topological cohomology theories such as [[singular cohomology]]. Especially in algebraic geometry and the theory of [[complex manifold]]s, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of [[D-module]]s, which provide applications to the theory of [[differential equation]]s. In addition, generalisations of sheaves to more general settings than topological spaces, such as [[Grothendieck topology]], have provided applications to [[mathematical logic]] and [[number theory]].

== Overview ==
In [[topology]], [[differential geometry]], and [[algebraic geometry]], several structures defined on a [[topological space]] (e.g., a [[differentiable manifold]]) can be naturally ''localised'' or ''restricted'' to [[open set|open]] [[subset]]s of the space: typical examples include [[continuous function|continuous]] [[real numbers|real]] or [[complex number|complex]]-valued functions, ''n'' times [[differentiable function|differentiable]] (real or complex-valued) functions, [[bounded function|bounded]] real-valued functions, [[vector field]]s, and [[section (fiber bundle)|section]]s of any [[vector bundle]] on the space.

''Presheaves'' formalise the situation common to the examples above: a presheaf (of sets) on a topological space is a structure that associates to each open set ''U'' of the space a set ''F''(''U'') of ''sections'' on ''U'', and to each open set ''V'' included in ''U'' a map ''F''(''U'') → ''F''(''V'') giving ''restrictions'' of sections over ''U'' to ''V''. Each of the examples above defines a presheaf by taking the restriction maps to be the usual restriction of functions, vector fields and sections of a vector bundle. Moreover, in each of these examples the sets of sections have additional [[algebraic structure]]: pointwise operations make them [[abelian group]]s, and in the examples of real and complex-valued functions the sets of sections even have a [[ring (mathematics)|ring]] structure. In addition, in each example the restriction maps are [[homomorphism]]s of the corresponding algebraic structure. This observation leads to the natural definition of presheaves with additional algebraic structure such as presheaves of groups, of abelian groups, of rings: sets of sections are required to have the specified algebraic structure, and the restrictions are required to be homomorphisms. Thus for example continuous real-valued functions on a topological space form a presheaf of rings on the space.

Given a presheaf, a natural question to ask is to what extent its sections over an open set ''U'' are specified by their restrictions to smaller open sets ''V''''i'' of an [[open cover]] of ''U''. A presheaf is ''separated'' if its sections are "locally determined": whenever two sections over ''U'' coincide when restricted to each of ''V''''i'', the two sections are identical. All examples of presheaves discussed above are separated, since in each case the sections are specified by their values at the points of the underlying space. Finally, a separated presheaf is a ''sheaf'' if ''compatible sections can be glued together'', i.e., whenever there is a section of the presheaf over each of the covering sets ''V''''i'', chosen so that they match on the overlaps of the covering sets, these sections correspond to a (unique) section on ''U'', of which they are restrictions. It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is ''local'' in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.

On the other hand, a function can be bounded on each set of an (infinite) open cover of a space without being bounded on all of the space; thus bounded functions provide an example of a presheaf that in general fails to be a sheaf. Another example of a presheaf that fails to be a sheaf is the ''constant presheaf'' that associates the same fixed set (or abelian group, or ring,...) to each open set: it follows from the gluing property of sheaves that the set of sections on a disjoint union of two open sets is the [[Cartesian product]] of the sets of sections over the two open sets. The correct way to define the [[constant sheaf]] ''FA'' (associated to for instance a set ''A'') on a topological space is to require sections on an open set ''U'' to be continuous maps from ''U'' to ''A'' equipped with the [[discrete topology]]; then in particular ''FA''(''U'') = ''A'' for [[connected space|connected]] ''U''.

Maps between sheaves or presheaves (called [[morphism]]s), consist of maps between the sets of sections over each open set of the underlying space, compatible with restrictions of sections. If the presheaves or sheaves considered are provided with additional algebraic structure, these maps are assumed to be homomorphisms. Sheaves endowed with nontrivial endomorphisms, such as the action of an [[algebraic torus]] or a [[Galois group]], are of particular interest.

Presheaves and sheaves are typically denoted by capital letters, ''F'' being particularly common, presumably for the [[French language|French]] word for sheaves, ''faisceaux''. Use of calligraphic letters such as <math>\mathcal{F}</math> is also common.

== Formal definitions ==
The first step in defining a sheaf is to define a ''presheaf'', which captures the idea of associating data and restriction maps to the open sets of a topological space. The second step is to require the normalisation and gluing axioms. A presheaf that satisfies these axioms is a sheaf.

=== Presheaves ===
{{See also|Presheaf (category theory)}}

Let ''X'' be a topological space, and let '''C''' be a [[category (category theory)|category]]. Usually '''C''' is the [[category of sets]], the [[category of groups]], the [[category of abelian groups]], or the [[category of commutative rings]]. A '''presheaf''' ''F'' on ''X'' is a functor with values in '''C''' given by the following data:
*For each open set ''U'' of ''X'', there corresponds an object ''F''(''U'') in '''C'''
*For each inclusion of open sets ''V'' ⊆ ''U'', there corresponds a [[morphism]] <math>\operatorname{res}_{V, U} \colon F(U) \rightarrow F(V)</math> in the category '''C'''.
The morphisms res''V'',''U'' are called '''restriction morphisms'''. If {{nowrap|''s'' ∈ ''F''(''U'')}}, then its restriction {{nowrap|res''V'',''U''(s)}} is often denoted ''s''|''V'' by analogy with restriction of functions. The restriction morphisms are required to satisfy two properties:
*For every open set ''U'' of ''X'', the restriction morphism res''U'',''U'' : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
*If we have three open sets ''W'' ⊆ ''V'' ⊆ ''U'', then the [[function composition|composite]] {{nowrap|1=res''W'',''V'' o res''V'',''U'' equals res''W'',''U''.}}
Informally, the second axiom says it doesn't matter whether we restrict to ''W'' in one step or restrict first to ''V'', then to ''W''.

There is a compact way to express the notion of a presheaf in terms of [[category theory]]. First we define the category of open sets on ''X'' to be the [[posetal category]] ''O''(''X'') whose objects are the open sets of ''X'' and whose morphisms are inclusions. Then a '''C'''-valued presheaf on ''X'' is the same as a [[contravariant functor]] from ''O''(''X'') to '''C'''. This definition can be generalized to the case when the source category is not of the form ''O''(''X'') for any ''X''; see [[presheaf (category theory)]].

If ''F'' is a '''C'''-valued presheaf on ''X'', and ''U'' is an open subset of ''X'', then ''F''(''U'') is called the '''sections of ''F'' over ''U'''''. If '''C''' is a [[concrete category]], then each element of ''F''(''U'') is called a '''section'''. A section over ''X'' is called a '''global section'''. A common notation (used also below) for the restriction res''V'',''U''(''s'') of a section is ''s''|''V''. This terminology and notation is by analogy with sections of [[fiber bundle]]s or sections of the étalé space of a sheaf; see below. ''F''(''U'') is also often denoted Γ(''U'',''F''), especially in contexts such as [[sheaf cohomology]] where ''U'' tends to be fixed and ''F'' tends to be variable.

=== Sheaves ===
For simplicity, consider first the case where the sheaf takes values in the category of sets. In fact, this definition applies more generally to the situation where the category is a [[concrete category]] whose underlying set functor is [[Conservative functor|conservative]], meaning that if the underlying map of sets is a bijection, then the original morphism is an isomorphism.

A ''sheaf'' is a presheaf with values in the category of sets that satisfies the following two axioms:
# (Locality) If (''U''''i'') is an open [[cover (topology)|covering]] of an open set ''U'', and if ''s'',''t'' ∈ ''F''(''U'') are such that ''s''|''U''''i'' = ''t''|''U''''i'' for each set ''U''''i'' of the covering, then ''s'' = ''t''; and
# ([[Gluing axiom|Gluing]]) If (''U''''i'') is an open covering of an open set ''U'', and if for each ''i'' a section ''s''''i'' ∈ ''F''(''U''''i'') is given such that for each pair ''U''''i'',''U''''j'' of the covering sets the restrictions of ''s''''i'' and ''s''''j'' agree on the overlaps: ''s''''i''|''U''''i''∩''U''''j'' = ''s''''j''|''U''''i''∩''U''''j'', then there is a section ''s'' ∈ ''F''(''U'') such that ''s''|''U''''i'' = ''s''''i'' for each ''i''.

The section ''s'' whose existence is guaranteed by axiom 2 is called the '''gluing''', '''concatenation''', or '''collation''' of the sections ''s''''i''. By axiom 1 it is unique. Sections ''s''''i'' satisfying the condition of axiom 2 are often called ''compatible''; thus axioms 1 and 2 together state that ''compatible sections can be uniquely glued together''. A '''separated presheaf''', or '''monopresheaf''', is a presheaf satisfying axiom 1.<ref>{{Citation | last1=Tennison | first1=B. R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975}}</ref>

If '''C''' has [[product (category theory)|products]], the sheaf axioms are equivalent to the requirement that, for any open covering ''U''''i'', the first arrow in the following diagram is an [[equalizer (mathematics)|equalizer]]:

:<math>F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i, j} F(U_i \cap U_j).</math>

Here the first map is the product of the restriction maps

:<math>\operatorname{res}_{U_i, U} \colon F(U) \rightarrow F(U_i)</math>

and the pair of arrows the products of the two sets of restrictions

:<math>\operatorname{res}_{U_i \cap U_j, U_i} \colon F(U_i) \rightarrow F(U_i \cap U_j)</math>

and

:<math>\operatorname{res}_{U_i \cap U_j, U_j} \colon F(U_j) \rightarrow F(U_i \cap U_j).</math>

For a separated presheaf, the first arrow need only be injective.

In general, for an open set ''U'' and open covering (''U''''i''), construct a category ''J'' whose objects are the sets ''Ui'' and the intersections {{nowrap|''Ui'' ∩ ''Uj''}} and whose morphisms are the inclusions of {{nowrap|''Ui'' ∩ ''Uj''}} in ''Ui'' and ''Uj''. The sheaf axioms for ''U'' and (''U''''i'') are that the [[limit (category theory)|limit]] of the functor ''F'' restricted to the category ''J'' must be isomorphic to ''F''(''U'').

Notice that the empty subset of a topological space is covered by the empty family of sets. The product of an empty family or the limit of an empty family is a terminal object, and consequently the value of a sheaf on the empty set must be a terminal object. If sheaf values are in the category of sets, applying the local identity axiom to the empty family shows that over the empty set, there is at most one section, and applying the gluing axiom to the empty family shows that there is at least one section. This property is called the '''normalisation axiom'''.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a [[basis (topology)|basis]] for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis. (see [[gluing axiom#Sheaves on a basis of open sets]].)

=== Morphisms ===
A morphism of sheaves is a clean function between them. Because sheaves contain data relative to every open set of a topological space, a morphism of sheaves is defined as a collection of functions, one for each open set. The collection satisfies a compatibility condition.

Let ''F'' and ''G'' be two sheaves on ''X'' with values in the category '''C'''. A ''[[morphism]]'' <math>\varphi:G\to F</math> consists of a morphism <math>\varphi_U:G(U)\to F(U)</math> for each open set {{mvar|U}} of {{mvar|X}}, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset ''V'' of an open set ''U'', the following diagram is [[commutative diagram|commutative]].

:<math>\begin{array}{rcl}
G(U) & \xrightarrow{\quad\varphi_U\quad} & F(U)\\
r_{V,U}\Biggl\downarrow & & \Biggl\downarrow r_{V,U}\\
G(V) & \xrightarrow[{\quad\varphi_V\quad}]{} & F(V)
\end{array}</math>

Sheaves are themselves functors, from "data" to a space. In this language, a morphism of sheaves is a [[natural transformation]] of the corresponding functors. With this notion of morphism, there is a category of '''C'''-valued sheaves on ''X'' for any '''C'''. The objects are the '''C'''-valued sheaves, and the morphisms are morphisms of sheaves. An ''[[isomorphism]]'' of sheaves is an isomorphism in this category.

An isomorphism of sheaves is an isomorphism on each open set ''U''. In other words, φ is an isomorphism if and only if for each ''U'', φ(''U'') is an isomorphism. A morphism of sheaves φ is an isomorphism if and only if there exists an open cover <math>\{U_\alpha\}</math> such that <math>\varphi|_{U_\alpha}</math> are isomorphisms of sheaves for all <math>\alpha</math>. The same facts are true of [[monomorphism]]s. However, they are false for [[epimorphism]]s, and their failure is measured by [[sheaf cohomology]].

Notice that we did not use the gluing axiom in defining a morphism of sheaves. Consequently, the above definition makes sense for presheaves as well. The category of '''C'''-valued presheaves is then a [[functor category]], the category of contravariant functors from ''O''(''X'') to '''C'''.

== Examples ==
Because sheaves encode exactly the data needed to pass between local and global situations, they have been used across mathematical subfields—wherever local and global facts [[Hairy ball theorem|differ]]. Here are some additional examples of sheaves:

* Any continuous map of topological spaces determines a sheaf of sets. Let ''f'' : ''Y'' → ''X'' be a continuous map. We define a sheaf Γ(''Y''/''X'') on ''X'' by setting Γ(''Y''/''X'')(U) equal to the sections ''U'' → ''Y'', that is, Γ(''Y''/''X'')(U) is the set of all continuous functions ''s'' : ''U'' → ''Y'' such that ''f ∘ s'' = ''id''''U''. Restriction is given by restriction of functions. This sheaf is called the '''sheaf of sections''' of ''f'', and it is especially important when ''f'' is the projection of a [[fiber bundle]] onto its base space. Notice that if the image of ''f'' does not contain ''U'', then Γ(''Y''/''X'')(''U'') is empty. For a concrete example, take ''X'' = '''C''' \ {0}, ''Y'' = '''C''', and ''f''(''z'') = exp(''z''). Γ(''Y''/''X'')(''U'') is the set of branches of the logarithm on ''U''.
* Fix a point ''x'' in ''X'' and an object ''S'' in a category '''C'''. The ''skyscraper sheaf over ''x'' with stalk'' ''S'' is the sheaf ''S''''x'' defined as follows: If ''U'' is an open set containing ''x'', then ''S''''x''(''U'') = ''S''. If ''U'' does not contain ''x'', then ''S''''x''(''U'') is the terminal object of '''C'''. The restriction maps are either the identity on ''S'', if both open sets contain ''x'', or the unique map from ''S'' to the terminal object of '''C'''.

=== Sheaves on manifolds ===
In the following examples ''M'' is an ''n''-dimensional ''C''''k''-manifold. The table lists the values of certain sheaves over open subsets ''U'' of ''M'' and their restriction maps.

{| class="wikitable"
|-
! Sheaf !! Sections over an open set ''U'' !! Restriction maps !! Remarks
|-
! Sheaf of ''j''-times continuously differentiable functions <math>\mathcal{O}^j_M</math>, ''j'' ≤ ''k''
| ''C''''j''-functions ''U'' → '''R'''
| Restriction of functions.
| This is a sheaf of rings with addition and multiplication given by pointwise addition and multiplication. When ''j'' = ''k'', this sheaf is called the '''structure sheaf''' and is denoted <math>\mathcal{O}_M</math>.
|-
! Sheaf of nonzero ''k''-times continuously differentiable functions <math>\mathcal{O}_X^\times</math>
| Nowhere zero ''C''''k''-functions ''U'' → '''R'''
| Restriction of functions.
| A sheaf of groups under pointwise multiplication.
|-
! '''Cotangent sheaves''' Ω''p''''M''
| [[Differential form]]s of degree ''p'' on ''U''
| Restriction of differential forms.
| Ω1''M'' and Ωn''M'' are commonly denoted Ω''M'' and ω''M'', respectively.
|-
! '''Sheaf of distributions''' {{mathcal|DB}}
| [[Distribution (mathematics)|Distribution]]s on ''U''
| The dual map to extension of smooth compactly supported functions by zero.
| Here ''M'' is assumed to be smooth.
|-
! '''Sheaf of differential operators''' <math>\mathcal{D}_M</math>
| Finite-order [[differential operator]]s on ''U''
| Restriction of differential operators.
| Here ''M'' is assumed to be smooth.
|}

=== Presheaves that are not sheaves ===
Here are two examples of presheaves that are not sheaves:
* Let ''X'' be the [[discrete two-point space|two-point topological space]] {''x'', ''y''} with the discrete topology. Define a presheaf ''F'' as follows: ''F''(∅) = {∅}, ''F''({''x''}) = '''R''', ''F''({''y''}) = '''R''', ''F''({''x'', ''y''}) = '''R''' × '''R''' × '''R'''. The restriction map ''F''({''x'', ''y''}) → ''F''({''x''}) is the projection of '''R''' × '''R''' × '''R''' onto its first coordinate, and the restriction map ''F''({''x'', ''y''}) → ''F''({''y''}) is the projection of '''R''' × '''R''' × '''R''' onto its second coordinate. ''F'' is a presheaf that is not separated: A global section is determined by three numbers, but the values of that section over {''x''} and {''y''} determine only two of those numbers. So while we can glue any two sections over {''x''} and {''y''}, we cannot glue them uniquely.
* Let ''X'' be the [[real line]], and let ''F''(''U'') be the set of [[bounded function|bounded]] continuous functions on ''U''. This is not a sheaf because it is not always possible to glue. For example, let ''U''''i'' be the set of all ''x'' such that |''x''| < ''i''. The identity function ''f''(''x'') = ''x'' is bounded on each ''U''''i''. Consequently we get a section ''s''''i'' on ''U''''i''. However, these sections do not glue, because the function ''f'' is not bounded on the real line. Consequently ''F'' is a presheaf, but not a sheaf. In fact, ''F'' is separated because it is a sub-presheaf of the sheaf of continuous functions.

== Turning a presheaf into a sheaf ==
It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf ''F'' and produces a new sheaf ''aF'' called the '''sheaving''', '''sheafification''' or '''sheaf associated to the presheaf''' ''F''. The functor ''a'' is called the '''sheaving functor''', '''sheafification functor''', or '''associated sheaf functor'''. There is a natural morphism of presheaves <math>i\colon F\to aF</math> that has the universal property that for any sheaf ''G'' and any morphism of presheaves <math>f\colon F\to G</math>, there is a unique morphism of sheaves <math>\tilde f \colon aF \rightarrow G</math> such that <math>f = \tilde f i</math>. In fact ''a'' is the left [[adjoint functor]] to the inclusion functor (or [[forgetful functor]]) from the category of sheaves to the category of presheaves, and ''i'' is the [[adjoint functor#Unit and co-unit|unit]] of the adjunction. In this way, the category of sheaves turns into a [[Giraud subcategory]] of presheaves.

One concrete way of constructing the sheaf ''aF'' is to identify it with the sheaf of sections of an appropriate topological space. This space is analogous to the [[#The étalé space of a sheaf|étalé space]] of a sheaf. Briefly, the underlying set of the topological space is the disjoint union of the [[#Stalks of a sheaf|stalks]] of ''F'', denoted {{nowrap|Spé ''F''}}. There is a natural map {{nowrap|φ : Spé ''F'' → ''X''}} that sends each germ to the point of ''X'' over which it lies. For each open set ''U'' and each section ''s'' of ''F'' over ''U'', we define a section <math>\bar s</math> of φ that sends ''x'' to the germ ''s''''x''. Then {{nowrap|Spé ''F''}} is given the finest topology for which all sections <math>\bar s</math> are continuous, and ''aF'' is the sheaf of continuous sections of φ for this topology.

There are other constructions of the sheaf ''aF''. In particular, [[Alexander Grothendieck]] and [[Jean-Louis Verdier]] ([[Séminaire de Géométrie Algébrique du Bois Marie#SGA 4|SGA 4]] II 3.0.5) define a functor ''L'' from presheaves to presheaves which, when applied to a presheaf, yields a separated presheaf and, when applied to a separated presheaf, yields a sheaf. Applying the functor ''L'' twice therefore turns a presheaf into a sheaf, and in fact ''LLF'' is the associated sheaf ''aF''.

== Operations ==
If ''K'' is a [[Subobject|subsheaf]] of a sheaf ''F'' of abelian groups, then the '''quotient sheaf''' ''Q'' is the sheaf associated to the presheaf <math>U \mapsto F(U)/K(U)</math>; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;
:<math>0 \to K \to F \to Q \to 0.</math>
(this is also called a [[sheaf extension]].)

Let ''F'', ''G'' be sheaves of abelian groups. The set of morphisms of sheaves from ''F'' to ''G'' forms an abelian group (by the abelian group structure of ''G''). The '''sheaf hom''' of ''F'' and ''G'', denoted by,
:<math>\mathcal{Hom}(F, G)</math>
is the sheaf of abelian groups <math>U \mapsto \operatorname{\mathcal{Hom}}(F|_U, G|_U)</math> where <math>F|_U</math> is the sheaf on ''U'' given by <math>(F|_U)(V) = F(V)</math> (Note sheafification is not needed here). The direct sum of ''F'' and ''G'' is the sheaf given by <math>U \mapsto F(U) \oplus G(U) </math>, and the tensor product of ''F'' and ''G'' is the sheaf associated to the presheaf <math>U \mapsto F(U) \otimes G(U)</math>.

All of these operations extend to [[sheaf of modules|sheaves of modules]] over a [[sheaf of rings]] ''A''; the above is the special case when ''A'' is the [[constant sheaf]] <math>\underline{\mathbf{Z}}</math>.

== Images of sheaves ==
{{Images of sheaves}}
{{Main|Image functors for sheaves}}

The definition of a morphism on sheaves makes sense only for sheaves on the same space ''X''. This is because the data contained in a sheaf is indexed by the open sets of the space. If we have two sheaves on different spaces, then their data is indexed differently. There is no way to go directly from one set of data to the other.

However, it is possible to move a sheaf from one space to another using a continuous function. Let ''f'' : ''X'' → ''Y'' be a continuous function from a topological space ''X'' to a topological space ''Y''. If we have a sheaf on ''X'', we can move it to ''Y'', and vice versa. There are four ways in which sheaves can be moved.
* A sheaf <math>\mathcal{F}</math> on ''X'' can be moved to ''Y'' using the [[direct image functor]] <math>f_*</math> or the [[direct image with proper support functor]] <math>f_!</math>.
* A sheaf <math>\mathcal{G}</math> on ''Y'' can be moved to ''X'' using the [[inverse image functor]] <math>f^{-1}</math> or the [[twisted inverse image functor]] <math>f^!</math>.

The twisted inverse image functor <math>f^!</math> is, in general, only defined as a functor between [[derived category|derived categories]]. These functors come in adjoint pairs: <math>f^{-1}</math> and <math>f_*</math> are left and right adjoints of each other, and <math>Rf_!</math> and <math>f^!</math> are left and right adjoints of each other. The functors are intertwined with each other by [[Coherent duality|Grothendieck duality]] and [[Verdier duality]].

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted <math>f^*</math> and it is distinct from <math>f^{-1}</math>. See [[inverse image functor]].

== Stalks of a sheaf ==
{{Main|Stalk (sheaf)}}

The '''stalk''' <math>\mathcal{F}_x</math> of a sheaf <math>\mathcal{F}</math> captures the properties of a sheaf "around" a point ''x'' ∈ ''X''.
Here, "around" means that, conceptually speaking, one looks at smaller and smaller [[neighborhood (mathematics)|neighborhoods]] of the point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.

The stalk is defined by
:<math>\mathcal{F}_x = \varinjlim_{U\ni x} \mathcal{F}(U),</math>
the [[direct limit]] being over all open subsets of ''X'' containing the given point ''x''. In other words, an element of the stalk is given by a section over some open neighborhood of ''x'', and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism ''F''(''U'') → ''F''''x'' takes a section ''s'' in ''F''(''U'') to its ''germ''. This generalises the usual definition of a [[germ (mathematics)|germ]].

A different way of defining the stalk is
:<math>\mathcal{F}_x := i^{-1}\mathcal{F}(\{x\}),</math>
where ''i'' is the inclusion of the one-point space {''x''} into ''X''. The equivalence follows from the definition of the [[inverse image functor|inverse image]].

In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. They also find use in constructions such as [[Godement resolution]]s.

== Ringed spaces and locally ringed spaces ==
{{Main|Ringed space}}

A pair <math>(X, \mathcal{O}_X)</math> consisting of a topological space ''X'' and a sheaf of rings on ''X'' is called a '''[[ringed space]]'''. Many types of spaces can be defined as certain types of ringed spaces. The sheaf <math>\mathcal{O}_X</math> is called the '''structure sheaf''' of the space. A very common situation is when all the stalks of the structure sheaf are [[local ring]]s, in which case the pair is called a '''locally ringed space'''. Here are examples of definitions made in this way:
* An ''n''-dimensional ''C''''k'' manifold ''M'' is a locally ringed space whose structure sheaf is an <math>\underline{\mathbf{R}}</math>-algebra and is locally isomorphic to the sheaf of ''C''k real-valued functions on '''R'''''n''.
* A [[complex analytic space]] is a locally ringed space whose structure sheaf is a <math>\underline{\mathbf{C}}</math>-algebra and is locally isomorphic to the vanishing locus of a finite set of holomorphic functions together with the restriction (to the vanishing locus) of the sheaf of holomorphic functions on '''C'''''n'' for some ''n''.
* A [[scheme (mathematics)|scheme]] is a locally ringed space that is locally isomorphic to the [[spectrum of a ring]].
* A [[semialgebraic space]] is a locally ringed space that is locally isomorphic to a [[semialgebraic set]] in Euclidean space together with its sheaf of semialgebraic functions.

== Sheaves of modules ==
{{main|Sheaf of modules}}
Let <math>(X, \mathcal{O}_X)</math> be a ringed space. A '''sheaf of modules''' is a sheaf <math>\mathcal{M}</math> such that on every open set ''U'' of ''X'', <math>\mathcal{M}(U)</math> is an <math>\mathcal{O}_X(U)</math>-module and for every inclusion of open sets ''V'' ⊆ ''U'', the restriction map <math>\mathcal{M}(U) \to \mathcal{M}(V)</math> is a homomorphism of <math>\mathcal{O}_X(U)</math>-modules.

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between [[vector bundle]]s and [[locally free sheaf|locally free sheaves]] of <math>\mathcal{O}_X</math>-modules. Sheaves of solutions to differential equations are [[D-module]]s, that is, modules over the sheaf of differential operators.

A particularly important case are [[abelian sheaf|abelian sheaves]], which are modules over the constant sheaf <math>\underline{\mathbf{Z}}</math>. Every sheaf of modules is an abelian sheaf.

=== Finiteness conditions for sheaves of modules ===
{{Further|Coherent sheaf}}
The condition that a module is finitely generated or finitely presented can also be formulated for a sheaf of modules. <math>\mathcal{M}</math> is '''finitely generated''' if, for every point ''x'' of ''X'', there exists an open neighborhood ''U'' of ''x'', a natural number ''n'' (possibly depending on ''U''), and a surjective morphism of sheaves <math>\mathcal{O}_X^n|_U \to \mathcal{M}|_U</math>. Similarly, <math>\mathcal{M}</math> is '''finitely presented''' if in addition there exists a natural number ''m'' (again possibly depending on ''U'') and a morphism of sheaves <math>\mathcal{O}_X^m|_U \to \mathcal{O}_X^n|_U</math> such that the sequence of morphisms <math>\mathcal{O}_X^m|_U \to \mathcal{O}_X^n|_U \to \mathcal{M}</math> is exact. Equivalently, the kernel of the morphism <math>\mathcal{O}_X^n|_U \to \mathcal{M}</math> is itself a finitely generated sheaf.

These, however, are not the only possible finiteness conditions on a sheaf. The most important finiteness condition for a sheaf is coherence. <math>\mathcal{M}</math> is '''coherent''' if it is of finite type and if, for every open set ''U'' and every morphism of sheaves <math>\phi : \mathcal{O}_X^n \to \mathcal{M}</math> (not necessarily surjective), the kernel of φ is of finite type. <math>\mathcal{O}_X</math> is '''coherent''' if it is coherent as a module over itself. Note that coherence is a strictly stronger condition than finite presentation: <math>\mathcal{O}_X</math> is always finitely presented as a module over itself, but it is not always coherent. For example, let ''X'' be a point, let <math>\mathcal{O}_X</math> be the ring {{nowrap begin}}''R'' = '''C'''[''x''1, ''x''2, ...]{{nowrap end}} of complex polynomials in countably many indeterminates. Choose {{nowrap begin}}''n'' = 1{{nowrap end}}, and for the morphism φ, take the map that sends every variable to zero. The kernel of this map is not finitely generated, so <math>\mathcal{O}_X</math> is not coherent.

== The étalé space of a sheaf ==
{{anchor|Etale space}}
In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called the ''étalé space'', from the French word étalé {{IPA-fr|etale|}}, meaning roughly "spread out". If ''F'' is a sheaf over ''X'', then the '''étalé space''' of ''F'' is a topological space ''E'' together with a [[local homeomorphism]] ''π'' : ''E'' → ''X'' such that the sheaf of sections of ''π'' is ''F''. The space ''E'' is usually very strange, and even if the sheaf ''F'' arises from a natural topological situation, ''E'' may not have any clear topological interpretation. For example, if ''F'' is the sheaf of sections of a continuous function ''f'' : ''Y'' → ''X'', then ''E'' = ''Y'' if and only if ''f'' is a [[local homeomorphism]].

The étalé space ''E'' is constructed from the stalks of ''F'' over ''X''. As a set, it is their [[disjoint union]] and ''π'' is the obvious map that takes the value ''x'' on the stalk of ''F'' over ''x'' ∈ ''X''. The topology of ''E'' is defined as follows. For each element ''s'' of ''F''(''U'') and each ''x'' in ''U'', we get a germ of ''s'' at ''x''. These germs determine points of ''E''. For any ''U'' and ''s'' ∈ ''F''(''U''), the union of these points (for all ''x'' ∈ ''U'') is declared to be open in ''E''. Notice that each stalk has the [[discrete topology]] as subspace topology. Two morphisms between sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.

The construction above determines an [[equivalence of categories]] between the category of sheaves of sets on ''X'' and the category of étalé spaces over ''X''. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf.

This construction makes all sheaves into [[representable functor]]s on certain categories of topological spaces. As above, let ''F'' be a sheaf on ''X'', let ''E'' be its étalé space, and let ''π'' : ''E'' → ''X'' be the natural projection. Consider the category '''Top'''/''X'' of topological spaces over ''X'', that is, the category of topological spaces together with fixed continuous maps to ''X''. Every object of this space is a continuous map ''f'' : ''Y'' → ''X'', and a morphism from ''Y'' → ''X'' to ''Z'' → ''X'' is a continuous map ''Y'' → ''Z'' that commutes with the two maps to ''X''. There is a functor Γ from '''Top'''/''X'' to the category of sets that takes an object ''f'' : ''Y'' → ''X'' to (''f''−1''F'')(''Y''). For example, if ''i'' : ''U'' → ''X'' is the inclusion of an open subset, then Γ(''i'') = (''i''−1''F'')(''U'') agrees with the usual ''F''(''U''), and if ''i'' : {''x''} → ''X'' is the inclusion of a point, then Γ({''x''}) = (''i''−1''F'')({''x''}) is the stalk of ''F'' at ''x''. There is a natural isomorphism
:<math>(f^{-1}F)(Y) \cong \operatorname{Hom}_{\mathbf{Top}/X}(f, \pi)</math>,
which shows that ''E'' represents the functor Γ.

''E'' is constructed so that the projection map π is a covering map. In algebraic geometry, the natural analog of a covering map is called an [[étale morphism]]. Despite its similarity to "étalé", the word étale {{IPA-fr|etal|}} has a different meaning in French. It is possible to turn ''E'' into a [[scheme (mathematics)|scheme]] and π into a morphism of schemes in such a way that π retains the same universal property, but π is ''not'' in general an étale morphism because it is not quasi-finite. It is, however, formally étale.

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as [[mathematical analysis]].

== Sheaf cohomology ==
{{Main|Sheaf cohomology}}

It was noted above that the functor <math>\Gamma(U,-)</math> preserves isomorphisms and monomorphisms, but not epimorphisms. If ''F'' is a sheaf of abelian groups, or more generally a sheaf with values in an [[abelian category]], then <math>\Gamma(U,-)</math> is actually a [[left exact functor]]. This means that it is possible to construct [[derived functor]]s of <math>\Gamma(U,-)</math>. These derived functors are called the ''cohomology groups'' (or ''modules'') of ''F'' and are written <math>H^i(U,-)</math>. Grothendieck proved in his "[[Tohoku paper]]" ({{harvtxt|Grothendieck|1957}}) that every category of sheaves of abelian groups contains enough [[injective object]]s, so these derived functors always exist.

However, computing sheaf cohomology using injective resolutions is nearly impossible. In practice, it is much more common to find a different and more tractable resolution of ''F''. A general construction is provided by [[Godement resolution]]s, and particular resolutions may be constructed using [[soft sheaf|soft sheaves]], [[fine sheaf|fine sheaves]], and [[flabby sheaf|flabby sheaves]] (also known as ''flasque sheaves'' from the French ''flasque'' meaning flabby). As a consequence, it can become possible to compare sheaf cohomology with other cohomology theories. For example, the [[de Rham complex]] is a resolution of the constant sheaf <math>\underline{\mathbf{R}}</math> on any smooth manifold, so the sheaf cohomology of <math>\underline{\mathbf{R}}</math> is equal to its [[de Rham cohomology]]. In fact, comparing sheaf cohomology to de Rham cohomology and singular cohomology provides a proof of de Rham's theorem that the two cohomology theories are isomorphic.

A different approach is by [[Čech cohomology]]. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct <math>H^1</math> but incorrect higher cohomology groups. To get around this, [[Jean-Louis Verdier]] developed [[hypercover]]ings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of [[Pierre Deligne]]'s [[mixed Hodge structure]]s.

A much cleaner approach to the computation of some cohomology groups is the [[Borel–Bott–Weil theorem]], which identifies the cohomology groups of some [[line bundle]]s on [[flag manifold]]s with [[irreducible representation]]s of [[Lie group]]s. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space.

In many cases there is a duality theory for sheaves that generalizes [[Poincaré duality]]. See [[Coherent duality|Grothendieck duality]] and [[Verdier duality]].

== Sites and topoi ==
{{Main|Grothendieck topology|Topos}}

[[André Weil]]'s [[Weil conjectures]] stated that there was a [[Weil cohomology theory|cohomology theory]] for [[algebraic variety|algebraic varieties]] over [[finite field]]s that would give an analogue of the [[Riemann hypothesis]]. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf <math>\underline{\mathbf{C}}</math> in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the [[Zariski topology]], and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). [[Alexandre Grothendieck]] solved this problem by introducing [[Grothendieck topology|Grothendieck topologies]], which axiomatize the notion of ''covering''. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to define [[étale cohomology]] and [[l-adic cohomology]], which eventually were used to prove the Weil conjectures.

A category with a Grothendieck topology is called a ''site''. A category of sheaves on a site is called a ''topos'' or a ''Grothendieck topos''. The notion of a topos was later abstracted by [[William Lawvere]] and Miles Tierney to define an [[elementary topos]], which has connections to [[mathematical logic]].

== History ==

{{unreferenced section|date=January 2016}}

The first origins of '''sheaf theory''' are hard to pin down — they may be co-extensive with the idea of [[analytic continuation]]{{Clarify|date=July 2010}}. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on [[cohomology]].
* 1936 [[Eduard Čech]] introduces the ''[[Nerve of an open covering|nerve]]'' construction, for associating a [[simplicial complex]] to an open covering.
* 1938 [[Hassler Whitney]] gives a 'modern' definition of cohomology, summarizing the work since [[James Waddell Alexander II|J. W. Alexander]] and [[Kolmogorov]] first defined ''[[cochain]]s''.
* 1943 [[Norman Steenrod]] publishes on homology ''with [[local coefficients]]''.
* 1945 [[Jean Leray]] publishes work carried out as a [[prisoner of war]], motivated by proving [[Fixed point (mathematics)|fixed point]] theorems for application to [[Partial differential equation|PDE]] theory; it is the start of sheaf theory and [[spectral sequence]]s.
* 1947 [[Henri Cartan]] reproves the [[de Rham theorem]] by sheaf methods, in correspondence with [[André Weil]] (see [[De Rham–Weil theorem]]). Leray gives a sheaf definition in his courses via closed sets (the later ''carapaces'').
* 1948 The Cartan seminar writes up sheaf theory for the first time.
* 1950 The "second edition" sheaf theory from the Cartan seminar: the [[sheaf space]] (''espace étalé'') definition is used, with stalkwise structure. [[Support (mathematics)|Support]]s are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time [[Kiyoshi Oka]] introduces an idea (adjacent to that) of a sheaf of ideals, in [[several complex variables]].
* 1951 The Cartan seminar proves [[Theorems A and B]], based on Oka's work.
* 1953 The finiteness theorem for [[coherent sheaf|coherent sheaves]] in the analytic theory is proved by Cartan and [[Jean-Pierre Serre]], as is [[Serre duality]].
* 1954 Serre's paper ''[[#CITEREFSerre1955|Faisceaux algébriques cohérents]]'' (published in 1955) introduces sheaves into [[algebraic geometry]]. These ideas are immediately exploited by [[Friedrich Hirzebruch]], who writes a major 1956 book on topological methods.
* 1955 [[Alexander Grothendieck]] in lectures in [[Kansas]] defines [[abelian category]] and ''presheaf'', and by using [[injective resolution]]s allows direct use of sheaf cohomology on all topological spaces, as [[derived functor]]s.
* 1956 [[Oscar Zariski]]'s report ''[[#CITEREFMartinChernZariski1956|Algebraic sheaf theory]]''
* 1957 Grothendieck's [[#CITEREFGrothendieck1957|''Tohoku'' paper]] rewrites [[homological algebra]]; he proves [[Coherent duality|Grothendieck duality]] (i.e., Serre duality for possibly [[Mathematical singularity|singular]] algebraic varieties).
* 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: [[Scheme (mathematics)|scheme]]s and general sheaves on them, [[local cohomology]], [[derived category|derived categories]] (with Verdier), and [[Grothendieck topologies]]. There emerges also his influential schematic idea of 'six operations' in homological algebra.
* 1958 [[Roger Godement]]'s book on sheaf theory is published. At around this time [[Mikio Sato]] proposes his [[hyperfunction]]s, which will turn out to have sheaf-theoretic nature.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to [[algebraic topology]]. It was later discovered that the logic in categories of sheaves is [[intuitionistic logic]] (this observation is now often referred to as [[Kripke–Joyal semantics]], but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as [[Gottfried Wilhelm Leibniz|Leibniz]].

== See also ==
* [[Coherent sheaf]]
* [[Cosheaf]]
* [[Gerbe]]
* [[Stack (mathematics)]]
* [[Sheaf of spectra]]
* [[Presheaf of spaces]]
* [[Base change theorems]]
* [[Locally constant sheaf]]
* [[Constructible sheaf]]

== Notes ==
{{Reflist}}

== References ==
* {{Citation | last1=Bredon | first1=Glen E. | author1-link = Glen Bredon | title=Sheaf theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94905-5 | mr=1481706 | edition=2nd | year=1997 | volume=170}} (oriented towards conventional topological applications)
* {{Citation | last1=Godement | first1=Roger | author1-link = Roger Godement | title=Topologie algébrique et théorie des faisceaux | publisher=Hermann | location=Paris | mr=0345092 | year=1973}}
* {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Sur quelques points d'algèbre homologique | mr=0102537 | year=1957 | journal=The Tohoku Mathematical Journal. Second Series | issn=0040-8735 | volume=9 | pages=119–221 | doi=10.2748/tmj/1178244839}}
* {{Citation | last1=Hirzebruch | first1=Friedrich | author1-link = Friedrich Hirzebruch | title=Topological methods in algebraic geometry | publisher=Springer-Verlag | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58663-0 | mr=1335917 | year=1995}} (updated edition of a classic using enough sheaf theory to show its power)
* {{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | last2=Schapira | first2=Pierre | title=Sheaves on manifolds | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-51861-7 | mr=1299726 | year=1994 | volume=292}}(advanced techniques such as the [[derived category]] and [[vanishing cycle]]s on the most reasonable spaces)
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link = Saunders Mac Lane | last2=Moerdijk | first2=Ieke | author2-link = Ieke Moerdijk | title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-0-387-97710-2 | mr=1300636 | year=1994}} (category theory and toposes emphasised)
* {{Citation | last1=Martin | first1=William T. | last2=Chern | first2=Shiing-Shen | author2-link=Shiing-Shen Chern | last3=Zariski | first3=Oscar | author3-link=Oscar Zariski | title=Scientific report on the Second Summer Institute, several complex variables | mr=0077995 | year=1956 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=62 | pages=79–141 | doi=10.1090/S0002-9904-1956-10013-X | issue=2}}
* [[J. Arthur Seebach]], Linda A. Seebach & [[Lynn A. Steen]] (1970) "What is a Sheaf", [[American Mathematical Monthly]] 77:681–703 {{MR|id=0263073}}.
* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Faisceaux algébriques cohérents | url=http://www.mat.uniroma1.it/people/arbarello/FAC.pdf | mr=0068874 | year=1955 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=61 | pages=197–278 | doi=10.2307/1969915 | jstor=1969915 | issue=2 }}
* {{Citation | last1=Swan | first1=Richard G. | authorlink=Richard Swan| title=The Theory of Sheaves | publisher=[[University of Chicago Press]]| year=1964}} (concise lecture notes)
* {{Citation | last1=Tennison | first1=Barry R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975}} (pedagogic treatment)

== External links ==
* {{planetmath reference|id=5648|title=Sheaf}}

[[Category:Sheaf theory|*]]
[[Category:Topological methods of algebraic geometry]]
[[Category:Algebraic topology]]

Sheaf (mathematics)

2018-12-07T10:52:04Z

Crasshopper: /* Morphisms */ not what the word “heuristic” means…

{{about|sheaves on [[topological space]]s|sheaves on a site|Grothendieck topology|and|Topos}}
In [[mathematics]], a '''sheaf''' is a tool for systematically tracking locally defined data attached to the [[open set]]s of a [[topological space]]. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the [[ring (mathematics)|ring]]s of [[continuous function|continuous]] or [[smooth function|smooth]] [[real numbers|real]]-valued [[function (mathematics)|function]]s defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of [[set (mathematics)|sets]] or sheaves of rings, depending on the type of data assigned to open sets.

There are also [[map (mathematics)|map]]s (or [[morphism]]s) from one sheaf to another; sheaves (of a specific type, such as sheaves of [[abelian group]]s) with their [[morphism]]s on a fixed topological space form a [[category (mathematics)|category]]. On the other hand, to each [[continuous map]] there is associated both a [[direct image functor]], taking sheaves and their morphisms on the [[domain (mathematics)|domain]] to sheaves and morphisms on the [[codomain]], and an [[inverse image functor]] operating in the opposite direction. These [[functor]]s, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in topology and especially in [[algebraic geometry|algebraic]] and [[differential geometry]]. First, geometric structures such as that of a [[differentiable manifold]] or a [[scheme (mathematics)|scheme]] can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as [[vector bundles]] or [[divisor (algebraic geometry)|divisors]] are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general [[sheaf cohomology|cohomology theory]], which encompasses also the "usual" topological cohomology theories such as [[singular cohomology]]. Especially in algebraic geometry and the theory of [[complex manifold]]s, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of [[D-module]]s, which provide applications to the theory of [[differential equation]]s. In addition, generalisations of sheaves to more general settings than topological spaces, such as [[Grothendieck topology]], have provided applications to [[mathematical logic]] and [[number theory]].

== Overview ==
In [[topology]], [[differential geometry]], and [[algebraic geometry]], several structures defined on a [[topological space]] (e.g., a [[differentiable manifold]]) can be naturally ''localised'' or ''restricted'' to [[open set|open]] [[subset]]s of the space: typical examples include [[continuous function|continuous]] [[real numbers|real]] or [[complex number|complex]]-valued functions, ''n'' times [[differentiable function|differentiable]] (real or complex-valued) functions, [[bounded function|bounded]] real-valued functions, [[vector field]]s, and [[section (fiber bundle)|section]]s of any [[vector bundle]] on the space.

''Presheaves'' formalise the situation common to the examples above: a presheaf (of sets) on a topological space is a structure that associates to each open set ''U'' of the space a set ''F''(''U'') of ''sections'' on ''U'', and to each open set ''V'' included in ''U'' a map ''F''(''U'') → ''F''(''V'') giving ''restrictions'' of sections over ''U'' to ''V''. Each of the examples above defines a presheaf by taking the restriction maps to be the usual restriction of functions, vector fields and sections of a vector bundle. Moreover, in each of these examples the sets of sections have additional [[algebraic structure]]: pointwise operations make them [[abelian group]]s, and in the examples of real and complex-valued functions the sets of sections even have a [[ring (mathematics)|ring]] structure. In addition, in each example the restriction maps are [[homomorphism]]s of the corresponding algebraic structure. This observation leads to the natural definition of presheaves with additional algebraic structure such as presheaves of groups, of abelian groups, of rings: sets of sections are required to have the specified algebraic structure, and the restrictions are required to be homomorphisms. Thus for example continuous real-valued functions on a topological space form a presheaf of rings on the space.

Given a presheaf, a natural question to ask is to what extent its sections over an open set ''U'' are specified by their restrictions to smaller open sets ''V''''i'' of an [[open cover]] of ''U''. A presheaf is ''separated'' if its sections are "locally determined": whenever two sections over ''U'' coincide when restricted to each of ''V''''i'', the two sections are identical. All examples of presheaves discussed above are separated, since in each case the sections are specified by their values at the points of the underlying space. Finally, a separated presheaf is a ''sheaf'' if ''compatible sections can be glued together'', i.e., whenever there is a section of the presheaf over each of the covering sets ''V''''i'', chosen so that they match on the overlaps of the covering sets, these sections correspond to a (unique) section on ''U'', of which they are restrictions. It is easy to verify that all examples above except the presheaf of bounded functions are in fact sheaves: in all cases the criterion of being a section of the presheaf is ''local'' in a sense that it is enough to verify it in an arbitrary neighbourhood of each point.

On the other hand, a function can be bounded on each set of an (infinite) open cover of a space without being bounded on all of the space; thus bounded functions provide an example of a presheaf that in general fails to be a sheaf. Another example of a presheaf that fails to be a sheaf is the ''constant presheaf'' that associates the same fixed set (or abelian group, or ring,...) to each open set: it follows from the gluing property of sheaves that the set of sections on a disjoint union of two open sets is the [[Cartesian product]] of the sets of sections over the two open sets. The correct way to define the [[constant sheaf]] ''FA'' (associated to for instance a set ''A'') on a topological space is to require sections on an open set ''U'' to be continuous maps from ''U'' to ''A'' equipped with the [[discrete topology]]; then in particular ''FA''(''U'') = ''A'' for [[connected space|connected]] ''U''.

Maps between sheaves or presheaves (called [[morphism]]s), consist of maps between the sets of sections over each open set of the underlying space, compatible with restrictions of sections. If the presheaves or sheaves considered are provided with additional algebraic structure, these maps are assumed to be homomorphisms. Sheaves endowed with nontrivial endomorphisms, such as the action of an [[algebraic torus]] or a [[Galois group]], are of particular interest.

Presheaves and sheaves are typically denoted by capital letters, ''F'' being particularly common, presumably for the [[French language|French]] word for sheaves, ''faisceaux''. Use of calligraphic letters such as <math>\mathcal{F}</math> is also common.

== Formal definitions ==
The first step in defining a sheaf is to define a ''presheaf'', which captures the idea of associating data and restriction maps to the open sets of a topological space. The second step is to require the normalisation and gluing axioms. A presheaf that satisfies these axioms is a sheaf.

=== Presheaves ===
{{See also|Presheaf (category theory)}}

Let ''X'' be a topological space, and let '''C''' be a [[category (category theory)|category]]. Usually '''C''' is the [[category of sets]], the [[category of groups]], the [[category of abelian groups]], or the [[category of commutative rings]]. A '''presheaf''' ''F'' on ''X'' is a functor with values in '''C''' given by the following data:
*For each open set ''U'' of ''X'', there corresponds an object ''F''(''U'') in '''C'''
*For each inclusion of open sets ''V'' ⊆ ''U'', there corresponds a [[morphism]] <math>\operatorname{res}_{V, U} \colon F(U) \rightarrow F(V)</math> in the category '''C'''.
The morphisms res''V'',''U'' are called '''restriction morphisms'''. If {{nowrap|''s'' ∈ ''F''(''U'')}}, then its restriction {{nowrap|res''V'',''U''(s)}} is often denoted ''s''|''V'' by analogy with restriction of functions. The restriction morphisms are required to satisfy two properties:
*For every open set ''U'' of ''X'', the restriction morphism res''U'',''U'' : ''F''(''U'') → ''F''(''U'') is the identity morphism on ''F''(''U'').
*If we have three open sets ''W'' ⊆ ''V'' ⊆ ''U'', then the [[function composition|composite]] {{nowrap|1=res''W'',''V'' o res''V'',''U'' equals res''W'',''U''.}}
Informally, the second axiom says it doesn't matter whether we restrict to ''W'' in one step or restrict first to ''V'', then to ''W''.

There is a compact way to express the notion of a presheaf in terms of [[category theory]]. First we define the category of open sets on ''X'' to be the [[posetal category]] ''O''(''X'') whose objects are the open sets of ''X'' and whose morphisms are inclusions. Then a '''C'''-valued presheaf on ''X'' is the same as a [[contravariant functor]] from ''O''(''X'') to '''C'''. This definition can be generalized to the case when the source category is not of the form ''O''(''X'') for any ''X''; see [[presheaf (category theory)]].

If ''F'' is a '''C'''-valued presheaf on ''X'', and ''U'' is an open subset of ''X'', then ''F''(''U'') is called the '''sections of ''F'' over ''U'''''. If '''C''' is a [[concrete category]], then each element of ''F''(''U'') is called a '''section'''. A section over ''X'' is called a '''global section'''. A common notation (used also below) for the restriction res''V'',''U''(''s'') of a section is ''s''|''V''. This terminology and notation is by analogy with sections of [[fiber bundle]]s or sections of the étalé space of a sheaf; see below. ''F''(''U'') is also often denoted Γ(''U'',''F''), especially in contexts such as [[sheaf cohomology]] where ''U'' tends to be fixed and ''F'' tends to be variable.

=== Sheaves ===
For simplicity, consider first the case where the sheaf takes values in the category of sets. In fact, this definition applies more generally to the situation where the category is a [[concrete category]] whose underlying set functor is [[Conservative functor|conservative]], meaning that if the underlying map of sets is a bijection, then the original morphism is an isomorphism.

A ''sheaf'' is a presheaf with values in the category of sets that satisfies the following two axioms:
# (Locality) If (''U''''i'') is an open [[cover (topology)|covering]] of an open set ''U'', and if ''s'',''t'' ∈ ''F''(''U'') are such that ''s''|''U''''i'' = ''t''|''U''''i'' for each set ''U''''i'' of the covering, then ''s'' = ''t''; and
# ([[Gluing axiom|Gluing]]) If (''U''''i'') is an open covering of an open set ''U'', and if for each ''i'' a section ''s''''i'' ∈ ''F''(''U''''i'') is given such that for each pair ''U''''i'',''U''''j'' of the covering sets the restrictions of ''s''''i'' and ''s''''j'' agree on the overlaps: ''s''''i''|''U''''i''∩''U''''j'' = ''s''''j''|''U''''i''∩''U''''j'', then there is a section ''s'' ∈ ''F''(''U'') such that ''s''|''U''''i'' = ''s''''i'' for each ''i''.

The section ''s'' whose existence is guaranteed by axiom 2 is called the '''gluing''', '''concatenation''', or '''collation''' of the sections ''s''''i''. By axiom 1 it is unique. Sections ''s''''i'' satisfying the condition of axiom 2 are often called ''compatible''; thus axioms 1 and 2 together state that ''compatible sections can be uniquely glued together''. A '''separated presheaf''', or '''monopresheaf''', is a presheaf satisfying axiom 1.<ref>{{Citation | last1=Tennison | first1=B. R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975}}</ref>

If '''C''' has [[product (category theory)|products]], the sheaf axioms are equivalent to the requirement that, for any open covering ''U''''i'', the first arrow in the following diagram is an [[equalizer (mathematics)|equalizer]]:

:<math>F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i, j} F(U_i \cap U_j).</math>

Here the first map is the product of the restriction maps

:<math>\operatorname{res}_{U_i, U} \colon F(U) \rightarrow F(U_i)</math>

and the pair of arrows the products of the two sets of restrictions

:<math>\operatorname{res}_{U_i \cap U_j, U_i} \colon F(U_i) \rightarrow F(U_i \cap U_j)</math>

and

:<math>\operatorname{res}_{U_i \cap U_j, U_j} \colon F(U_j) \rightarrow F(U_i \cap U_j).</math>

For a separated presheaf, the first arrow need only be injective.

In general, for an open set ''U'' and open covering (''U''''i''), construct a category ''J'' whose objects are the sets ''Ui'' and the intersections {{nowrap|''Ui'' ∩ ''Uj''}} and whose morphisms are the inclusions of {{nowrap|''Ui'' ∩ ''Uj''}} in ''Ui'' and ''Uj''. The sheaf axioms for ''U'' and (''U''''i'') are that the [[limit (category theory)|limit]] of the functor ''F'' restricted to the category ''J'' must be isomorphic to ''F''(''U'').

Notice that the empty subset of a topological space is covered by the empty family of sets. The product of an empty family or the limit of an empty family is a terminal object, and consequently the value of a sheaf on the empty set must be a terminal object. If sheaf values are in the category of sets, applying the local identity axiom to the empty family shows that over the empty set, there is at most one section, and applying the gluing axiom to the empty family shows that there is at least one section. This property is called the '''normalisation axiom'''.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a [[basis (topology)|basis]] for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. Thus a sheaf can often be defined by giving its values on the open sets of a basis, and verifying the sheaf axioms relative to the basis. (see [[gluing axiom#Sheaves on a basis of open sets]].)

=== Morphisms ===
A morphism of sheaves is a clean function between them. Because sheaves contain data relative to every open set of a topological space, a morphism of sheaves is defined as a collection of functions, one for each open set. The collection satisfies a compatibility condition.

Let ''F'' and ''G'' be two sheaves on ''X'' with values in the category '''C'''. A ''[[morphism]]'' <math>\varphi:G\to F</math> consists of a morphism <math>\varphi_U:G(U)\to F(U)</math> for each open set {{mvar|U}} of {{mvar|X}}, subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset ''V'' of an open set ''U'', the following diagram is [[commutative diagram|commutative]].

:<math>\begin{array}{rcl}
G(U) & \xrightarrow{\quad\varphi_U\quad} & F(U)\\
r_{V,U}\Biggl\downarrow & & \Biggl\downarrow r_{V,U}\\
G(V) & \xrightarrow[{\quad\varphi_V\quad}]{} & F(V)
\end{array}</math>

Sheaves are themselves functors, from "data" to a space. In this language, a morphism of sheaves is a [[natural transformation]] of the corresponding functors. With this notion of morphism, there is a category of '''C'''-valued sheaves on ''X'' for any '''C'''. The objects are the '''C'''-valued sheaves, and the morphisms are morphisms of sheaves. An ''[[isomorphism]]'' of sheaves is an isomorphism in this category.

An isomorphism of sheaves is an isomorphism on each open set ''U''. In other words, φ is an isomorphism if and only if for each ''U'', φ(''U'') is an isomorphism. A morphism of sheaves φ is an isomorphism if and only if there exists an open cover <math>\{U_\alpha\}</math> such that <math>\varphi|_{U_\alpha}</math> are isomorphisms of sheaves for all <math>\alpha</math>. The same facts are true of [[monomorphism]]s. However, they are false for [[epimorphism]]s, and their failure is measured by [[sheaf cohomology]].

Notice that we did not use the gluing axiom in defining a morphism of sheaves. Consequently, the above definition makes sense for presheaves as well. The category of '''C'''-valued presheaves is then a [[functor category]], the category of contravariant functors from ''O''(''X'') to '''C'''.

== Examples ==
Because sheaves encode exactly the data needed to pass between local and global situations, there are many examples of sheaves occurring throughout mathematics. Here are some additional examples of sheaves:

* Any continuous map of topological spaces determines a sheaf of sets. Let ''f'' : ''Y'' → ''X'' be a continuous map. We define a sheaf Γ(''Y''/''X'') on ''X'' by setting Γ(''Y''/''X'')(U) equal to the sections ''U'' → ''Y'', that is, Γ(''Y''/''X'')(U) is the set of all continuous functions ''s'' : ''U'' → ''Y'' such that ''f ∘ s'' = ''id''''U''. Restriction is given by restriction of functions. This sheaf is called the '''sheaf of sections''' of ''f'', and it is especially important when ''f'' is the projection of a [[fiber bundle]] onto its base space. Notice that if the image of ''f'' does not contain ''U'', then Γ(''Y''/''X'')(''U'') is empty. For a concrete example, take ''X'' = '''C''' \ {0}, ''Y'' = '''C''', and ''f''(''z'') = exp(''z''). Γ(''Y''/''X'')(''U'') is the set of branches of the logarithm on ''U''.
* Fix a point ''x'' in ''X'' and an object ''S'' in a category '''C'''. The ''skyscraper sheaf over ''x'' with stalk'' ''S'' is the sheaf ''S''''x'' defined as follows: If ''U'' is an open set containing ''x'', then ''S''''x''(''U'') = ''S''. If ''U'' does not contain ''x'', then ''S''''x''(''U'') is the terminal object of '''C'''. The restriction maps are either the identity on ''S'', if both open sets contain ''x'', or the unique map from ''S'' to the terminal object of '''C'''.

=== Sheaves on manifolds ===
In the following examples ''M'' is an ''n''-dimensional ''C''''k''-manifold. The table lists the values of certain sheaves over open subsets ''U'' of ''M'' and their restriction maps.

{| class="wikitable"
|-
! Sheaf !! Sections over an open set ''U'' !! Restriction maps !! Remarks
|-
! Sheaf of ''j''-times continuously differentiable functions <math>\mathcal{O}^j_M</math>, ''j'' ≤ ''k''
| ''C''''j''-functions ''U'' → '''R'''
| Restriction of functions.
| This is a sheaf of rings with addition and multiplication given by pointwise addition and multiplication. When ''j'' = ''k'', this sheaf is called the '''structure sheaf''' and is denoted <math>\mathcal{O}_M</math>.
|-
! Sheaf of nonzero ''k''-times continuously differentiable functions <math>\mathcal{O}_X^\times</math>
| Nowhere zero ''C''''k''-functions ''U'' → '''R'''
| Restriction of functions.
| A sheaf of groups under pointwise multiplication.
|-
! '''Cotangent sheaves''' Ω''p''''M''
| [[Differential form]]s of degree ''p'' on ''U''
| Restriction of differential forms.
| Ω1''M'' and Ωn''M'' are commonly denoted Ω''M'' and ω''M'', respectively.
|-
! '''Sheaf of distributions''' {{mathcal|DB}}
| [[Distribution (mathematics)|Distribution]]s on ''U''
| The dual map to extension of smooth compactly supported functions by zero.
| Here ''M'' is assumed to be smooth.
|-
! '''Sheaf of differential operators''' <math>\mathcal{D}_M</math>
| Finite-order [[differential operator]]s on ''U''
| Restriction of differential operators.
| Here ''M'' is assumed to be smooth.
|}

=== Presheaves that are not sheaves ===
Here are two examples of presheaves that are not sheaves:
* Let ''X'' be the [[discrete two-point space|two-point topological space]] {''x'', ''y''} with the discrete topology. Define a presheaf ''F'' as follows: ''F''(∅) = {∅}, ''F''({''x''}) = '''R''', ''F''({''y''}) = '''R''', ''F''({''x'', ''y''}) = '''R''' × '''R''' × '''R'''. The restriction map ''F''({''x'', ''y''}) → ''F''({''x''}) is the projection of '''R''' × '''R''' × '''R''' onto its first coordinate, and the restriction map ''F''({''x'', ''y''}) → ''F''({''y''}) is the projection of '''R''' × '''R''' × '''R''' onto its second coordinate. ''F'' is a presheaf that is not separated: A global section is determined by three numbers, but the values of that section over {''x''} and {''y''} determine only two of those numbers. So while we can glue any two sections over {''x''} and {''y''}, we cannot glue them uniquely.
* Let ''X'' be the [[real line]], and let ''F''(''U'') be the set of [[bounded function|bounded]] continuous functions on ''U''. This is not a sheaf because it is not always possible to glue. For example, let ''U''''i'' be the set of all ''x'' such that |''x''| < ''i''. The identity function ''f''(''x'') = ''x'' is bounded on each ''U''''i''. Consequently we get a section ''s''''i'' on ''U''''i''. However, these sections do not glue, because the function ''f'' is not bounded on the real line. Consequently ''F'' is a presheaf, but not a sheaf. In fact, ''F'' is separated because it is a sub-presheaf of the sheaf of continuous functions.

== Turning a presheaf into a sheaf ==
It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf ''F'' and produces a new sheaf ''aF'' called the '''sheaving''', '''sheafification''' or '''sheaf associated to the presheaf''' ''F''. The functor ''a'' is called the '''sheaving functor''', '''sheafification functor''', or '''associated sheaf functor'''. There is a natural morphism of presheaves <math>i\colon F\to aF</math> that has the universal property that for any sheaf ''G'' and any morphism of presheaves <math>f\colon F\to G</math>, there is a unique morphism of sheaves <math>\tilde f \colon aF \rightarrow G</math> such that <math>f = \tilde f i</math>. In fact ''a'' is the left [[adjoint functor]] to the inclusion functor (or [[forgetful functor]]) from the category of sheaves to the category of presheaves, and ''i'' is the [[adjoint functor#Unit and co-unit|unit]] of the adjunction. In this way, the category of sheaves turns into a [[Giraud subcategory]] of presheaves.

One concrete way of constructing the sheaf ''aF'' is to identify it with the sheaf of sections of an appropriate topological space. This space is analogous to the [[#The étalé space of a sheaf|étalé space]] of a sheaf. Briefly, the underlying set of the topological space is the disjoint union of the [[#Stalks of a sheaf|stalks]] of ''F'', denoted {{nowrap|Spé ''F''}}. There is a natural map {{nowrap|φ : Spé ''F'' → ''X''}} that sends each germ to the point of ''X'' over which it lies. For each open set ''U'' and each section ''s'' of ''F'' over ''U'', we define a section <math>\bar s</math> of φ that sends ''x'' to the germ ''s''''x''. Then {{nowrap|Spé ''F''}} is given the finest topology for which all sections <math>\bar s</math> are continuous, and ''aF'' is the sheaf of continuous sections of φ for this topology.

There are other constructions of the sheaf ''aF''. In particular, [[Alexander Grothendieck]] and [[Jean-Louis Verdier]] ([[Séminaire de Géométrie Algébrique du Bois Marie#SGA 4|SGA 4]] II 3.0.5) define a functor ''L'' from presheaves to presheaves which, when applied to a presheaf, yields a separated presheaf and, when applied to a separated presheaf, yields a sheaf. Applying the functor ''L'' twice therefore turns a presheaf into a sheaf, and in fact ''LLF'' is the associated sheaf ''aF''.

== Operations ==
If ''K'' is a [[Subobject|subsheaf]] of a sheaf ''F'' of abelian groups, then the '''quotient sheaf''' ''Q'' is the sheaf associated to the presheaf <math>U \mapsto F(U)/K(U)</math>; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;
:<math>0 \to K \to F \to Q \to 0.</math>
(this is also called a [[sheaf extension]].)

Let ''F'', ''G'' be sheaves of abelian groups. The set of morphisms of sheaves from ''F'' to ''G'' forms an abelian group (by the abelian group structure of ''G''). The '''sheaf hom''' of ''F'' and ''G'', denoted by,
:<math>\mathcal{Hom}(F, G)</math>
is the sheaf of abelian groups <math>U \mapsto \operatorname{\mathcal{Hom}}(F|_U, G|_U)</math> where <math>F|_U</math> is the sheaf on ''U'' given by <math>(F|_U)(V) = F(V)</math> (Note sheafification is not needed here). The direct sum of ''F'' and ''G'' is the sheaf given by <math>U \mapsto F(U) \oplus G(U) </math>, and the tensor product of ''F'' and ''G'' is the sheaf associated to the presheaf <math>U \mapsto F(U) \otimes G(U)</math>.

All of these operations extend to [[sheaf of modules|sheaves of modules]] over a [[sheaf of rings]] ''A''; the above is the special case when ''A'' is the [[constant sheaf]] <math>\underline{\mathbf{Z}}</math>.

== Images of sheaves ==
{{Images of sheaves}}
{{Main|Image functors for sheaves}}

The definition of a morphism on sheaves makes sense only for sheaves on the same space ''X''. This is because the data contained in a sheaf is indexed by the open sets of the space. If we have two sheaves on different spaces, then their data is indexed differently. There is no way to go directly from one set of data to the other.

However, it is possible to move a sheaf from one space to another using a continuous function. Let ''f'' : ''X'' → ''Y'' be a continuous function from a topological space ''X'' to a topological space ''Y''. If we have a sheaf on ''X'', we can move it to ''Y'', and vice versa. There are four ways in which sheaves can be moved.
* A sheaf <math>\mathcal{F}</math> on ''X'' can be moved to ''Y'' using the [[direct image functor]] <math>f_*</math> or the [[direct image with proper support functor]] <math>f_!</math>.
* A sheaf <math>\mathcal{G}</math> on ''Y'' can be moved to ''X'' using the [[inverse image functor]] <math>f^{-1}</math> or the [[twisted inverse image functor]] <math>f^!</math>.

The twisted inverse image functor <math>f^!</math> is, in general, only defined as a functor between [[derived category|derived categories]]. These functors come in adjoint pairs: <math>f^{-1}</math> and <math>f_*</math> are left and right adjoints of each other, and <math>Rf_!</math> and <math>f^!</math> are left and right adjoints of each other. The functors are intertwined with each other by [[Coherent duality|Grothendieck duality]] and [[Verdier duality]].

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted <math>f^*</math> and it is distinct from <math>f^{-1}</math>. See [[inverse image functor]].

== Stalks of a sheaf ==
{{Main|Stalk (sheaf)}}

The '''stalk''' <math>\mathcal{F}_x</math> of a sheaf <math>\mathcal{F}</math> captures the properties of a sheaf "around" a point ''x'' ∈ ''X''.
Here, "around" means that, conceptually speaking, one looks at smaller and smaller [[neighborhood (mathematics)|neighborhoods]] of the point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.

The stalk is defined by
:<math>\mathcal{F}_x = \varinjlim_{U\ni x} \mathcal{F}(U),</math>
the [[direct limit]] being over all open subsets of ''X'' containing the given point ''x''. In other words, an element of the stalk is given by a section over some open neighborhood of ''x'', and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism ''F''(''U'') → ''F''''x'' takes a section ''s'' in ''F''(''U'') to its ''germ''. This generalises the usual definition of a [[germ (mathematics)|germ]].

A different way of defining the stalk is
:<math>\mathcal{F}_x := i^{-1}\mathcal{F}(\{x\}),</math>
where ''i'' is the inclusion of the one-point space {''x''} into ''X''. The equivalence follows from the definition of the [[inverse image functor|inverse image]].

In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. They also find use in constructions such as [[Godement resolution]]s.

== Ringed spaces and locally ringed spaces ==
{{Main|Ringed space}}

A pair <math>(X, \mathcal{O}_X)</math> consisting of a topological space ''X'' and a sheaf of rings on ''X'' is called a '''[[ringed space]]'''. Many types of spaces can be defined as certain types of ringed spaces. The sheaf <math>\mathcal{O}_X</math> is called the '''structure sheaf''' of the space. A very common situation is when all the stalks of the structure sheaf are [[local ring]]s, in which case the pair is called a '''locally ringed space'''. Here are examples of definitions made in this way:
* An ''n''-dimensional ''C''''k'' manifold ''M'' is a locally ringed space whose structure sheaf is an <math>\underline{\mathbf{R}}</math>-algebra and is locally isomorphic to the sheaf of ''C''k real-valued functions on '''R'''''n''.
* A [[complex analytic space]] is a locally ringed space whose structure sheaf is a <math>\underline{\mathbf{C}}</math>-algebra and is locally isomorphic to the vanishing locus of a finite set of holomorphic functions together with the restriction (to the vanishing locus) of the sheaf of holomorphic functions on '''C'''''n'' for some ''n''.
* A [[scheme (mathematics)|scheme]] is a locally ringed space that is locally isomorphic to the [[spectrum of a ring]].
* A [[semialgebraic space]] is a locally ringed space that is locally isomorphic to a [[semialgebraic set]] in Euclidean space together with its sheaf of semialgebraic functions.

== Sheaves of modules ==
{{main|Sheaf of modules}}
Let <math>(X, \mathcal{O}_X)</math> be a ringed space. A '''sheaf of modules''' is a sheaf <math>\mathcal{M}</math> such that on every open set ''U'' of ''X'', <math>\mathcal{M}(U)</math> is an <math>\mathcal{O}_X(U)</math>-module and for every inclusion of open sets ''V'' ⊆ ''U'', the restriction map <math>\mathcal{M}(U) \to \mathcal{M}(V)</math> is a homomorphism of <math>\mathcal{O}_X(U)</math>-modules.

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between [[vector bundle]]s and [[locally free sheaf|locally free sheaves]] of <math>\mathcal{O}_X</math>-modules. Sheaves of solutions to differential equations are [[D-module]]s, that is, modules over the sheaf of differential operators.

A particularly important case are [[abelian sheaf|abelian sheaves]], which are modules over the constant sheaf <math>\underline{\mathbf{Z}}</math>. Every sheaf of modules is an abelian sheaf.

=== Finiteness conditions for sheaves of modules ===
{{Further|Coherent sheaf}}
The condition that a module is finitely generated or finitely presented can also be formulated for a sheaf of modules. <math>\mathcal{M}</math> is '''finitely generated''' if, for every point ''x'' of ''X'', there exists an open neighborhood ''U'' of ''x'', a natural number ''n'' (possibly depending on ''U''), and a surjective morphism of sheaves <math>\mathcal{O}_X^n|_U \to \mathcal{M}|_U</math>. Similarly, <math>\mathcal{M}</math> is '''finitely presented''' if in addition there exists a natural number ''m'' (again possibly depending on ''U'') and a morphism of sheaves <math>\mathcal{O}_X^m|_U \to \mathcal{O}_X^n|_U</math> such that the sequence of morphisms <math>\mathcal{O}_X^m|_U \to \mathcal{O}_X^n|_U \to \mathcal{M}</math> is exact. Equivalently, the kernel of the morphism <math>\mathcal{O}_X^n|_U \to \mathcal{M}</math> is itself a finitely generated sheaf.

These, however, are not the only possible finiteness conditions on a sheaf. The most important finiteness condition for a sheaf is coherence. <math>\mathcal{M}</math> is '''coherent''' if it is of finite type and if, for every open set ''U'' and every morphism of sheaves <math>\phi : \mathcal{O}_X^n \to \mathcal{M}</math> (not necessarily surjective), the kernel of φ is of finite type. <math>\mathcal{O}_X</math> is '''coherent''' if it is coherent as a module over itself. Note that coherence is a strictly stronger condition than finite presentation: <math>\mathcal{O}_X</math> is always finitely presented as a module over itself, but it is not always coherent. For example, let ''X'' be a point, let <math>\mathcal{O}_X</math> be the ring {{nowrap begin}}''R'' = '''C'''[''x''1, ''x''2, ...]{{nowrap end}} of complex polynomials in countably many indeterminates. Choose {{nowrap begin}}''n'' = 1{{nowrap end}}, and for the morphism φ, take the map that sends every variable to zero. The kernel of this map is not finitely generated, so <math>\mathcal{O}_X</math> is not coherent.

== The étalé space of a sheaf ==
{{anchor|Etale space}}
In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called the ''étalé space'', from the French word étalé {{IPA-fr|etale|}}, meaning roughly "spread out". If ''F'' is a sheaf over ''X'', then the '''étalé space''' of ''F'' is a topological space ''E'' together with a [[local homeomorphism]] ''π'' : ''E'' → ''X'' such that the sheaf of sections of ''π'' is ''F''. The space ''E'' is usually very strange, and even if the sheaf ''F'' arises from a natural topological situation, ''E'' may not have any clear topological interpretation. For example, if ''F'' is the sheaf of sections of a continuous function ''f'' : ''Y'' → ''X'', then ''E'' = ''Y'' if and only if ''f'' is a [[local homeomorphism]].

The étalé space ''E'' is constructed from the stalks of ''F'' over ''X''. As a set, it is their [[disjoint union]] and ''π'' is the obvious map that takes the value ''x'' on the stalk of ''F'' over ''x'' ∈ ''X''. The topology of ''E'' is defined as follows. For each element ''s'' of ''F''(''U'') and each ''x'' in ''U'', we get a germ of ''s'' at ''x''. These germs determine points of ''E''. For any ''U'' and ''s'' ∈ ''F''(''U''), the union of these points (for all ''x'' ∈ ''U'') is declared to be open in ''E''. Notice that each stalk has the [[discrete topology]] as subspace topology. Two morphisms between sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.

The construction above determines an [[equivalence of categories]] between the category of sheaves of sets on ''X'' and the category of étalé spaces over ''X''. The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf.

This construction makes all sheaves into [[representable functor]]s on certain categories of topological spaces. As above, let ''F'' be a sheaf on ''X'', let ''E'' be its étalé space, and let ''π'' : ''E'' → ''X'' be the natural projection. Consider the category '''Top'''/''X'' of topological spaces over ''X'', that is, the category of topological spaces together with fixed continuous maps to ''X''. Every object of this space is a continuous map ''f'' : ''Y'' → ''X'', and a morphism from ''Y'' → ''X'' to ''Z'' → ''X'' is a continuous map ''Y'' → ''Z'' that commutes with the two maps to ''X''. There is a functor Γ from '''Top'''/''X'' to the category of sets that takes an object ''f'' : ''Y'' → ''X'' to (''f''−1''F'')(''Y''). For example, if ''i'' : ''U'' → ''X'' is the inclusion of an open subset, then Γ(''i'') = (''i''−1''F'')(''U'') agrees with the usual ''F''(''U''), and if ''i'' : {''x''} → ''X'' is the inclusion of a point, then Γ({''x''}) = (''i''−1''F'')({''x''}) is the stalk of ''F'' at ''x''. There is a natural isomorphism
:<math>(f^{-1}F)(Y) \cong \operatorname{Hom}_{\mathbf{Top}/X}(f, \pi)</math>,
which shows that ''E'' represents the functor Γ.

''E'' is constructed so that the projection map π is a covering map. In algebraic geometry, the natural analog of a covering map is called an [[étale morphism]]. Despite its similarity to "étalé", the word étale {{IPA-fr|etal|}} has a different meaning in French. It is possible to turn ''E'' into a [[scheme (mathematics)|scheme]] and π into a morphism of schemes in such a way that π retains the same universal property, but π is ''not'' in general an étale morphism because it is not quasi-finite. It is, however, formally étale.

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such as [[mathematical analysis]].

== Sheaf cohomology ==
{{Main|Sheaf cohomology}}

It was noted above that the functor <math>\Gamma(U,-)</math> preserves isomorphisms and monomorphisms, but not epimorphisms. If ''F'' is a sheaf of abelian groups, or more generally a sheaf with values in an [[abelian category]], then <math>\Gamma(U,-)</math> is actually a [[left exact functor]]. This means that it is possible to construct [[derived functor]]s of <math>\Gamma(U,-)</math>. These derived functors are called the ''cohomology groups'' (or ''modules'') of ''F'' and are written <math>H^i(U,-)</math>. Grothendieck proved in his "[[Tohoku paper]]" ({{harvtxt|Grothendieck|1957}}) that every category of sheaves of abelian groups contains enough [[injective object]]s, so these derived functors always exist.

However, computing sheaf cohomology using injective resolutions is nearly impossible. In practice, it is much more common to find a different and more tractable resolution of ''F''. A general construction is provided by [[Godement resolution]]s, and particular resolutions may be constructed using [[soft sheaf|soft sheaves]], [[fine sheaf|fine sheaves]], and [[flabby sheaf|flabby sheaves]] (also known as ''flasque sheaves'' from the French ''flasque'' meaning flabby). As a consequence, it can become possible to compare sheaf cohomology with other cohomology theories. For example, the [[de Rham complex]] is a resolution of the constant sheaf <math>\underline{\mathbf{R}}</math> on any smooth manifold, so the sheaf cohomology of <math>\underline{\mathbf{R}}</math> is equal to its [[de Rham cohomology]]. In fact, comparing sheaf cohomology to de Rham cohomology and singular cohomology provides a proof of de Rham's theorem that the two cohomology theories are isomorphic.

A different approach is by [[Čech cohomology]]. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations. It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct <math>H^1</math> but incorrect higher cohomology groups. To get around this, [[Jean-Louis Verdier]] developed [[hypercover]]ings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction of [[Pierre Deligne]]'s [[mixed Hodge structure]]s.

A much cleaner approach to the computation of some cohomology groups is the [[Borel–Bott–Weil theorem]], which identifies the cohomology groups of some [[line bundle]]s on [[flag manifold]]s with [[irreducible representation]]s of [[Lie group]]s. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space.

In many cases there is a duality theory for sheaves that generalizes [[Poincaré duality]]. See [[Coherent duality|Grothendieck duality]] and [[Verdier duality]].

== Sites and topoi ==
{{Main|Grothendieck topology|Topos}}

[[André Weil]]'s [[Weil conjectures]] stated that there was a [[Weil cohomology theory|cohomology theory]] for [[algebraic variety|algebraic varieties]] over [[finite field]]s that would give an analogue of the [[Riemann hypothesis]]. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf <math>\underline{\mathbf{C}}</math> in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is the [[Zariski topology]], and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero). [[Alexandre Grothendieck]] solved this problem by introducing [[Grothendieck topology|Grothendieck topologies]], which axiomatize the notion of ''covering''. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to define [[étale cohomology]] and [[l-adic cohomology]], which eventually were used to prove the Weil conjectures.

A category with a Grothendieck topology is called a ''site''. A category of sheaves on a site is called a ''topos'' or a ''Grothendieck topos''. The notion of a topos was later abstracted by [[William Lawvere]] and Miles Tierney to define an [[elementary topos]], which has connections to [[mathematical logic]].

== History ==

{{unreferenced section|date=January 2016}}

The first origins of '''sheaf theory''' are hard to pin down — they may be co-extensive with the idea of [[analytic continuation]]{{Clarify|date=July 2010}}. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on [[cohomology]].
* 1936 [[Eduard Čech]] introduces the ''[[Nerve of an open covering|nerve]]'' construction, for associating a [[simplicial complex]] to an open covering.
* 1938 [[Hassler Whitney]] gives a 'modern' definition of cohomology, summarizing the work since [[James Waddell Alexander II|J. W. Alexander]] and [[Kolmogorov]] first defined ''[[cochain]]s''.
* 1943 [[Norman Steenrod]] publishes on homology ''with [[local coefficients]]''.
* 1945 [[Jean Leray]] publishes work carried out as a [[prisoner of war]], motivated by proving [[Fixed point (mathematics)|fixed point]] theorems for application to [[Partial differential equation|PDE]] theory; it is the start of sheaf theory and [[spectral sequence]]s.
* 1947 [[Henri Cartan]] reproves the [[de Rham theorem]] by sheaf methods, in correspondence with [[André Weil]] (see [[De Rham–Weil theorem]]). Leray gives a sheaf definition in his courses via closed sets (the later ''carapaces'').
* 1948 The Cartan seminar writes up sheaf theory for the first time.
* 1950 The "second edition" sheaf theory from the Cartan seminar: the [[sheaf space]] (''espace étalé'') definition is used, with stalkwise structure. [[Support (mathematics)|Support]]s are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time [[Kiyoshi Oka]] introduces an idea (adjacent to that) of a sheaf of ideals, in [[several complex variables]].
* 1951 The Cartan seminar proves [[Theorems A and B]], based on Oka's work.
* 1953 The finiteness theorem for [[coherent sheaf|coherent sheaves]] in the analytic theory is proved by Cartan and [[Jean-Pierre Serre]], as is [[Serre duality]].
* 1954 Serre's paper ''[[#CITEREFSerre1955|Faisceaux algébriques cohérents]]'' (published in 1955) introduces sheaves into [[algebraic geometry]]. These ideas are immediately exploited by [[Friedrich Hirzebruch]], who writes a major 1956 book on topological methods.
* 1955 [[Alexander Grothendieck]] in lectures in [[Kansas]] defines [[abelian category]] and ''presheaf'', and by using [[injective resolution]]s allows direct use of sheaf cohomology on all topological spaces, as [[derived functor]]s.
* 1956 [[Oscar Zariski]]'s report ''[[#CITEREFMartinChernZariski1956|Algebraic sheaf theory]]''
* 1957 Grothendieck's [[#CITEREFGrothendieck1957|''Tohoku'' paper]] rewrites [[homological algebra]]; he proves [[Coherent duality|Grothendieck duality]] (i.e., Serre duality for possibly [[Mathematical singularity|singular]] algebraic varieties).
* 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: [[Scheme (mathematics)|scheme]]s and general sheaves on them, [[local cohomology]], [[derived category|derived categories]] (with Verdier), and [[Grothendieck topologies]]. There emerges also his influential schematic idea of 'six operations' in homological algebra.
* 1958 [[Roger Godement]]'s book on sheaf theory is published. At around this time [[Mikio Sato]] proposes his [[hyperfunction]]s, which will turn out to have sheaf-theoretic nature.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to [[algebraic topology]]. It was later discovered that the logic in categories of sheaves is [[intuitionistic logic]] (this observation is now often referred to as [[Kripke–Joyal semantics]], but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as [[Gottfried Wilhelm Leibniz|Leibniz]].

== See also ==
* [[Coherent sheaf]]
* [[Cosheaf]]
* [[Gerbe]]
* [[Stack (mathematics)]]
* [[Sheaf of spectra]]
* [[Presheaf of spaces]]
* [[Base change theorems]]
* [[Locally constant sheaf]]
* [[Constructible sheaf]]

== Notes ==
{{Reflist}}

== References ==
* {{Citation | last1=Bredon | first1=Glen E. | author1-link = Glen Bredon | title=Sheaf theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94905-5 | mr=1481706 | edition=2nd | year=1997 | volume=170}} (oriented towards conventional topological applications)
* {{Citation | last1=Godement | first1=Roger | author1-link = Roger Godement | title=Topologie algébrique et théorie des faisceaux | publisher=Hermann | location=Paris | mr=0345092 | year=1973}}
* {{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Sur quelques points d'algèbre homologique | mr=0102537 | year=1957 | journal=The Tohoku Mathematical Journal. Second Series | issn=0040-8735 | volume=9 | pages=119–221 | doi=10.2748/tmj/1178244839}}
* {{Citation | last1=Hirzebruch | first1=Friedrich | author1-link = Friedrich Hirzebruch | title=Topological methods in algebraic geometry | publisher=Springer-Verlag | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-58663-0 | mr=1335917 | year=1995}} (updated edition of a classic using enough sheaf theory to show its power)
* {{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | last2=Schapira | first2=Pierre | title=Sheaves on manifolds | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-51861-7 | mr=1299726 | year=1994 | volume=292}}(advanced techniques such as the [[derived category]] and [[vanishing cycle]]s on the most reasonable spaces)
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link = Saunders Mac Lane | last2=Moerdijk | first2=Ieke | author2-link = Ieke Moerdijk | title=Sheaves in Geometry and Logic: A First Introduction to Topos Theory | publisher=Springer-Verlag | location=Berlin, New York | series=Universitext | isbn=978-0-387-97710-2 | mr=1300636 | year=1994}} (category theory and toposes emphasised)
* {{Citation | last1=Martin | first1=William T. | last2=Chern | first2=Shiing-Shen | author2-link=Shiing-Shen Chern | last3=Zariski | first3=Oscar | author3-link=Oscar Zariski | title=Scientific report on the Second Summer Institute, several complex variables | mr=0077995 | year=1956 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=62 | pages=79–141 | doi=10.1090/S0002-9904-1956-10013-X | issue=2}}
* [[J. Arthur Seebach]], Linda A. Seebach & [[Lynn A. Steen]] (1970) "What is a Sheaf", [[American Mathematical Monthly]] 77:681–703 {{MR|id=0263073}}.
* {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Faisceaux algébriques cohérents | url=http://www.mat.uniroma1.it/people/arbarello/FAC.pdf | mr=0068874 | year=1955 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=61 | pages=197–278 | doi=10.2307/1969915 | jstor=1969915 | issue=2 }}
* {{Citation | last1=Swan | first1=Richard G. | authorlink=Richard Swan| title=The Theory of Sheaves | publisher=[[University of Chicago Press]]| year=1964}} (concise lecture notes)
* {{Citation | last1=Tennison | first1=Barry R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975}} (pedagogic treatment)

== External links ==
* {{planetmath reference|id=5648|title=Sheaf}}

[[Category:Sheaf theory|*]]
[[Category:Topological methods of algebraic geometry]]
[[Category:Algebraic topology]]

Bellman–Ford algorithm

2018-11-20T04:46:47Z

Crasshopper: /* Algorithm */

{{Infobox Algorithm
|class=[[Single-source shortest path problem]] (for weighted directed graphs)
|image=
|caption =
|data=[[Graph (data structure)|Graph]]
|time=<math>\Theta (|V| |E|)</math>
|best-time=<math>\Theta (|E|)</math>
|space=<math>\Theta (|V|)</math>
}}

{{Tree search algorithm}}

The '''Bellman–Ford algorithm''' is an [[algorithm]] that computes [[shortest path]]s from a single source [[vertex (graph theory)|vertex]] to all of the other vertices in a [[weighted digraph]].<ref name=Bang>{{harvtxt|Bang-Jensen|Gutin|2000}}</ref>
It is slower than [[Dijkstra's algorithm]] for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers.
The algorithm was first proposed by {{harvs|first=Alfonso|last=Shimbel|year=1955|txt}}, but is instead named after [[Richard Bellman]] and [[L. R. Ford, Jr.|Lester Ford, Jr.]], who published it in [[#{{harvid|Bellman|1958}}|1958]] and [[#{{harvid|Ford|1956}}|1956]], respectively.<ref name="Schrijver">{{harvtxt|Schrijver|2005}}</ref> [[Edward F. Moore]] also published the same algorithm in 1957, and for this reason it is also sometimes called the '''Bellman–Ford–Moore algorithm'''.<ref name=Bang />

Negative edge weights are found in various applications of graphs, hence the usefulness of this algorithm.{{sfnp|Sedgewick|2002}}
If a graph contains a "negative cycle" (i.e. a [[cycle (graph theory)|cycle]] whose edges sum to a negative value) that is reachable from the source, then there is no ''cheapest'' path: any path that has a point on the negative cycle can be made cheaper by one more [[Walk (graph theory)|walk]] around the negative cycle. In such a case, the Bellman–Ford algorithm can detect negative cycles and report their existence.<ref name=Bang />{{sfnp|Kleinberg|Tardos|2006}}

== Algorithm ==
[[File:Bellman-Ford worst-case example.svg|thumb|In this example graph, assuming that A is the source and edges are processed in the worst order, from right to left, it requires the full {{math||''V''|−1}} or 4 iterations for the distance estimates to converge. Conversely, if the edges are processed in the best order, from left to right, the algorithm converges in a single iteration.]]

Like [[Dijkstra's Algorithm]], Bellman–Ford proceeds by [[Relaxation (iterative method)|relaxation]], in which approximations to the correct distance are replaced by better ones until eventually reaching the solution. In both algorithms, the approximate distance to each vertex is always an overestimate of the true distance, and is replaced by the minimum of its old value and the length of a newly found path.
However, Dijkstra's algorithm uses a priority queue to [[Greedy algorithm|greedily]] select the closest vertex that has not yet been processed, and performs this relaxation process on all of its outgoing edges; by contrast, the Bellman–Ford algorithm simply relaxes ''all'' the edges, and does this <math>|V|-1</math> times, where <math>|V|</math> is the number of vertices in the graph. In each of these repetitions, the number of vertices with correctly calculated distances grows, from which it follows that eventually all vertices will have their correct distances. This method allows the Bellman–Ford algorithm to be applied to a wider class of inputs than Dijkstra.

Bellman–Ford runs in <math>O(|V|\cdot |E|)</math> [[Big O notation|time]], where <math>|V|</math> and <math>|E|</math> are the number of vertices and edges respectively.

'''function''' BellmanFord(''list'' vertices, ''list'' edges, ''vertex'' source)
::distance[],predecessor[]
''
''// This implementation takes in a graph, represented as''
''// lists of vertices and edges, and fills two arrays''
''// (distance and predecessor) about the shortest path''
''// from the source to each vertex''
''
''// Step 1: initialize graph''
'''for each''' vertex v '''in''' vertices:
distance[v] := '''inf''' // Initialize the distance to all vertices to infinity
predecessor[v] := '''null''' // And having a null predecessor
''
distance[source] := 0 // Except the source vertex, which is zero
''
''// Step 2: relax edges repeatedly''

'''for''' i '''from''' 1 '''to''' size(vertices)-1:
'''for each''' edge (u, v) '''with''' weight w '''in''' edges:
'''if''' distance[u] + w < distance[v]:
distance[v] := distance[u] + w
predecessor[v] := u
''
''// Step 3: check for negative-weight cycles''
'''for each''' edge (u, v) '''with''' weight w '''in''' edges:
'''if''' distance[u] + w < distance[v]:
'''error''' "Graph contains a negative-weight cycle"
''
'''return''' distance[], predecessor[]

Simply put, the algorithm initializes the distance to the source to 0 and all other nodes to infinity. Then for all edges, if the distance to the destination can be shortened by taking the edge, the distance is updated to the new lower value. At each iteration {{mvar|i}} that the edges are scanned, the algorithm finds all shortest paths of at most length {{mvar|i}} edges (and possibly some paths longer than {{mvar|i}} edges). Since the longest possible path without a cycle can be <math>|V|-1</math> edges, the edges must be scanned <math>|V|-1</math> times to ensure the shortest path has been found for all nodes. A final scan of all the edges is performed and if any distance is updated, then a path of length <math>|V|</math> edges has been found which can only occur if at least one negative cycle exists in the graph.

== Proof of correctness ==

The correctness of the algorithm can be shown by [[mathematical induction|induction]]:

'''Lemma'''. After ''i'' repetitions of ''for'' loop,
* if Distance(''u'') is not infinity, it is equal to the length of some path from ''s'' to ''u''; and
* if there is a path from ''s'' to ''u'' with at most ''i'' edges, then Distance(u) is at most the length of the shortest path from ''s'' to ''u'' with at most ''i'' edges.

'''Proof'''. For the base case of induction, consider <code>i=0</code> and the moment before ''for'' loop is executed for the first time. Then, for the source vertex, <code>source.distance = 0</code>, which is correct. For other vertices ''u'', <code>u.distance = '''infinity'''</code>, which is also correct because there is no path from ''source'' to ''u'' with 0 edges.

For the inductive case, we first prove the first part. Consider a moment when a vertex's distance is updated by
<code>v.distance := u.distance + uv.weight</code>. By inductive assumption, <code>u.distance</code> is the length of some path from ''source'' to ''u''. Then <code>u.distance + uv.weight</code> is the length of the path from ''source'' to ''v'' that follows the path from ''source'' to ''u'' and then goes to ''v''.

For the second part, consider a shortest path ''P'' (there may be more than one) from ''source'' to ''u'' with at most ''i'' edges. Let ''v'' be the last vertex before ''u'' on this path. Then, the part of the path from ''source'' to ''v'' is a shortest path from ''source'' to ''v'' with at most ''i-1'' edges, since if it were not, then there must be some strictly shorter path from ''source'' to ''v'' with at most ''i-1'' edges, and we could then append the edge ''uv'' to this path to obtain a path with at most ''i'' edges that is strictly shorter than ''P''—a contradiction. By inductive assumption, <code>v.distance</code> after ''i''−1 iterations is at most the length of this path from ''source'' to ''v''. Therefore, <code>uv.weight + v.distance</code> is at most the length of ''P''. In the ''ith'' iteration, <code>u.distance</code> gets compared with <code>uv.weight + v.distance</code>, and is set equal to it if <code>uv.weight + v.distance</code> is smaller. Therefore, after ''i'' iterations, <code>u.distance</code> is at most the length of ''P'', i.e., the length of the shortest path from ''source'' to ''u'' that uses at most ''i'' edges.

If there are no negative-weight cycles, then every shortest path visits each vertex at most once, so at step 3 no further improvements can be made. Conversely, suppose no improvement can be made. Then for any cycle with vertices ''v''[0], ..., ''v''[''k''−1],

<code>v[i].distance <= v[i-1 (mod k)].distance + v[i-1 (mod k)]v[i].weight</code>

Summing around the cycle, the ''v''[''i''].distance and ''v''[''i''−1 (mod ''k'')].distance terms cancel, leaving

<code>0 <= sum from 1 to k of v[i-1 (mod k)]v[i].weight</code>

I.e., every cycle has nonnegative weight.

== Finding negative cycles ==
When the algorithm is used to find shortest paths, the existence of negative cycles is a problem, preventing the algorithm from finding a correct answer. However, since it terminates upon finding a negative cycle, the Bellman–Ford algorithm can be used for applications in which this is the target to be sought – for example in [[cycle-cancelling]] techniques in [[Flow network|network flow]] analysis.<ref name="Bang" />

== Applications in routing ==

A distributed variant of the Bellman–Ford algorithm is used in [[distance-vector routing protocol]]s, for example the [[Routing Information Protocol]] (RIP). The algorithm is distributed because it involves a number of nodes (routers) within an [[autonomous system (Internet)|Autonomous system]], a collection of IP networks typically owned by an ISP.
It consists of the following steps:

# Each node calculates the distances between itself and all other nodes within the AS and stores this information as a table.
# Each node sends its table to all neighboring nodes.
# When a node receives distance tables from its neighbors, it calculates the shortest routes to all other nodes and updates its own table to reflect any changes.
The main disadvantages of the Bellman–Ford algorithm in this setting are as follows:

* It does not scale well.
* Changes in [[network topology]] are not reflected quickly since updates are spread node-by-node.
* [[Count to infinity#Count-to-infinity problem|Count to infinity]] if link or node failures render a node unreachable from some set of other nodes, those nodes may spend forever gradually increasing their estimates of the distance to it, and in the meantime there may be routing loops.

== Improvements ==
The Bellman–Ford algorithm may be improved in practice (although not in the worst case) by the observation that, if an iteration of the main loop of the algorithm terminates without making any changes, the algorithm can be immediately terminated, as subsequent iterations will not make any more changes. With this early termination condition, the main loop may in some cases use many fewer than {{math|{{abs|''V''}} − 1}} iterations, even though the worst case of the algorithm remains unchanged.

{{harvtxt|Yen|1970}} described two more improvements to the Bellman–Ford algorithm for a graph without negative-weight cycles; again, while making the algorithm faster in practice, they do not change its <math>O(|V|\cdot |E|)</math> worst case time bound. His first improvement reduces the number of relaxation steps that need to be performed within each iteration of the algorithm. If a vertex ''v'' has a distance value that has not changed since the last time the edges out of ''v'' were relaxed, then there is no need to relax the edges out of ''v'' a second time. In this way, as the number of vertices with correct distance values grows, the number whose outgoing edges that need to be relaxed in each iteration shrinks, leading to a constant-factor savings in time for [[dense graph]]s.

Yen's second improvement first assigns some arbitrary linear order on all vertices and then partitions the set of all edges into two subsets. The first subset, ''Ef'', contains all edges (''vi'', ''vj'') such that ''i'' < ''j''; the second, ''Eb'', contains edges (''vi'', ''vj'') such that ''i'' > ''j''. Each vertex is visited in the order ''v1'', ''v2'', ..., ''v''|''V''|, relaxing each outgoing edge from that vertex in ''Ef''. Each vertex is then visited in the order ''v''|''V''|, ''v''|''V''|−1, ..., ''v''1, relaxing each outgoing edge from that vertex in ''Eb''. Each iteration of the main loop of the algorithm, after the first one, adds at least two edges to the set of edges whose relaxed distances match the correct shortest path distances: one from ''Ef'' and one from ''Eb''. This modification reduces the worst-case number of iterations of the main loop of the algorithm from {{math|{{abs|''V''}} − 1}} to <math>|V|/2</math>.<ref>Cormen et al., 2nd ed., Problem 24-1, pp. 614–615.</ref><ref name=Sedweb />

Another improvement, by {{harvtxt|Bannister|Eppstein|2012}}, replaces the arbitrary linear order of the vertices used in Yen's second improvement by a [[random permutation]]. This change makes the worst case for Yen's improvement (in which the edges of a shortest path strictly alternate between the two subsets ''Ef'' and ''Eb'') very unlikely to happen. With a randomly permuted vertex ordering, the [[expected value|expected]] number of iterations needed in the main loop is at most <math>|V|/3</math>.<ref name=Sedweb>See Sedgewick's [http://algs4.cs.princeton.edu/44sp/ web exercises] for ''Algorithms'', 4th ed., exercises 5 and 12 (retrieved 2013-01-30).</ref>

In China, an algorithm which adds a first-in first-out queue to the Bellman–Ford algorithm, known as [[SPFA]], published by Fanding Duan in 1994, is popular with students who take part in [[:zh:全国青少年信息学奥林匹克联赛|NOIP]] and [[ACM-ICPC]].<ref name="duan">{{Citation
| last=Duan
| first=Fanding
| year=1994
| title=关于最短路径的SPFA快速算法
| journal=Journal of Southwest Jiaotong University
| volume=29
| issue=2
| pages=207–212
| url=http://wenku.baidu.com/view/3b8c5d778e9951e79a892705.html
}}</ref>

== Notes ==
{{Reflist}}

== References ==

=== Original sources ===
*{{cite conference
| last = Shimbel | first = A.
| title = Structure in communication nets
| location = New York, New York
| pages = 199–203
| publisher = Polytechnic Press of the Polytechnic Institute of Brooklyn
| conference = Proceedings of the Symposium on Information Networks
| year = 1955
| ref = harv}}
*{{cite journal
| last = Bellman | first = Richard | authorlink = Richard Bellman
| mr = 0102435
| journal = Quarterly of Applied Mathematics
| pages = 87–90
| title = On a routing problem
| volume = 16
| year = 1958
| ref = harv}}
*{{cite book
|authorlink=L. R. Ford, Jr. | last=Ford | first=Lester R. Jr.
|title=Network Flow Theory
|date=August 14, 1956
|series=Paper P-923
|publisher=RAND Corporation
|location=Santa Monica, California
|url=http://www.rand.org/pubs/papers/P923.html
|ref = harv}}
*{{cite conference
| last = Moore | first = Edward F. | authorlink = Edward F. Moore
| title = The shortest path through a maze
| location = Cambridge, Massachusetts
| mr = 0114710
| pages = 285–292
| publisher = Harvard Univ. Press
| conference = Proc. Internat. Sympos. Switching Theory 1957, Part II
| year = 1959
| ref = harv}}
*{{cite journal
| last = Yen | first = Jin Y.
| mr = 0253822
| journal = Quarterly of Applied Mathematics
| pages = 526–530
| title = An algorithm for finding shortest routes from all source nodes to a given destination in general networks
| volume = 27
| year = 1970
| ref = harv}}
*{{cite conference|title=Randomized speedup of the Bellman–Ford algorithm|first1=M. J.|last1=Bannister|first2=D.|last2=Eppstein|author2-link=David Eppstein|arxiv=1111.5414|conference=Analytic Algorithmics and Combinatorics (ANALCO12), Kyoto, Japan|year=2012|pages=41–47|url=https://arxiv.org/pdf/1111.5414.pdf|ref=harv|bibcode=2011arXiv1111.5414B}}

=== Secondary sources ===
*{{Cite book|first1=Jørgen |last1=Bang-Jensen|first2=Gregory|last2=Gutin|year=2000|title=Digraphs: Theory, Algorithms and Applications|edition=First |isbn=978-1-84800-997-4|chapter=Section 2.3.4: The Bellman-Ford-Moore algorithm|url=http://www.cs.rhul.ac.uk/books/dbook/|ref=harv}}
*{{cite journal|first=Alexander|last=Schrijver|title=On the history of combinatorial optimization (till 1960)|pages=1–68|publisher=Elsevier|journal=Handbook of Discrete Optimization|year=2005|url=http://homepages.cwi.nl/~lex/files/histco.pdf|ref=harv}}
*{{Introduction to Algorithms}}, Second Edition. MIT Press and McGraw-Hill, 2001. {{ISBN|0-262-03293-7}}. Section 24.1: The Bellman–Ford algorithm, pp. 588–592. Problem 24-1, pp. 614–615. Third Edition. MIT Press, 2009. {{ISBN|978-0-262-53305-8}}. Section 24.1: The Bellman–Ford algorithm, pp. 651–655.
*{{cite book | first1 = George T. | last1 = Heineman | first2 = Gary | last2 = Pollice | first3 = Stanley | last3 = Selkow | title= Algorithms in a Nutshell | publisher=[[O'Reilly Media]] | year=2008 | chapter=Chapter 6: Graph Algorithms | pages = 160–164 | isbn=978-0-596-51624-6 | ref = harv }}
*{{cite book|last1=Kleinberg|first1=Jon|author1-link=Jon Kleinberg|last2=Tardos|first2=Éva|author2-link=Éva Tardos|year=2006|title=Algorithm Design|location=New York|publisher=Pearson Education, Inc.|ref=harv}}
*{{Cite book|first=Robert |last=Sedgewick|authorlink=Robert Sedgewick (computer scientist)|year=2002|title= Algorithms in Java|edition=3rd |isbn= 0-201-36121-3|chapter=Section 21.7: Negative Edge Weights|url=http://safari.oreilly.com/0201361213/ch21lev1sec7|ref=harv}}

{{DEFAULTSORT:Bellman-Ford algorithm}}
[[Category:Graph algorithms]]
[[Category:Polynomial-time problems]]
[[Category:Articles with example C code]]
[[Category:Articles with example pseudocode]]
[[Category:Dynamic programming]]

Idris (programming language)

2018-10-08T05:58:34Z

Crasshopper:

{{primary sources|date=June 2016}}
{{Infobox programming language
| name = Idris
| paradigm = [[functional programming|Functional]]
| year =
| designer = Edwin Brady
| developer =
| latest_release_version = 1.3.0<ref>{{cite web|title=Idris 1.3.0 released|url=https://www.idris-lang.org/idris-1-3-0-released/|accessdate=2018-05-26}}</ref>
| latest_release_date = {{Start date and age|2018|05|26}}
| latest_test_version =
| latest_test_date =
| typing =
| implementations =
| dialects =
| influenced_by = [[Agda (programming language)|Agda]], [[Coq]],<ref name="idris-web">{{cite web|title=Idris, a language with dependent types|url=http://www.idris-lang.org/|accessdate=2014-10-26}}</ref> [[Epigram (programming language)|Epigram]], [[Haskell (programming language)|Haskell]],<ref name="idris-web"/> [[ML (programming language)|ML]],<ref name="idris-web"/> [[Rust (programming language)|Rust]], [[Perl]]
| influenced =
| operating_system = [[Cross-platform]]
| license = BSD-3
| file ext = .idr, .lidr
| website = {{URL|idris-lang.org}}
}}

'''Idris''' is a general-purpose [[purely functional programming language]] with [[dependent type]]s, strict or optional [[lazy evaluation]] and features such as a [[Termination analysis|totality checker]].

Even before its possible usage for [[proof assistant|interactive theorem-proving]], the focus of Idris is on [[general-purpose language|general-purpose programming]], like the purely functional [[Haskell (programming language)|Haskell]], and with sufficient performance. The [[type system]] of Idris is similar to the one used by [[Agda (programming language)|Agda]] and theorem-proving in it is similar to [[Coq]], including [[tactics (theorem proving)|tactics]]. In comparison, Idris has a priority on easy management of [[Side effect (computer science)|side-effects]] and support for implementing [[EDSL#Usage patterns|embedded domain specific languages]].

{{As of|2017|05}}, Idris compiles to [[C (programming language)|C]] (relying on a custom copying [[Garbage collection (computer science)|garbage collector]] using [[Cheney's algorithm]]) and [[JavaScript]] (both browser- and [[Node.js]]-based). There are also a number of third-party code generators for other platforms, including [[Java (programming language)|Java]], [[Java virtual machine|JVM]], [[Common Intermediate Language|CIL]], [[OCaml]], and a partial [[LLVM]] backend.<ref>{{cite web|url=http://docs.idris-lang.org/en/latest/reference/codegen.html|title=Code Generation Targets — Idris 1.1.1 documentation|website=docs.idris-lang.org}}</ref>

The name ''Idris'' goes back to the character of the singing dragon in the 1970s UK children's television program ''[[Ivor the Engine#Idris the Dragon|Ivor the Engine]]''.<ref>{{cite web|title=Frequently Asked Questions|url=http://docs.idris-lang.org/en/latest/faq/faq.html#what-does-the-name-idris-mean|accessdate=2015-07-19}}</ref>

==Features==
Idris combines a number of features from relatively mainstream functional programming languages with features borrowed from [[proof assistant]]s.

===Functional programming===
The syntax of Idris shows many similarities with that of Haskell. A [[hello world program]] in Idris might look like this:
<source lang="idris">
module Main

main : IO ()
main = putStrLn "Hello, World!"
</source>

The only differences between this program and its [[Haskell (programming language)#Code examples|Haskell equivalent]] are the single colon (instead of two) in the [[Type signature|signature]] of the main function and the omission of the word "where" in the module declaration.

====Inductive and parametric data types====
Like most modern functional programming languages, Idris supports a notion of [[inductively-defined data type]] and [[parametric polymorphism]]. Such types can be defined both in traditional "Haskell98" syntax:

<source lang="idris">
data Tree a = Node (Tree a) (Tree a) | Leaf a
</source>

or in the more general [[Generalized algebraic data type|GADT]] syntax:

<source lang="idris">
data Tree : Type -> Type where
Node : Tree a -> Tree a -> Tree a
Leaf : a -> Tree a
</source>

====Dependent types====
With [[dependent type]]s, it is possible for values to appear in the types; in effect, any value-level computation can be performed during [[typechecking]]. The following defines a type of lists of statically known length, traditionally called 'vectors':

<source lang="idris">
data Vect : Nat -> Type -> Type where
Nil : Vect 0 a
(::) : (x : a) -> (xs : Vect n a) -> Vect (n + 1) a
</source>

This type can be used as follows:

<source lang="idris">
total
append : Vect n a -> Vect m a -> Vect (n + m) a
append Nil ys = ys
append (x :: xs) ys = x :: append xs ys
</source>

The functions append a vector of m elements of type a to a vector of n elements of type a. Since the precise types of the input vectors depend on a value, it is possible to be certain at compile-time that the resulting vector will have exactly (n + m) elements of type a.
The word "total" invokes the [[Termination analysis|totality checker]] which will report an error if the function [[Partial function|doesn't cover all possible cases]] or cannot be (automatically) proven to not enter an [[infinite loop]].

Another common example is pairwise addition of two vectors that are parameterized over their length:

<source lang="idris">
total
pairAdd : Num a => Vect n a -> Vect n a -> Vect n a
pairAdd Nil Nil = Nil
pairAdd (x :: xs) (y :: ys) = x + y :: pairAdd xs ys
</source>

Num a signifies that the type a belongs to the [[type class]] Num. Note that this function still typechecks successfully as total, even though there is no case matching Nil in one vector and a number in the other. Since both vectors are ensured by the type system to have exactly the same length, we can be sure at compile time that this case will not occur. Hence it does not need to be mentioned for the function to be total.

===Proof assistant features===
Dependent types are powerful enough to encode most properties of programs, and an Idris program can prove invariants at compile-time. This makes Idris into a proof assistant.

There are two standard ways of interacting with proof assistants: by writing a series of tactic invocations ([[Coq]] style), or by interactively elaborating a proof term ([[Epigram (programming language)|Epigram]]/[[Agda (programming language)|Agda]] style). Idris supports both modes of interaction, although the set of available tactics is not yet as useful as that of Coq.

===Code generation===
Because Idris contains a proof assistant, Idris programs can be written to pass proofs around. If treated naïvely, such proofs remain around at runtime. Idris aims to avoid this pitfall by aggressively erasing unused terms,<ref>{{cite web|url=http://idris.readthedocs.org/en/latest/reference/erasure.html|title=Erasure By Usage Analysis — Idris 1.1.1 documentation|website=idris.readthedocs.org}}</ref> with promising{{Vague|date=May 2018}} results.<ref>{{cite web|url=http://ziman.functor.sk/erasure-bm/|title=Benchmark results|website=ziman.functor.sk}}</ref>

By default, Idris generates native code by going through C. Other backends are available for generating JavaScript and Java.

==See also==
* [[Total functional programming]]

==References==
{{reflist}}

==External links==
* [http://idris-lang.org/ The Idris homepage], including documentation, frequently asked questions and examples
* [http://hackage.haskell.org/package/idris Idris at the Hackage repository]
* [http://docs.idris-lang.org/en/latest/index.html Documentation for the Idris Language (tutorial, language reference, etc.)]

{{Programming languages}}

[[Category:Dependently typed languages]]
[[Category:Functional languages]]
[[Category:Free software programmed in Haskell]]
[[Category:Haskell programming language family]]
[[Category:Cross-platform free software]]
[[Category:Free compilers and interpreters]]
[[Category:Software using the BSD license]]
[[Category:Programming languages created in 2009]]
[[Category:High-level programming languages]]
[[Category:2009 software]]

Content-addressable storage

2018-09-16T20:06:39Z

Crasshopper: use simple words

{{Use dmy dates|date=March 2013}}
{{multiple issues|
{{confusing|date=February 2012}}
{{overly detailed|date=February 2012}}
{{cleanup-rewrite|date=February 2012}}
}}
'''Content-addressable storage''', also referred to as '''associative storage''' or abbreviated '''CAS''', is a way to store information so it can be retrieved based on its content, not its location. It has been used for high-speed storage and [[information retrieval|retrieval]] of [[fixed content]], such as documents stored for compliance with government regulations. Content-addressable storage is like [[content-addressable memory]].

==CAS and FCS==

Content Addressable Storage (CAS) and Fixed Content Storage (FCS) are different acronyms for the same type of technology. The CAS / FCS technology is intended to store data that does not change (fixed) in time. The difference is that typically CAS exposes a digest generated by a [[cryptographic hash function]] (such as [[SHA-1]] or [[MD5]]) from the document it refers to. If the hash function is weak, this method could be subject to collisions in an adversarial environment (different documents returning the same hash). The main advantages of CAS / FCS technology is that the location of the actual data and the number of copies is unknown to the user.

== Content-addressed vs. location-addressed ==

When being contrasted with content-addressed storage, a typical local or networked [[Object storage device|storage device]] is referred to as {{em|location-addressed}}. In a location-addressed storage device, each element of data is stored onto the physical medium, and its location recorded for later use. The storage device often keeps a list, or directory, of these locations. When a future request is made for a particular item, the request includes only the location (for example, path and file names) of the data. The storage device can then use this information to locate the data on the physical medium, and retrieve it. When new information is written into a location-addressed device, it is simply stored in some available free space, without regard to its content. The information at a given location can usually be altered or completely overwritten without any special action on the part of the storage device.

Within the scope of this discussion, a good way to think of the above is as {{em|container-addressed}} storage.

The [[Content Addressable File Store]] (CAFS) was a hardware device developed and sold by [[International Computers Limited]] (ICL) in the 1970s and 1980s that provided location-addressed disk storage with built-in search capability. The search logic was incorporated into the disk controller. A query expressed in a high-level query language could be compiled into a search specification that was then sent to the disk controller for execution. Files could also be accessed via the conventional location-addressing mechanism, permitting CAFS to support an [[IDMS]] CODASYL database and also support content addressing of the same records.

In contrast, when information is stored into a CAS system, the system will record a {{em|content address}}, which is an [[identifier]] uniquely and permanently linked to the information content itself. A request to retrieve information from a CAS system must provide the content identifier, from which the system can determine the physical location of the data and retrieve it. Because the identifiers are based on content, any change to a data element will necessarily change its content address. In nearly all cases, a CAS device will not permit editing information once it has been stored. Whether it can be deleted is often controlled by a policy.

While the idea of content-addressed storage is not new, production-quality systems were not readily available until roughly 2003.<ref name="usenix">[http://www.usenix.org/events/usenix03/tech/tolia.html USENIX Annual Technical Conference 2003, General Track - Abstract]</ref> In mid-2004, the industry group [[Storage Networking Industry Association|SNIA]] began working with a number of CAS providers to create standard behavior and interoperability guidelines for CAS systems.<ref name="cassi">CAS Industry standardization activities - XAM: http://www.snia.org/forums/xam</ref>

== Pros and cons ==

CAS storage works most efficiently on data that does not change often. It is of particular interest to large organizations that must comply with document-retention laws, such as [[Sarbanes-Oxley]]. In these corporations a large volume of documents will be stored for as much as a decade, with no changes and infrequent access. CAS is designed to make the searching for a given document content very quick, and provides an assurance that the retrieved document is identical to the one originally stored. (If the documents were different, their content addresses would differ.) In addition, since data is stored into a CAS system by what it contains, there is never a situation where more than one copy of an identical document exists in storage. By definition, two identical documents have the same content address, and so point to the same storage location.

For data that changes frequently, CAS is not as efficient as location-based addressing. In these cases, the CAS device would need to continually recompute the address of data as it was changed, and the client systems would be forced to continually update information regarding where a given document exists. For random access systems, a CAS would also need to handle the possibility of two initially identical documents diverging, requiring a copy of one document to be created on demand.{{Further|Copy on write}}

== Typical implementation ==
{{advert|section|date=June 2017}}
Paul Carpentier and Jan van Riel coined the term CAS while working at a company called FilePool in the late 1990s. FilePool was acquired in 2001 and became the underpinnings of the first commercially available CAS system, which was introduced as [[EMC Corporation|EMC's]] Centera platform.<ref name="centera">[http://www.findarticles.com/p/articles/mi_m0BRZ/is_10_22/ai_98977101 Content-addressable storage - Storage as I See it], by Mark Ferelli, Oct, 2002, BNET.com</ref> The Centera CAS system consists of a series of networked nodes (1-U servers running [[Linux]]), divided between storage nodes and access nodes. The access nodes maintain a synchronized directory of content addresses, and the corresponding storage node where each address can be found. When a new data element, or blob ([[Binary large object]]), is added, the device calculates a [[hash value|hash]] of the content and returns this hash as the blob's content address.<ref name="techworld.com">[http://www.techworld.com/features/index.cfm?featureID=235&printerfriendly=1 Making a hash of file content Content-addressable storage uses hash algorithms.], By Chris Mellor, Published: 9 December 2003, Techworld{{deadlink|date=December 2017}} Article moved to https://www.techworld.com/data/making-a-hash-of-file-content-235/</ref> As mentioned above, the hash is searched for to verify that identical content is not already present. If the content already exists, the device does not need to perform any additional steps; the content address already points to the proper content. Otherwise, the data is passed off to a storage node and written to the physical media.

When a content address is provided to the device, it first queries the directory for the physical location of the specified content address. The information is then retrieved from a storage node, and the actual hash of the data recomputed and verified. Once this is complete, the device can supply the requested data to the client. Within the Centera system, each content address actually represents a number of distinct data blobs, as well as optional [[metadata]]. Whenever a client adds an additional blob to an existing content block, the system recomputes the content address.

To provide additional data security, the Centera access nodes, when no read or write operation is in progress, constantly communicate with the storage nodes, checking the presence of at least two copies of each blob as well as their integrity. Additionally, they can be configured to exchange data with a different, e.g. off-site, Centera system, thereby strengthening the precautions against accidental data loss.

IBM has another flavor of CAS which can be software based, Tivoli Storage manager 5.3, or hardware based, the IBM DR550. The architecture is different in that it is based on a [[hierarchical storage management]] (HSM) design which provides some additional flexibility such as being able to support not only [[Write Once Read Many|WORM]] disk but WORM tape and the migration of data from WORM disk to WORM tape and vice versa. This provides for additional flexibility in disaster recovery situations as well as the ability to reduce storage costs by moving data off disk to tape.

Another typical implementation is iCAS from iTernity. The concept of iCAS is based on containers. Each container is addressed by its hash value. A container holds different numbers of fixed content documents. The container is not changeable and the hash value is fixed after the write process.

== Open-source implementations ==

One of the very first content-addressed storage servers, [[Venti]],<ref>[http://doc.cat-v.org/plan_9/4th_edition/papers/venti/ Venti: a new approach to archival storage]</ref> was originally developed for [[Plan 9 from Bell Labs]] and is now also available for Unix-like systems as part of [[Plan 9 from User Space]].

A first step towards an open source CAS+ implementation is Twisted Storage.<ref name="twistedstorage">[http://twistedstorage.sourceforge.net Twisted Storage]</ref>

[[Tahoe-LAFS|Tahoe Least-Authority File Store]] is an open source implementation of CAS.

[[Git (software)#Implementations|Git]] is a [[userspace]] CAS filesystem. However it is primarily used as a source code control system.

[[git-annex]] is a distributed file synchronization system which uses content-addressable storage for files it manages. It relies on Git and [[symbolic links]] to index their filesystem location.

[[Project Honeycomb]] is an open source [[Application programming interface|API]] for CAS systems.<ref>{{cite web|url=http://www.opensolaris.org/os/project/honeycomb/ |title=Archived copy |accessdate=2007-10-01 |deadurl=yes |archiveurl=https://web.archive.org/web/20071012085111/http://www.opensolaris.org/os/project/honeycomb/ |archivedate=12 October 2007 |df= }}</ref>

The [[XAM]] interface being developed under the auspices of the [[Storage Networking Industry Association]] is an attempt to create a standard interface for archiving on CAS (and CAS like) products and projects.{{Citation needed|date=March 2012}}

[[Bitcache]] is an open source distributed implementation of CAS written in Ruby.<ref>[http://bitcache.org Bitcache - Distributed, content-addressable data storage]</ref> Bitcache server has an implementation for Drupal as well.<ref>[http://drupal.org/project/bitcache a module that provides a Bitcache-compatible data storage repository for Drupal and implements the Bitcache REST API]</ref>

[[Perkeep]] is a recent project to bring the advantages of content-addressable storage "to the masses". It is intended to be used for a wide variety of use cases, including distributed backup; a snapshotted-by-default, version-controlled filesystem; and decentralised, permission-controlled filesharing.

Irmin is an [[ocaml]] "library for persistent stores with built-in snapshot, branching and reverting mechanisms"; the same design principles as Git.

Cassette is an open source CAS implementation for C#/.NET.<ref>[https://github.com/drewnoakes/cassette Cassette - A simple content-addressable storage system for .NET 4.5]</ref>

Arvados Keep is an open source content-addressable distributed storage system.<ref>[https://arvados.org/projects/arvados/wiki/Keep/ Keep - Content-Addressable Distributed Storage]</ref> It is designed for large-scale, computationally intensive data science work such as storing and processing genomic data.

Infinit is a content-addressable and decentralized (peer-to-peer) storage platform that was acquired by [[Docker (software)|Docker]] Inc.

[[InterPlanetary File System]] (IPFS), is a content-addressable, peer-to-peer hypermedia distribution protocol.

[[casync]] is a Linux software utility by Lennart Poettering to distribute frequently-updated file system images over the Internet.<ref name="phoronix">{{cite web|url=https://www.phoronix.com/scan.php?page=news_item&px=Lennart-casync|title=Lennart Poettering Announces New Project: casync - Phoronix|website=[[Phoronix]]}}</ref>

==See also==
*[[Content Addressable File Store]]
*[[Content-centric networking]] / Named data networking
* [[Data Defined Storage]]

== References ==
{{Reflist}}

== External links ==
* [http://doc.cat-v.org/plan_9/misc/foundation/ Fast, Inexpensive Content-Addressed Storage in Foundation]
* [http://doc.cat-v.org/plan_9/4th_edition/papers/venti/ Venti: a new approach to archival storage]

[[Category:Associative arrays]]
[[Category:Computer storage devices]]

Anti–Corn Law League

2018-09-08T21:51:40Z

Crasshopper: /* Corn Laws */

[[File:1846 - Anti-Corn Law League Meeting.jpg|thumb|right|250px|A meeting of the Anti-Corn Law League in [[Exeter Hall]] in 1846]]
The '''Anti-Corn Law League''' was a successful political movement in [[United Kingdom of Great Britain and Ireland|Great Britain]] aimed at the abolition of the unpopular [[Corn Laws]], which protected landowners’ interests by levying taxes on imported wheat, thus raising the price of bread at a time when factory-owners were trying to cut wages. The League was a middle-class nationwide organization that held many well-attended rallies on the premise that a crusade was needed to convince parliament to repeal the corn laws. Its long-term goals included the removal of feudal privileges, which it denounced as impeding progress, lowering economic well-being, and restricting freedom. The League played little role in the final act in 1846 when Sir [[Robert Peel]] led the successful battle for repeal. However, its experience provided a model that was widely adopted in Britain and other democratic nations to demonstrate the organization of a political pressure group with the popular base.

==Corn Laws==

The Corn Laws were taxes on imported grain introduced in 1815,<ref>E Halévy, ''The Liberal Awakening'' (London 1961) p. 4</ref> and designed to keep prices high{{fact}} for cereal producers in Great Britain. The laws indeed did raise food prices, and became the focus of opposition from urban groups who had less political power than rural Britain. The corn laws initially prohibited foreign corn completely from being imported at below 80s a quarter,<ref>E Halévy, ''The Liberal Awakening'' (London 1961) p. 5</ref> a process replaced by a sliding scale in 1828.<ref>E Halévy, ''The Liberal Awakening'' (London 1961) p. 249</ref> Such import duties still made it expensive for anyone to import grain from other countries, even when food supplies were short. The laws were supported by [[Tory|Conservative]] (and [[Whigs (British political party)|Whig]]) landowners, and opposed by urban industrialists and workers.{{fact}} The League was responsible for turning public and elite opinion against the laws. It was a large, nationwide middle-class moral crusade with a utopian vision. Its leading advocate [[Richard Cobden]], according to historian [[Asa Briggs]], promised that repeal would settle four great problems simultaneously:
: First, it would guarantee the prosperity of the manufacturer by affording him outlets for his products. Second, it would relieve the '[[condition of England question]]' by cheapening the price of food and ensuring more regular employment. Third, it would make English agriculture more efficient by stimulating demand for its products in urban and industrial areas. Fourth, it would introduce through mutually advantageous international trade a new era of international fellowship and peace. The only barrier to these four beneficent solutions was the ignorant self-interest of the landlords, the 'bread-taxing oligarchy, unprincipled, unfeeling, rapacious and plundering.'<ref> Asa Briggs, ''The Making of Modern England 1783–1867: The Age of Improvement'' (1959) p. 314 </ref>

==The League==
The first Anti-Corn Law Association was set up in London in 1836; but it was not until 1838 that the nation-wide League, combining all such local associations, was founded, with [[Richard Cobden]] and [[John Bright]] among its leaders.<ref>E Halévy, ''The Triumph of Reform'' (London 1961) p. 330-4</ref> Cobden was the chief strategist; Bright was its great orator. A representative activist was [[Thomas Perronet Thompson]], who specialized in the grass-roots mobilisation of opinion through pamphlets, newspaper articles, correspondence, speeches, and endless local planning meetings.<ref>Michael J. Turner, "The ‘Bonaparte of free trade’ and the Anti-Corn Law League." ''Historical Journal'' 41.4 (1998): 1011-1034. </ref> The League was based in Manchester and had support from numerous industrialists, especially in the textile industry.<ref> Spall, 1988.</ref>

The League borrowed many of the tactics first developed by the anti-slavery crusaders, while also attempting to replicate its mantle of moral reform.<ref>Simon Morgan, "The Anti-Corn Law League and British anti-slavery in transatlantic perspective, 1838–1846." ''Historical Journal"" 52.1 (2009): 87-107. </ref> Among these were the use of emotionally charged meetings and closely argued tracts: nine million were distributed by a staff of 800 in 1843 alone.<ref>G M Trevelyan, ''British History in the 19th Century'' (London 1922) p. 270</ref> The League also used its financial strength and campaign resources to defeat protectionists at by-elections by enfranchising League supporters through giving them a 40 shilling freehold:<ref> Eric J. Evans, ''The Forging of the Modern State: Early Industrial Britain 1783–1870 '' (2nd ed. 1996, pp. 280–81) </ref> the strategy certainly alarmed the [[Tories]],<ref>E Halévy, ''Victorian Years'' (London 1961) p. 110-1</ref> but was expensive and led to numerous defeats, which the League blamed on the tyrannical power of the landlords.{{citeneeded|date=April 2018}} One of the most nationally visible efforts came in the 1843 election in Salisbury. Its candidate was defeated and it was unable to convince voters regarding free trade. However, the League did learn lessons that helped to transform its political tactics. It learned to concentrate on elections where there was a good expectation of victory.<ref>Ronald K. Huch, "The Anti-Corn Law League and the Salisbury Election of November 1843." ''Canadian Journal of History'' 6.3 (1971): 247-256.</ref>

Nevertheless the League had a restricted capability for contesting electoral seats, and its role in the final act of 1846 was largely that of creating a favourable climate of opinion. 1845 saw [[Lord John Russell]], the Whig leader, declare for complete repeal of the corn duty as the only way to satisfy the League;<ref>E Halévy, ''Victorian Years'' (London 1961) p. 115</ref> while the Tory leader, Sir [[Robert Peel]], had also been privately won over by Cobden's reasoning to the league's way of thinking.<ref>G M Trevelyan, ''British History in the 19th Century'' (London 1922) p. 268</ref> When the crunch came, Peel put through a (staggered) repeal through Parliament without a general election,<ref>Norman Gash, ''Sir Robert Peel: The Life of Sir Robert Peel after 1830'' (1972) pp. 575–76.</ref> to the applause of Cobden and Bright.<ref>E Halévy, ''Victorian Years'' (London 1961) p. 123-5</ref>

The League then prepared to dissolve itself.<ref>«As no other gentleman has anything to address to this meeting, it is now my duty to say that the Anti-Corn-Law League stands conditionally dissolved» [George Wilson at a meeting of the Council of the Anti-Corn Law League held in Manchester Town Hall (Thursday 2 July 1846)]</ref> The Tory victory of 1852 saw preparations to revive the League, however, in order to keep a watching brief on Protectionist forces; and it was only after [[Disraeli]]’s 1852 budget that Cobden felt able to write to [[George Wilson (reformer)|George Wilson]]: “The Budget has finally closed the controversy with Protection...The League may be dissolved when you like”.<ref>E Halévy, ''Victorian Years'' (London 1961) p. 325-8</ref> Many of its members thereafter continued their political activism in the [[Liberal Party (UK)|Liberal Party]], with the goal of establishing a fully free-trade economy.

W.H. Chaloner argues that the repeal in 1846 marked a major turning point, making free trade the national policy into the 20th century, and demonstrating the power of "Manchester-school" industrial interests over protectionist agricultural interests. He says repeal stabilized wheat prices in the 1850s and 1860s; however other technical developments caused the fall of wheat prices from 1870-1894.<ref>W. H. Chaloner, "The Anti-Corn Law League," ''History Today'' (1968) 18#3 pp 196-204</ref>

==Model for other lobbying organisations==
The League marked the emergence of the first powerful national lobbying group into politics, one with a centralized office, consistency of purpose, rich funding, very strong local and national organization, and single-minded dedicated leaders. It elected men to Parliament. Many of its procedures were innovative, while others were borrowed from the anti-slavery movement. It became the model for later reform movements.<ref>Briggs, ''The Making of Modern England,'' p. 116</ref>

The model of the League led to the formation of the Lancashire Public School Association to campaign for free, locally-financed and controlled secular education in Lancashire. It later became the National Public School Association. It had little success because national secular education, was a divisive issue even among the radical groups However it did help convert the Liberal Party from its laissez-faire philosophy to that of a more interventionist character.<ref>Donald K. Jones, "The Educational Legacy of the Anti‐Corn Law League." ''History of Education'' 3.1 (1974): 18-35. </ref>

Historian A. C. Howe argues:
:Although historians remain divided on the impact of the league on Peel's decision to abandon the corn laws it was undoubtedly, in appearance, the most successful of nineteenth-century single-issue pressure groups, in its ability to generate enthusiasm, support, and unparalleled financial backing. Although its potential was not realized, it had shown the capacity for an extra-parliamentary middle-class organization to reshape politics so as to reflect the anti-aristocratic objectives of a determined band of entrepreneurial politicians.

:It remained the model for many diverse pressure groups, for example the United Kingdom Alliance, the National Educational League, the Navy League, the Tenant League in Ireland, and the National Society in Piedmont, as well as those specifically related to free trade, including the Edwardian Tariff Reform League and Free Trade Union, and in the 1950s S. W. Alexander's Anti-Dear Food League. It also inspired imitators in France, Germany, the Low Countries, Spain, and the United States. The league had only temporarily reshaped the landscape of parliamentary politics but it had helped create a vibrant popular attachment to free trade within British political culture that would last well into the twentieth century.<ref>A. C. Howe, ‘Anti-Corn Law League (act. 1839–1846)’, ''Oxford Dictionary of National Biography,'' Oxford University Press. [http://www.oxforddnb.com/view/theme/42282, accessed 8 Nov 2017]</ref>

==Critics==
*[[R. S. Surtees]] satirized the league in his 1845 novel, [[Hillingdon Hall]]. His cockney protagonist refers to “the ‘umbuggery of its ways...strong symptoms of utilitarian self-interest”; while a roguish actor is shown being couched as a paid lecturer for the League: “you have nothing to do but repeat the same old story over and over again….Whatever is wrong, lay it to the corn tax. If a man can’t pay his Christmas bills, attribute it to the bread tax”.<ref>R S Surtees, ''Hillingdon Hall'' (Stroud 2006) p. 44-7 and p. 39</ref>

==See also==
* [[Manchester Liberalism]]
* [[Canada Corn Act]]
* [[Meat riots]]

==Notes==
{{reflist}}

==Further reading==
===Scholarly studies===
* Briggs, Asa. ''The Making of Modern England 1783–1867: The Age of Improvement'' (1959) pp. 312–25, short interpretive history
* Edsall, Nicholas C. ''Richard Cobden, independent radical'' (Harvard University Press, 1986)
* Halévy, Elie. ''Victorian years, 1841–1895'' (Vol. 4) (Barnes & Noble, 1961) pp. 3–150; narrative history
* Hinde, Wendy. ''Richard Cobden: A Victorian Outsider'' (Yale University Press, 1987.)
* [[Anthony Howe (historian)|Howe, Anthony]]. ''Free Trade and Liberal England. 1846–1946'' (Oxford: Clarendon Press, 1997).
* Lawson-Tancred, Mary. "The Anti-League and the Corn Law Crisis of 1846." ''Historical Journal'' (1960) 3#2 pp. 162–83.
* McCord, Norman: ''The Anti-Corn Law League 1838–1846''. (Allen & Unwin, 1958)
* Mosse, George L. "The Anti-League: 1844–1846." ''Economic History Review'' (1947) 17#2 pp. 134–42. [https://www.jstor.org/stable/2590555 in JSTOR]; the organized opposition to the League
* Pickering, Paul A and Alex Tyrrell. ''The people's bread, a history of the Anti-Corn Law League''. (Leicester University Press, 2000, {{ISBN|0-7185-0218-3}})
* Spall, Richard Francis Spall Jr. "Free Trade, Foreign Relations, and the Anti-Corn-Law League," ''International History Review'' 10#3 (1988), pp. 405-432 [https://www.jstor.org/stable/40105891 online]
* {{cite encyclopedia |last=Steelman |first=Aaron |authorlink= |editor-first=Ronald |editor-last=Hamowy |editor-link=Ronald Hamowy |encyclopedia=The Encyclopedia of Libertarianism |title=Anti-Corn Law League |url= https://books.google.com/books?id=yxNgXs3TkJYC |year=2008 |publisher= [[SAGE Publications|SAGE]]; [[Cato Institute]] |location= Thousand Oaks, CA |isbn= 978-1-4129-6580-4 |doi=10.4135/9781412965811.n9 |pages=14–15|oclc=750831024| lccn = 2008009151 }}
* Trentmann, Frank. ''Free Trade Nation. Commerce, Consumption, and Civil Society in Modern Britain'' (Oxford University Press, 2008).

===Historiography===
* Loades, David Michael, ed. ''Reader's guide to British history'' (Fitzroy Dearborn Publishers, 2003) vol 1. pp. 56–57, 185–86, 283–84

===Contemporary publications===
* Ashworth, Henry: ''Recollections of Richard Cobden and the Anti-Corn Law League'', 2 editions, London 1876 and 1881
* Bright, John: ''Speeches of John Bright, M.P., on the American Question.'' With an introduction by [[Frank Moore (journalist)|Frank Moore]]. [With a portrait.]. Boston: Little, Brown & Co., 1865.
* Leech, H. J. (ed.): ''The public letters of the Right Hon. John Bright''. London: Low, Marston & Co., 1895. Reprint New York, NY: Kraus Reprint, 1969.
* [[Archibald Prentice|Prentice, Archibald]]: ''History of the Anti-cornlaw-league''. (London. 1853, 2 vol.). 2. ed. with a new introduction. by W. H. Chaloner. London: Cass, 1968.
* [[Thorold Rogers|Rogers, Thorold]] (ed.): ''Speeches on Questions of Public Policy, by John Bright, M.P.''. 1868.
* [[Thorold Rogers|Rogers, Thorold]] (ed.): ''Public Addresses''. 1879.
* Archibald Philipp Primrose (Earl of Rosebery): ''Lord Rosebery's Speech on the Anti-Corn Law League and Free Trade, Manchester 1897''. London: Cobden Club, 1898.
* [[George Barnett Smith|Smith, George Barnett]]: ''The Life and Speeches of the Right Hon. John Bright, M.P.'', 2 vols., 1881.
* [[Charles Anthony Vince|Vince, Charles]]: ''John Bright'' (1898); ''Speeches on Parliamentary Reform by John Bright, M.P.'', revised by Himself (1866).

==External links==
* [http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=2092 The Online Library of Liberty]
** [http://oll.libertyfund.org/pages/cobden-and-the-anti-corn-law-league 66 contemporary British ilustrations about free trade, 1830s-1910s]

[[Category:1838 establishments in the United Kingdom]]
[[Category:19th century in the United Kingdom]]
[[Category:Economic history of the United Kingdom]]

Anti–Corn Law League

2018-09-08T21:51:12Z

Crasshopper: /* Corn Laws */ needs citation

[[File:1846 - Anti-Corn Law League Meeting.jpg|thumb|right|250px|A meeting of the Anti-Corn Law League in [[Exeter Hall]] in 1846]]
The '''Anti-Corn Law League''' was a successful political movement in [[United Kingdom of Great Britain and Ireland|Great Britain]] aimed at the abolition of the unpopular [[Corn Laws]], which protected landowners’ interests by levying taxes on imported wheat, thus raising the price of bread at a time when factory-owners were trying to cut wages. The League was a middle-class nationwide organization that held many well-attended rallies on the premise that a crusade was needed to convince parliament to repeal the corn laws. Its long-term goals included the removal of feudal privileges, which it denounced as impeding progress, lowering economic well-being, and restricting freedom. The League played little role in the final act in 1846 when Sir [[Robert Peel]] led the successful battle for repeal. However, its experience provided a model that was widely adopted in Britain and other democratic nations to demonstrate the organization of a political pressure group with the popular base.

==Corn Laws==

The Corn Laws were taxes on imported grain introduced in 1815,<ref>E Halévy, ''The Liberal Awakening'' (London 1961) p. 4</ref> and designed to keep prices high{{fact}} for cereal producers in Great Britain. The laws indeed did raise food prices, and became the focus of opposition from urban groups who had less political power than rural Britain. The corn laws initially prohibited foreign corn completely from being imported at below 80s a quarter,<ref>E Halévy, ''The Liberal Awakening'' (London 1961) p. 5</ref> a process replaced by a sliding scale in 1828.<ref>E Halévy, ''The Liberal Awakening'' (London 1961) p. 249</ref> Such import duties still made it expensive for anyone to import grain from other countries, even when food supplies were short. The laws were supported by [[Tory|Conservative]] (and [[Whigs (British political party)|Whig]]) landowners, and opposed by urban industrialists and workers. The League was responsible for turning public and elite opinion against the laws. It was a large, nationwide middle-class moral crusade with a utopian vision. Its leading advocate [[Richard Cobden]], according to historian [[Asa Briggs]], promised that repeal would settle four great problems simultaneously:
: First, it would guarantee the prosperity of the manufacturer by affording him outlets for his products. Second, it would relieve the '[[condition of England question]]' by cheapening the price of food and ensuring more regular employment. Third, it would make English agriculture more efficient by stimulating demand for its products in urban and industrial areas. Fourth, it would introduce through mutually advantageous international trade a new era of international fellowship and peace. The only barrier to these four beneficent solutions was the ignorant self-interest of the landlords, the 'bread-taxing oligarchy, unprincipled, unfeeling, rapacious and plundering.'<ref> Asa Briggs, ''The Making of Modern England 1783–1867: The Age of Improvement'' (1959) p. 314 </ref>

==The League==
The first Anti-Corn Law Association was set up in London in 1836; but it was not until 1838 that the nation-wide League, combining all such local associations, was founded, with [[Richard Cobden]] and [[John Bright]] among its leaders.<ref>E Halévy, ''The Triumph of Reform'' (London 1961) p. 330-4</ref> Cobden was the chief strategist; Bright was its great orator. A representative activist was [[Thomas Perronet Thompson]], who specialized in the grass-roots mobilisation of opinion through pamphlets, newspaper articles, correspondence, speeches, and endless local planning meetings.<ref>Michael J. Turner, "The ‘Bonaparte of free trade’ and the Anti-Corn Law League." ''Historical Journal'' 41.4 (1998): 1011-1034. </ref> The League was based in Manchester and had support from numerous industrialists, especially in the textile industry.<ref> Spall, 1988.</ref>

The League borrowed many of the tactics first developed by the anti-slavery crusaders, while also attempting to replicate its mantle of moral reform.<ref>Simon Morgan, "The Anti-Corn Law League and British anti-slavery in transatlantic perspective, 1838–1846." ''Historical Journal"" 52.1 (2009): 87-107. </ref> Among these were the use of emotionally charged meetings and closely argued tracts: nine million were distributed by a staff of 800 in 1843 alone.<ref>G M Trevelyan, ''British History in the 19th Century'' (London 1922) p. 270</ref> The League also used its financial strength and campaign resources to defeat protectionists at by-elections by enfranchising League supporters through giving them a 40 shilling freehold:<ref> Eric J. Evans, ''The Forging of the Modern State: Early Industrial Britain 1783–1870 '' (2nd ed. 1996, pp. 280–81) </ref> the strategy certainly alarmed the [[Tories]],<ref>E Halévy, ''Victorian Years'' (London 1961) p. 110-1</ref> but was expensive and led to numerous defeats, which the League blamed on the tyrannical power of the landlords.{{citeneeded|date=April 2018}} One of the most nationally visible efforts came in the 1843 election in Salisbury. Its candidate was defeated and it was unable to convince voters regarding free trade. However, the League did learn lessons that helped to transform its political tactics. It learned to concentrate on elections where there was a good expectation of victory.<ref>Ronald K. Huch, "The Anti-Corn Law League and the Salisbury Election of November 1843." ''Canadian Journal of History'' 6.3 (1971): 247-256.</ref>

Nevertheless the League had a restricted capability for contesting electoral seats, and its role in the final act of 1846 was largely that of creating a favourable climate of opinion. 1845 saw [[Lord John Russell]], the Whig leader, declare for complete repeal of the corn duty as the only way to satisfy the League;<ref>E Halévy, ''Victorian Years'' (London 1961) p. 115</ref> while the Tory leader, Sir [[Robert Peel]], had also been privately won over by Cobden's reasoning to the league's way of thinking.<ref>G M Trevelyan, ''British History in the 19th Century'' (London 1922) p. 268</ref> When the crunch came, Peel put through a (staggered) repeal through Parliament without a general election,<ref>Norman Gash, ''Sir Robert Peel: The Life of Sir Robert Peel after 1830'' (1972) pp. 575–76.</ref> to the applause of Cobden and Bright.<ref>E Halévy, ''Victorian Years'' (London 1961) p. 123-5</ref>

The League then prepared to dissolve itself.<ref>«As no other gentleman has anything to address to this meeting, it is now my duty to say that the Anti-Corn-Law League stands conditionally dissolved» [George Wilson at a meeting of the Council of the Anti-Corn Law League held in Manchester Town Hall (Thursday 2 July 1846)]</ref> The Tory victory of 1852 saw preparations to revive the League, however, in order to keep a watching brief on Protectionist forces; and it was only after [[Disraeli]]’s 1852 budget that Cobden felt able to write to [[George Wilson (reformer)|George Wilson]]: “The Budget has finally closed the controversy with Protection...The League may be dissolved when you like”.<ref>E Halévy, ''Victorian Years'' (London 1961) p. 325-8</ref> Many of its members thereafter continued their political activism in the [[Liberal Party (UK)|Liberal Party]], with the goal of establishing a fully free-trade economy.

W.H. Chaloner argues that the repeal in 1846 marked a major turning point, making free trade the national policy into the 20th century, and demonstrating the power of "Manchester-school" industrial interests over protectionist agricultural interests. He says repeal stabilized wheat prices in the 1850s and 1860s; however other technical developments caused the fall of wheat prices from 1870-1894.<ref>W. H. Chaloner, "The Anti-Corn Law League," ''History Today'' (1968) 18#3 pp 196-204</ref>

==Model for other lobbying organisations==
The League marked the emergence of the first powerful national lobbying group into politics, one with a centralized office, consistency of purpose, rich funding, very strong local and national organization, and single-minded dedicated leaders. It elected men to Parliament. Many of its procedures were innovative, while others were borrowed from the anti-slavery movement. It became the model for later reform movements.<ref>Briggs, ''The Making of Modern England,'' p. 116</ref>

The model of the League led to the formation of the Lancashire Public School Association to campaign for free, locally-financed and controlled secular education in Lancashire. It later became the National Public School Association. It had little success because national secular education, was a divisive issue even among the radical groups However it did help convert the Liberal Party from its laissez-faire philosophy to that of a more interventionist character.<ref>Donald K. Jones, "The Educational Legacy of the Anti‐Corn Law League." ''History of Education'' 3.1 (1974): 18-35. </ref>

Historian A. C. Howe argues:
:Although historians remain divided on the impact of the league on Peel's decision to abandon the corn laws it was undoubtedly, in appearance, the most successful of nineteenth-century single-issue pressure groups, in its ability to generate enthusiasm, support, and unparalleled financial backing. Although its potential was not realized, it had shown the capacity for an extra-parliamentary middle-class organization to reshape politics so as to reflect the anti-aristocratic objectives of a determined band of entrepreneurial politicians.

:It remained the model for many diverse pressure groups, for example the United Kingdom Alliance, the National Educational League, the Navy League, the Tenant League in Ireland, and the National Society in Piedmont, as well as those specifically related to free trade, including the Edwardian Tariff Reform League and Free Trade Union, and in the 1950s S. W. Alexander's Anti-Dear Food League. It also inspired imitators in France, Germany, the Low Countries, Spain, and the United States. The league had only temporarily reshaped the landscape of parliamentary politics but it had helped create a vibrant popular attachment to free trade within British political culture that would last well into the twentieth century.<ref>A. C. Howe, ‘Anti-Corn Law League (act. 1839–1846)’, ''Oxford Dictionary of National Biography,'' Oxford University Press. [http://www.oxforddnb.com/view/theme/42282, accessed 8 Nov 2017]</ref>

==Critics==
*[[R. S. Surtees]] satirized the league in his 1845 novel, [[Hillingdon Hall]]. His cockney protagonist refers to “the ‘umbuggery of its ways...strong symptoms of utilitarian self-interest”; while a roguish actor is shown being couched as a paid lecturer for the League: “you have nothing to do but repeat the same old story over and over again….Whatever is wrong, lay it to the corn tax. If a man can’t pay his Christmas bills, attribute it to the bread tax”.<ref>R S Surtees, ''Hillingdon Hall'' (Stroud 2006) p. 44-7 and p. 39</ref>

==See also==
* [[Manchester Liberalism]]
* [[Canada Corn Act]]
* [[Meat riots]]

==Notes==
{{reflist}}

==Further reading==
===Scholarly studies===
* Briggs, Asa. ''The Making of Modern England 1783–1867: The Age of Improvement'' (1959) pp. 312–25, short interpretive history
* Edsall, Nicholas C. ''Richard Cobden, independent radical'' (Harvard University Press, 1986)
* Halévy, Elie. ''Victorian years, 1841–1895'' (Vol. 4) (Barnes & Noble, 1961) pp. 3–150; narrative history
* Hinde, Wendy. ''Richard Cobden: A Victorian Outsider'' (Yale University Press, 1987.)
* [[Anthony Howe (historian)|Howe, Anthony]]. ''Free Trade and Liberal England. 1846–1946'' (Oxford: Clarendon Press, 1997).
* Lawson-Tancred, Mary. "The Anti-League and the Corn Law Crisis of 1846." ''Historical Journal'' (1960) 3#2 pp. 162–83.
* McCord, Norman: ''The Anti-Corn Law League 1838–1846''. (Allen & Unwin, 1958)
* Mosse, George L. "The Anti-League: 1844–1846." ''Economic History Review'' (1947) 17#2 pp. 134–42. [https://www.jstor.org/stable/2590555 in JSTOR]; the organized opposition to the League
* Pickering, Paul A and Alex Tyrrell. ''The people's bread, a history of the Anti-Corn Law League''. (Leicester University Press, 2000, {{ISBN|0-7185-0218-3}})
* Spall, Richard Francis Spall Jr. "Free Trade, Foreign Relations, and the Anti-Corn-Law League," ''International History Review'' 10#3 (1988), pp. 405-432 [https://www.jstor.org/stable/40105891 online]
* {{cite encyclopedia |last=Steelman |first=Aaron |authorlink= |editor-first=Ronald |editor-last=Hamowy |editor-link=Ronald Hamowy |encyclopedia=The Encyclopedia of Libertarianism |title=Anti-Corn Law League |url= https://books.google.com/books?id=yxNgXs3TkJYC |year=2008 |publisher= [[SAGE Publications|SAGE]]; [[Cato Institute]] |location= Thousand Oaks, CA |isbn= 978-1-4129-6580-4 |doi=10.4135/9781412965811.n9 |pages=14–15|oclc=750831024| lccn = 2008009151 }}
* Trentmann, Frank. ''Free Trade Nation. Commerce, Consumption, and Civil Society in Modern Britain'' (Oxford University Press, 2008).

===Historiography===
* Loades, David Michael, ed. ''Reader's guide to British history'' (Fitzroy Dearborn Publishers, 2003) vol 1. pp. 56–57, 185–86, 283–84

===Contemporary publications===
* Ashworth, Henry: ''Recollections of Richard Cobden and the Anti-Corn Law League'', 2 editions, London 1876 and 1881
* Bright, John: ''Speeches of John Bright, M.P., on the American Question.'' With an introduction by [[Frank Moore (journalist)|Frank Moore]]. [With a portrait.]. Boston: Little, Brown & Co., 1865.
* Leech, H. J. (ed.): ''The public letters of the Right Hon. John Bright''. London: Low, Marston & Co., 1895. Reprint New York, NY: Kraus Reprint, 1969.
* [[Archibald Prentice|Prentice, Archibald]]: ''History of the Anti-cornlaw-league''. (London. 1853, 2 vol.). 2. ed. with a new introduction. by W. H. Chaloner. London: Cass, 1968.
* [[Thorold Rogers|Rogers, Thorold]] (ed.): ''Speeches on Questions of Public Policy, by John Bright, M.P.''. 1868.
* [[Thorold Rogers|Rogers, Thorold]] (ed.): ''Public Addresses''. 1879.
* Archibald Philipp Primrose (Earl of Rosebery): ''Lord Rosebery's Speech on the Anti-Corn Law League and Free Trade, Manchester 1897''. London: Cobden Club, 1898.
* [[George Barnett Smith|Smith, George Barnett]]: ''The Life and Speeches of the Right Hon. John Bright, M.P.'', 2 vols., 1881.
* [[Charles Anthony Vince|Vince, Charles]]: ''John Bright'' (1898); ''Speeches on Parliamentary Reform by John Bright, M.P.'', revised by Himself (1866).

==External links==
* [http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=2092 The Online Library of Liberty]
** [http://oll.libertyfund.org/pages/cobden-and-the-anti-corn-law-league 66 contemporary British ilustrations about free trade, 1830s-1910s]

[[Category:1838 establishments in the United Kingdom]]
[[Category:19th century in the United Kingdom]]
[[Category:Economic history of the United Kingdom]]

History of the Federal Reserve System

2018-09-06T13:13:15Z

Crasshopper: /* The Federal Reserve Act, 1913 */ growing?

[[Image:United States Federal Reserve Board, 1917.jpg|right|thumb|400px|Federal Reserve Board, 1917]]This article is about the history of the [[United States]] [[Federal Reserve System]] from its creation to the present.

== Central banking prior to the Federal Reserve ==
{{main|History of central banking in the United States}}

The Federal Reserve System is the third central banking system in United States history. The [[First Bank of the United States]] (1791–1811) and the [[Second Bank of the United States]] (1817–1836) each had a 20-year charter. Both banks issued currency, made commercial loans, accepted deposits, purchased securities, maintained multiple branches and acted as fiscal agents for the U.S. Treasury.<ref name="FedBeginnings">{{cite web |url=http://www.bos.frb.org/about/pubs/begin.pdf |title= Historical Beginnings ... The Federal Reserve |author= Johnson, Roger T. |publisher= Federal Reserve Bank of Boston}}</ref> The U.S. Federal Government was required to purchase 20% of the bank [[capital stock]] shares and to appoint 20% of the board members (directors) of each of those first two banks "of the United States." Therefore, each bank's majority control was placed squarely in the hands of wealthy investors who purchased the remaining 80% of the stock. These banks were opposed by state-chartered banks, who saw them as very large competitors, and by many who insisted that they were in reality banking cartels compelling the [[commoner|common man]] to maintain and support them. President [[Andrew Jackson]] vetoed legislation to renew the Second Bank of the United States, starting a period of [[free banking]]. Jackson staked the legislative success of his second presidential term on the issue of central banking. "Every monopoly and all exclusive privileges are granted at the expense of the public, which ought to receive a fair equivalent. The many millions which this act proposes to bestow on the stockholders of the existing bank must come directly or indirectly out of the earnings of the American people," Jackson said in 1832.<ref>Andrew Jackson, "Veto Message, Washington, July 10, 1832," in Richardson, ed., Messages and Papers of the Presidents, II, 576–591.</ref> Jackson's second term in office ended in March 1837 without the Second Bank of the United States's charter being renewed.

In 1863, as a means to help finance the [[American Civil War|Civil War]], a system of national banks was instituted by the [[National Currency Act]]. The banks each had the power to issue standardized national bank notes based on United States bonds held by the bank. The Act was totally revised in 1864 and later named as the National-Bank Act, or [[National Banking Act]], as it is popularly known. The administration of the new national banking system was vested in the newly created Office of the Comptroller of the Currency and its chief administrator, the [[Comptroller of the Currency]]. The Office, which still exists today, examines and supervises all banks chartered nationally and is a part of the U.S. Treasury Department.

== The Federal Reserve Act, 1913 ==
{{main|The Federal Reserve Act}}
National bank currency was considered inelastic because it was based on the fluctuating value of U.S. Treasury bonds. If Treasury bond prices declined, a national bank had to reduce the amount of currency it had in circulation by either refusing to make new loans or by calling in loans it had made already. The related liquidity problem was largely caused by an immobile, pyramidal reserve system, in which nationally chartered rural/agriculture-based banks were required to set aside their reserves in federal reserve city banks, which in turn were required to have reserves in central city banks. During the planting seasons, rural banks would exploit their reserves to finance full plantings, and during the harvest seasons they would use profits from loan interest payments to restore and grow their reserves. A national bank whose reserves were being drained would replace its reserves by selling stocks and bonds, by borrowing from a [[Clearing house (finance)|clearing house]] or by calling in loans. As there was little in the way of deposit insurance, if a bank was rumored to be having liquidity problems then this might cause many people to remove their funds from the bank. Because of the crescendo effect of banks which lent more than their assets could cover, during the last quarter of the 19th century and the beginning of the 20th century, the United States economy went through a series of financial panics.<ref name="EFlaherty">{{cite web |url= http://odur.let.rug.nl/~usa/E/usbank/bank00.htm|title= A Brief History of Central Banking in the United States |author= Flaherty, Edward|work= |publisher= University of Groningen, Netherlands}}</ref>

===The National Monetary Commission, 1907-1913===
{{very long| section|date=July 2017}}
Prior to a particularly severe [[Panic of 1907|panic in 1907]], there was a motivation for renewed demands for banking and currency reform.<ref name="herrickpanic">{{cite journal |jstor= 1010701|title= The Panic of 1907 and Some of Its Lessons |author= Herrick, Myron|date=March 1908|journal= |publisher= Annals of the American Academy of Political and Social Science}}</ref> The following year, Congress enacted the [[Aldrich-Vreeland Act]] which provided for an emergency currency and established the [[National Monetary Commission]] to study banking and currency reform.<ref name="mnwarburg">{{cite web |url= http://www.minneapolisfed.org/pubs/region/89-05/reg895d.cfm|title= Paul Warburg's Crusade to Establish a Central Bank in the United States |author= Whithouse, Michael|date=May 1989|work= |publisher= Minnesota Federal Reserve}}</ref>

[[Image:Fed Reserve.JPG|thumb|350px]]
The chief of the bipartisan National Monetary Commission was financial expert and Senate Republican leader [[Nelson Aldrich]]. Aldrich set up two commissions – one to study the American monetary system in depth and the other, headed by Aldrich, to study the European central-banking systems and report on them.<ref name="mnwarburg"/>

Aldrich went to Europe opposed to centralized banking but, after viewing [[Germany]]'s banking system, he came away believing that a centralized bank was better than the government-issued bond system that he had previously supported. Centralized banking was met with much opposition from politicians, who were suspicious of a central bank and who charged that Aldrich was biased due to his close ties to wealthy bankers such as [[J.P. Morgan]] and his daughter's marriage to [[John D. Rockefeller, Jr.]]<ref name="mnwarburg"/>

In 1910, Aldrich and executives representing the banks of J.P. Morgan, Rockefeller, and [[Kuhn, Loeb & Co.]], secluded themselves for ten days at [[Jekyll Island]], [[Georgia (U.S. state)|Georgia]].<ref name="mnwarburg"/> The executives included [[Frank A. Vanderlip]], president of the National City Bank of New York, associated with the Rockefellers; Henry Davison, senior partner of J.P. Morgan Company; Charles D. Norton, president of the First National Bank of New York; and Col. [[Edward M. House]], who would later become President Woodrow Wilson's closest adviser and founder of the [[Council on Foreign Relations]].<ref name="ecresearch">{{cite web |url= http://www.cooperativeindividualism.org/aier_on_conspiracy_04.html|title= America's Unknown Enemy: Beyond Conspiracy |publisher= American Institute of Economic Research}}</ref> There, [[Paul Warburg]] of Kuhn, Loeb, & Co. directed the proceedings and wrote the primary features of what would be called the Aldrich Plan. Warburg would later write that "The matter of a uniform discount rate (interest rate) was discussed and settled at Jekyll Island." Vanderlip wrote in his 1935 autobiography ''From Farmboy to Financier'':<ref name=Vanderlip>{{cite book
|title=From farm boy to financier|chapter=XXI. A Conclave on Jekyl Island
|author1=Frank Arthur Vanderlip|authorlink1=Frank A. Vanderlip|author2=Boyden Sparkes
|publisher=D. Appleton-Century Co.|year=1935|oclc=1000045|pages=210–219}} Excerpts at
{{cite web
|title=Frank Vanderlip And The Creation Of The Federal Reserve
|author=Eric deCarbonnel|date=June 19, 2009|work=Market Skeptics|accessdate=2012-02-10
|url=http://www.marketskeptics.com/2009/06/frank-vanderlip-and-creation-of-federal.html}}</ref>

<blockquote>Despite my views about the value to society of greater publicity for the affairs of corporations, there was an occasion, near the close of 1910, when I was as secretive, indeed, as furtive as any conspirator. None of us who participated felt that we were conspirators; on the contrary we felt we were engaged in a patriotic work. We were trying to plan a mechanism that would correct the weaknesses of our banking system as revealed under the strains and pressures of the panic of 1907. I do not feel it is any exaggeration to speak of our secret expedition to Jekyl Island as the occasion of the actual conception of what eventually became the Federal Reserve System. … Discovery, we knew, simply must not happen, or else all our time and effort would be wasted. If it were to be exposed publicly that our particular group had gotten together and written a banking bill, that bill would have no chance whatever of passage by Congress. Yet, who was there in Congress who might have drafted a sound piece of legislation dealing with the purely banking problem with which we were concerned?</blockquote>

Despite meeting in secret, from both the public and the government, the importance of the Jekyll Island meeting was revealed three years after the Federal Reserve Act was passed, when journalist [[Bertie Charles Forbes]] in 1916 wrote an article about the "hunting trip".<ref name=Forbes>''Leslie's Weekly'', Oct. 19, 1916, p. 423. Collected in {{cite book
|title=Men who are making America
|author=B. C. Forbes|authorlink=Bertie Charles Forbes
|publisher=B. C. Forbes Publishing Co.|year=1917|oclc=629297|pages=398–400
|url=https://books.google.com/books?id=x0EEAAAAYAAJ&pg=PA398}}</ref>

The 1911–12 Republican plan was proposed by Aldrich to solve the banking dilemma, a goal which was supported by the American Bankers' Association. The plan provided for one great central bank, the National Reserve Association, with a capital of at least $100 million and with 15 branches in various sections. The branches were to be controlled by the member banks on a basis of their capitalization. The National Reserve Association would issue currency, based on gold and commercial paper, that would be the liability of the bank and not of the government. The Association would also carry a portion of member banks' reserves, determine discount reserves, buy and sell on the open market, and hold the deposits of the federal government. The branches and businessmen of each of the 15 districts would elect thirty out of the 39 members of the board of directors of the National Reserve Association.<ref>Link, Arthur. ''Wilson and the Progressive Era'' New York: Harper, (1954) pp. 44–45</ref>

Aldrich fought for a private monopoly with little government influence, but conceded that the government should be represented on the board of directors. Aldrich then presented what was commonly called the "Aldrich Plan" – which called for establishment of a "National Reserve Association" – to the National Monetary Commission.<ref name="mnwarburg"/> Most [[Republican Party (United States)|Republicans]] and [[Wall Street]] bankers favored the Aldrich Plan,<ref name="ecresearch"/> but it lacked enough support in the bipartisan Congress to pass.<ref name="mnglass">{{cite web|url= http://www.minneapolisfed.org/pubs/region/88-08/reg888a.cfm|title= Born of a panic: Forming the Federal Reserve System|date= August 1988|publisher= Minnesota Federal Reserve|deadurl= yes|archiveurl= https://web.archive.org/web/20070620235302/http://minneapolisfed.org/pubs/region/88-08/reg888a.cfm|archivedate= 2007-06-20|df= }}</ref>

Because the bill was introduced by Aldrich, who was considered{{By whom|date=May 2013}} the epitome of the "Eastern establishment", the bill received little support. It was derided by southerners and westerners who believed that wealthy families and large corporations ran the country and would thus run the proposed National Reserve Association.<ref name="mnglass"/> The National Board of Trade appointed Warburg as head of a committee to persuade Americans to support the plan. The committee set up offices in the then-45 states and distributed printed materials about the proposed central bank.<ref name="mnwarburg"/> The [[Nebraska]]n populist and frequent Democratic presidential candidate [[William Jennings Bryan]] said of the plan: "Big financiers are back of the Aldrich currency scheme." He asserted that if it passed, big bankers would "then be in complete control of everything through the control of our national finances."<ref name="boshistory">{{cite web |url= http://www.bos.frb.org/about/pubs/begin.pdf|title= Historical Beginnings… The Federal Reserve|author= Johnson, Roger|date=December 1999 |publisher= Federal Reserve Bank of Boston}}</ref>

There was also Republican opposition to the Aldrich Plan. Republican Sen. [[Robert M. La Follette Sr.|Robert M. La Follette]] and Rep. [[Charles August Lindbergh|Charles Lindbergh Sr.]] both spoke out against the favoritism that they contended the bill granted to Wall Street. "The Aldrich Plan is the Wall Street Plan…I have alleged that there is a 'Money Trust'", said Lindbergh. "The Aldrich plan is a scheme plainly in the interest of the Trust". In response, Rep. [[Arsène Pujo]], a Democrat from Louisiana, obtained congressional authorization to form and chair a subcommittee (the [[Pujo Committee]]) within the House Committee Banking Committee, to conduct investigative hearings on the alleged "Money Trust". The hearings continued for a full year and were led by the subcommittee's counsel, Democratic lawyer [[Samuel Untermyer]], who later also assisted in drafting the [[Federal Reserve Act]]. The "Pujo hearings"<ref name="Pujo">{{cite web |url= http://www.public.asu.edu/~icprv/courses/hst315/secret315/biographies/Prog/Pujo,%20Arsene%20PROG.txt |title=Pujo, Arsene, a brief biography}}</ref> convinced much of the populace that America's money largely rested in the hands of a select few on Wall Street. The Subcommittee issued a report saying:<ref>{{cite web |url=http://college.hmco.com/polisci/wilson/am_gov/7e/students/portfolio/1913pujo_b.html |title=U.S. Congress, Excerpts from the Report of the Committee Appointed to Investigate the Concentration of Money and Credit, House Report No. 1593, 3 vols. (Washington, D.C., 1913), III: pp. 55–56, 89, 129, 140.}}</ref>

<blockquote>If by a 'money trust' is meant an established and well-defined identity and community of interest between a few leaders of finance…which has resulted in a vast and growing concentration of control of money and credit in the hands of a comparatively few men…the condition thus described exists in this country today...To us the peril is manifest...When we find...the same man a director in a half dozen or more banks and trust companies all located in the same section of the same city, doing the same class of business and with a like set of associates similarly situated all belonging to the same group and representing the same class of interests, all further pretense of competition is useless....<ref name="boshistory"/></blockquote>

Seen as a "Money Trust" plan, the Aldrich Plan was opposed by the Democratic Party as was stated in its 1912 campaign platform, but the platform also supported a revision of banking laws intended to protect the public from financial panics and "the domination of what is known as the "Money Trust." During the 1912 election, the Democratic Party took control of the presidency and both chambers of Congress. The newly elected president, [[Woodrow Wilson]], was committed to banking and currency reform, but it took a great deal of his political influence to get an acceptable plan passed as the Federal Reserve Act in 1913.<ref name="mnglass"/> Wilson thought the Aldrich plan was perhaps "60–70% correct".<ref name="mnwarburg"/> When Virginia Rep. Carter Glass, chairman of the House Committee on Banking and Currency, presented his bill to President-elect Wilson, Wilson said that the plan must be amended to contain a Federal Reserve Board appointed by the executive branch to maintain control over the bankers.<ref name="boshistory"/>

After Wilson presented the bill to Congress, a group of Democratic congressmen revolted. The group, led by Representative [[Robert Lee Henry|Robert Henry]] of Texas, demanded that the "Money Trust" be destroyed before it could undertake major currency reforms. The opponents particularly objected to the idea of regional banks having to operate without the implicit government protections that large, so-called money-center banks would enjoy. The group almost succeeded in killing the bill, but were mollified by Wilson's promises to propose antitrust legislation after the bill had passed, and by Bryan's support of the bill.<ref name="boshistory"/>

===Enactment of the Federal Reserve Act (1913)===
{{Undue weight section|date=July 2017}}
{{main|The Federal Reserve Act}}
After months of hearings, amendments, and debates the Federal Reserve Act passed Congress in December, 1913. The bill passed the House by an overwhelming majority of 298 to 60 on December
22, 1913<ref>{{cite journal
|title=Money Bill Goes to Wilson To-day
|journal=New York Times|date=December 23, 1913 |page=1
|url=https://www.nytimes.com/1913/12/23/archives/money-bill-goes-to-wilson-today-house-accepts-the-conference-report.html}}</ref> and passed the Senate the next day by a vote of 43 to 25.<ref>{{cite journal
|title=Wilson signs the currency bill
|journal=New York Times|date=December 24, 1913 |page=1
|url=https://query.nytimes.com/gst/abstract.html?res=9B04E3DB173DE633A25757C2A9649D946296D6CF}}</ref>
An earlier version of the bill had passed the Senate 54 to 34,<ref>{{cite journal
|title=Currency Bill Passes Senate
|journal=New York Times|date=December 20, 1913 |page=1
|url=https://www.nytimes.com/1913/12/20/archives/currency-bill-passes-senate-owen-measure-adopted-54-to-34-six.html}}</ref> but almost 30 senators had left for Christmas vacation by the time the final bill came to a vote.
Most every Democrat was in support of and most Republicans were against it.<ref name="boshistory"/> As noted in a paper by the American Institute of Economic Research:

<blockquote>In its final form, the Federal Reserve Act represented a compromise among three political groups. Most Republicans (and the Wall Street bankers) favored the Aldrich Plan that came out of Jekyll Island. Progressive Democrats demanded a reserve system and currency supply owned and controlled by the Government in order to counter the "money trust" and destroy the existing concentration of credit resources in Wall Street. Conservative Democrats proposed a decentralized reserve system, owned and controlled privately but free of Wall Street domination. No group got exactly what it wanted. But the Aldrich plan more nearly represented the compromise position between the two Democrat extremes, and it was closest to the final legislation passed.<ref name="ecresearch"/></blockquote>

[[Frank Vanderlip]], one of the Jekyll Island attendees and the president of National City Bank, wrote in his autobiography:<ref name=Vanderlip />

<blockquote>Although the Aldrich Federal Reserve Plan was defeated when it bore the name Aldrich, nevertheless its essential points were all contained in the plan that was finally adopted.</blockquote>

Ironically, in October 1913, two months before the enactment of the Federal Reserve Act, Frank Vanderlip proposed before the Senate Banking Committee his own competing plan to the Federal Reserve System, one with a single central bank controlled by the Federal government, which almost derailed the legislation then being considered and already passed by the U.S. House of Representatives.<ref name="VanderlipPlan">{{cite news |url=https://timesmachine.nytimes.com/timesmachine/1913/10/25/100283245.pdf|title= Wilson Upholds Glass Money Bill; But Senators Think His Statement Offers a Loophole for His Accepting Vanderlip Plan |date=October 25, 1913 |publisher= New York Times | format=PDF}}</ref> Even Aldrich stated strong opposition to the currency plan passed by the House.<ref name="AldrichNYT">{{cite news |url=https://www.nytimes.com/1913/10/16/archives/aldrich-sees-bryan-back-of-money-bill-socialistic-unconstitutional.html|title= Aldrich Sees Bryan Back of Money Bill; Socialist, Unconstitutional Measure, Says Ex-Senator |date=October 18, 1913 |publisher=New York Times | accessdate=May 4, 2010}}</ref>

However, the former point was also made by Republican Representative [[Charles August Lindbergh|Charles Lindbergh Sr.]] of Minnesota, one of the most vocal opponents of the bill, who on the day the House agreed to the Federal Reserve Act told his colleagues:

<blockquote>But the Federal reserve board have no power whatever to regulate the rates of interest that bankers may charge borrowers of money. This is the Aldrich bill in disguise, the difference being that by this bill the Government issues the money, whereas by the Aldrich bill the issue was controlled by the banks...Wall Street will control the money as easily through this bill as they have heretofore.(Congressional Record, v. 51, page 1447, Dec. 22, 1913)</blockquote>

Republican Congressman [[Victor Murdock]] of Kansas, who voted for the bill, told Congress on that same day:

<blockquote>I do not blind myself to the fact that this measure will not be effectual as a remedy for a great national evil – the concentrated control of credit...The Money Trust has not passed [died]...You rejected the specific remedies of the Pujo committee, chief among them, the prohibition of interlocking directorates. He [your enemy] will not cease fighting...at some half-baked enactment...You struck a weak half-blow, and time will show that you have lost. You could have struck a full blow and you would have won.<ref>(Congressional Record, v. 51, pp. 1443–44, Dec. 22, 1913)</ref></blockquote>

In order to get the Federal Reserve Act passed, Wilson needed the support of populist [[William Jennings Bryan]], who was credited with ensuring Wilson's nomination by dramatically throwing his support Wilson's way at the 1912 Democratic convention.<ref name="boshistory"/> Wilson appointed Bryan as his Secretary of State.<ref name="mnglass"/> Bryan served as leader of the agrarian wing of the party and had argued for unlimited coinage of silver in his "[[Cross of Gold]] Speech" at the 1896 Democratic convention.<ref name="glassbio">{{cite web|url= http://www.minneapolisfed.org/pubs/region/97-12/glass-bio.cfm|title= Carter Glass: A Brief Biography|author= Page, Dave|date= December 1997|publisher= Minnesota Federal Reserve|deadurl= yes|archiveurl= https://web.archive.org/web/20080516215926/http://minneapolisfed.org/pubs/region/97-12/glass-bio.cfm|archivedate= 2008-05-16|df= }}</ref> Bryan and the agrarians wanted a government-owned central bank which could print paper money whenever Congress wanted, and thought the plan gave bankers too much power to print the government's currency. Wilson sought the advice of prominent lawyer [[Louis Brandeis]] to make the plan more amenable to the agrarian wing of the party; Brandeis agreed with Bryan. Wilson convinced them that because Federal Reserve notes were obligations of the government and because the president would appoint the members of the Federal Reserve Board, the plan fit their demands.<ref name="boshistory"/> However, Bryan soon became disillusioned with the system. In the November 1923 issue of ''"Hearst's Magazine"'' Bryan wrote that "The Federal Reserve Bank that should have been the farmer's greatest protection has become his greatest foe."

Southerners and westerners learned from Wilson that the system was decentralized into 12 districts and surely would weaken New York and strengthen the hinterlands. Sen. [[Robert L. Owen]] of [[Oklahoma]] eventually relented to speak in favor of the bill, arguing that the nation's currency was already under too much control by New York elites, whom he alleged had singlehandedly conspired to cause the 1907 Panic.<ref name="ecresearch"/>

Large bankers thought the legislation gave the government too much control over markets and private business dealings. The ''[[New York Times]]'' called the Act the "Oklahoma idea, the Nebraska idea" – referring to Owen and Bryan's involvement.<ref name="boshistory"/>

However, several Congressmen, including Owen, Lindbergh, La Follette, and Murdock claimed that the New York bankers feigned their disapproval of the bill in hopes of inducing Congress to pass it. The day before the bill was passed, Murdock told Congress:

<blockquote>You allowed the special interests by pretended dissatisfaction with the measure to bring about a sham battle, and the sham battle was for the purpose of diverting you people from the real remedy, and they diverted you. The Wall Street bluff has worked.<ref>(Congressional Record, 22 December 1913)</ref></blockquote>

When Wilson signed the Federal Reserve Act on December 23, 1913, he said he felt grateful for having had a part "in completing a work ... of lasting benefit for the country,"<ref name= "Wilson-signs">{{cite news |url= https://query.nytimes.com/gst/abstract.html?res=9B04E3DB173DE633A25757C2A9649D946296D6CF|title= Wilson Signs Currency Bill| publisher= New York Times, December 24, 1913 | date=December 24, 1913 | accessdate=May 4, 2010}}</ref> knowing that it took a great deal of compromise and expenditure of his own political capital to get it enacted. This was in keeping with the general plan of action he made in his First Inaugural Address on March 4, 1913, in which he stated:

<blockquote>We shall deal with our economic system as it is and as it may be modified, not as it might be if we had a clean sheet of paper to write upon; and step-by-step we shall make it what it should be, in the spirit of those who question their own wisdom and seek counsel and knowledge, not shallow self-satisfaction or the excitement of excursions we can not tell.
<ref name="1st-inaugural">{{cite web |url= http://www.civicwebs.com/cwvlib/constitutions/usa/e_wilson_1st_inaug_address.htm|title= President Wilson's First Inaugural Address|publisher= Civic Webs Virtual Library}}</ref></blockquote>

While a system of 12 regional banks was designed so as not to give eastern bankers too much influence over the new bank, in practice, the [[Federal Reserve Bank of New York]] became "[[first among equals]]". The New York Fed, for example, is solely responsible for conducting [[open market operations]], at the direction of the Federal Open Market Committee.<ref name="jecreport">{{cite web|url=http://www.house.gov/jec/fed/fed/fed-impt.htm |title=The Importance of the Federal Reserve |author=Keleher, Robert |date=March 1997 |work=Joint Economic Committee |publisher=US House of Representatives |deadurl=yes |archiveurl=https://web.archive.org/web/20080228130550/http://www.house.gov/jec/fed/fed/fed-impt.htm |archivedate=2008-02-28 |df= }}</ref> Democratic Congressman [[Carter Glass]] sponsored and wrote the eventual legislation,<ref name="mnglass"/> and his home state capital of Richmond, Virginia, was made a district headquarters. Democratic Senator [[James A. Reed]] of Missouri obtained two districts for his state.<ref>[http://stlouisfed.org/publications/foregone/chapter_three.htm A Foregone Conclusion: The Founding of the Federal Reserve Bank of St. Louis by James Neal Primm – stlouisfed.org – Retrieved January 1, 2007]</ref> However, the 1914 report of the Federal Reserve Organization Committee, which clearly laid out the rationale for their decisions on establishing Reserve Bank districts in 1914, showed that it was based almost entirely upon current correspondent banking relationships.<ref name=ResBankOrgCmte>{{cite web |url=https://fraser.stlouisfed.org/scribd/?title_id=603&filepath=/docs/historical/Misc/ReserveBankDecision1914.pdf |title= Decision of the Reserve Bank Organization Committee Determining the Federal Reserve Districts and the Location of Federal Reserve Banks under the Federal Reserve Act approved December 23, 1913 |author= Reserve Bank Organization Committee |date= April 14, 1914 |publisher= U.S. Government Printing Office}}</ref> To quell [[Elihu Root|Elihu Root's]] objections to possible inflation, the passed bill included provisions that the bank must hold at least 40% of its outstanding loans in gold. (In later years, to stimulate short-term economic activity, Congress would amend the act to allow more discretion in the amount of gold that must be redeemed by the Bank.)<ref name="ecresearch"/> Critics of the time (later joined by economist [[Milton Friedman]]) suggested that Glass's legislation was almost entirely based on the Aldrich Plan that had been derided as giving too much power to elite bankers. Glass denied copying Aldrich's plan. In 1922, he told Congress, "no greater misconception was ever projected in this Senate Chamber."<ref name="glassbio"/>

==Operations, 1915-1951==
{{expand section|date=July 2017}}
Wilson named Warburg and other prominent experts to direct the new system, which began operations in 1915 and played a major role in financing the Allied and American war efforts.<ref>Arthur Link, ''Wilson: The New Freedom''; pp. 199–240 (1956).</ref> Warburg at first refused the appointment, citing America's opposition to a "Wall Street man", but when [[World War I]] broke out he accepted. He was the only appointee asked to appear before the Senate, whose members questioned him about his interests in the central bank and his ties to [[Kuhn, Loeb, & Co.]]'s "money trusts".<ref name="mnwarburg"/>

== Accord of 1951 between the Federal Reserve and the Treasury Department ==
{{main|1951 Accord}}

== Post Bretton-Woods era ==

In July 1979, [[Paul Volcker]] was nominated, by [[Jimmy Carter|President Carter]], as Chairman of the Federal Reserve Board amid roaring inflation. He tightened the money supply, and by 1986 inflation had fallen sharply.<ref name="nationalreview">{{cite web |url= http://www.nationalreview.com/nrof_bartlett/bartlett200406140846.asp|title= Warriors Against Inflation |author= Bartlett, Bruce|date=2004-06-14 |publisher= ''National Review''}}</ref> In October 1979 the Federal Reserve announced a policy of "targeting" [[Money supply|money aggregates]] and bank reserves in its struggle with double-digit inflation.<ref>''Source: A Monetary Chronology of the United States, [[American Institute for Economic Research]], July 2006''</ref>

In January 1987, with retail inflation at only 1%, the Federal Reserve announced it was no longer going to use money-supply aggregates, such as M2, as guidelines for controlling inflation, even though this method had been in use from 1979, apparently with great success. Before 1980, interest rates were used as guidelines; inflation was severe. The Fed complained that the aggregates were confusing. Volcker was chairman until August 1987, whereupon [[Alan Greenspan]] assumed the mantle, seven months after monetary aggregate policy had changed.<ref>A Monetary Chronology of the United States, [[American Institute for Economic Research]], July 2006</ref>

== 2001 recession to present ==

From early 2001 to mid-2003 the Federal Reserve lowered its interest rates 13 times, from 6.25 to 1.00%, to fight [[recession]]. In November 2002, rates were cut to 1.75, and many interest rates went below the [[inflation]] rate. On June 25, 2003, the [[Fed funds rate|federal funds rate]] was lowered to 1.00%, its lowest nominal rate since July, 1958, when the overnight rate averaged 0.68%. Starting at the end of June 2004, the Federal Reserve System raised the target interest rate and then continued to do so 17 straight times.

In February 2006, [[Ben Bernanke]] was appointed by President [[George W. Bush]] as the chairman of the Federal Reserve.<ref>{{cite web |url=http://www.federalreserve.gov/aboutthefed/bios/board/bernanke.htm |title=Bernanke Biography |publisher=US Federal Reserve Bank |accessdate=2010-01-30 |archiveurl=https://web.archive.org/web/20100124045445/http://www.federalreserve.gov/aboutthefed/bios/board/bernanke.htm |archivedate=January 24, 2010 |deadurl=yes |df= }}</ref>

In March 2006, the Federal Reserve ceased to make public M3, because the costs of collecting this data outweighed the benefits.<ref>{{cite web|url=http://www.federalreserve.gov/Releases/h6/discm3.htm|accessdate=2010-12-28|title=FRB: H.6 Release--Discontinuance of M3}}</ref> M3 includes all of M2 (which includes M1) plus large-denomination ($100,000 +) [[time deposit]]s, balances in institutional money funds, repurchase liabilities issued by depository institutions, and Eurodollars held by U.S. residents at foreign branches of U.S. banks as well as at all banks in the United Kingdom and Canada.

===2008 subprime mortgage crisis===
{{main|Federal Reserve responses to the subprime crisis }}

Due to a credit crunch caused by [[Subprime mortgage crisis|the sub-prime mortgage crisis]] in September 2007, the Federal Reserve began cutting the federal funds rate. The Fed cut rates by 0.25% after its December 11, 2007 meeting and disappointed many individual investors who expected a higher rate cut: the Dow Jones Industrial Average dropped by nearly 300 points at its close that day. The Fed slashed the rate 0.75% in an emergency action on January 22, 2008 to assist in reversing a significant market slide influenced by weakening international markets. The Dow Jones Industrial Average initially fell nearly 4% (465 points) at the start of trading and then rebounded to a more tolerable 1.06% (128 point) loss. On January 30, 2008, eight days after the 75 points decrease, the Fed lowered its rate again, this time by 50 points.<ref>{{cite news|author=Michael M. Grynbaum and John Holusha|title=Fed Cuts Rate 0.75% and Stocks Swing|url= https://www.nytimes.com/2008/01/22/business/23cnd-fed.html |work=[[The New York Times]]|publisher=[[The New York Times Company]]|date=2008-01-22 |accessdate=2008-01-22}}</ref>

On August 25, 2009, President [[Barack Obama]] announced he would nominate Bernanke to a second term as chairman of the Federal Reserve.<ref>{{cite news|last=Hilsenrath |first=Jon |url=https://www.wsj.com/articles/SB125120274221856591 |first2=Elizabeth|last2=Williamson|first3=Jonathan|last3=Weisman|title=Calm in Crisis Won Fed Job|work=The Wall Street Journal|date=August 26, 2009|accessdate=January 30, 2010| archiveurl= https://web.archive.org/web/20100204034453/http://online.wsj.com/article/SB125120274221856591.html| archivedate= February 4, 2010 | deadurl= no}}</ref>

In October 2013, [[Janet Yellen]] was nominated to succeed [[Ben Bernanke]] as the chairperson of the [[Federal Reserve]].

In December 2015, the Fed raised its benchmark interest rates by a quarter of a percentage point to between 0.25 and 0.50 percent, after 9 years of an unchanged and stable very low interest rate.<ref>{{Cite news|url=https://www.reuters.com/article/usa-fed-idUSKBN0TZ1Y220151217|title=Fed raises interest rates, citing ongoing U.S. recovery|date=2015-12-17|newspaper=Reuters|access-date=2016-05-05}}</ref>

== Key laws affecting the Federal Reserve ==
Key laws affecting the Federal Reserve have been:<ref name="paf">ebook: The Federal Reserve – Purposes and Functions:http://www.federalreserve.gov/pf/pf.htm
:for info on government regulations, see pages 13 and 14. Addressing bank panics on page 83. Implementation of monetary policy on page 12 and 36. Board and reserve banks responsibility on page 12. Key laws affecting the federal reserve on page 11. Monetary policy uncertainties on pages 18–19.</ref>
* Banking Act of 1935
* [[Employment Act of 1946]]
* [[1951 Accord|Federal Reserve-Treasury Department Accord of 1951]]
* [[Bank Holding Company Act of 1956]] and the amendments of 1970
* [[Federal Reserve Reform Act of 1977]]
* International Banking Act of 1978
* [[Full Employment and Balanced Growth Act]] (1978)
* [[Depository Institutions Deregulation and Monetary Control Act]] (1980)
* [[Financial Institutions Reform, Recovery and Enforcement Act of 1989]]
* [[Federal Deposit Insurance Corporation Improvement Act of 1991]]
* [[Gramm-Leach-Bliley Act]] (1999)

== References ==
{{Reflist|2}}

==External links==
*[https://fraser.stlouisfed.org/archival/1344 Records of the Federal Reserve System, Record Group 82], materials held at the National Archives and Records Center, digitized and made available on [[FRASER]]
*[https://fraser.stlouisfed.org/archival/1342 Committee on the History of the Federal Reserve System materials], collected for the 50th anniversary of the Federal Reserve System, are available on [[FRASER]]

{{Federal Reserve System}}

{{DEFAULTSORT:History Of The Federal Reserve System}}
[[Category:Federal Reserve System]]
[[Category:Financial history of the United States]]
[[Category:History of finance]]
[[Category:History of the Federal Reserve System| ]]

User talk:LiberatorG

2018-07-01T13:56:27Z

Crasshopper: /* Welcome */

== Welcome ==


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Hello LiberatorG, and [[Wikipedia:Welcoming committee/Welcome to Wikipedia|Welcome to Wikipedia!]]</div>
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'''''[[Wikipedia:Welcoming committee/Welcome to Wikipedia|Welcome to Wikipedia!]]''''' I hope you enjoy the encyclopedia and want to [[Wikipedia:Wikipedians|stay]]. As a first step, you may wish to read the [[Wikipedia:Introduction|Introduction]].

If you have any questions, feel free to ask me at [[User talk:Aboutmovies|my talk page]] – I'm happy to help. Or, you can ask your question at the [[Wikipedia:New contributors' help page|New contributors' help page]].

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''LiberatorG, '''good luck, and have fun.''''' – [[User:Aboutmovies|Aboutmovies]] ([[User talk:Aboutmovies|talk]]) 06:22, 6 July 2014 (UTC)</div>hi

= URI schemes ==

Hi, thanks for explaining the reasoning behind your reverts on [[URI]].

My reasoning for adding strange URI’s is to show the difference between webpage addresses and URL’s.

In practice, most people glance at Wikipedia and move on; they don’t follow all links. That’s why I want strange URI’s in the article.

Happy to reword or pick another place in the document to put these. I think they make the encyclopedia friendlier, funnier, and clearer.

There should be a way to do this that does not harm the clarity of your text or the rest of the exposition.
[[User:Crasshopper|Crasshopper]] ([[User talk:Crasshopper|talk]]) 13:56, 1 July 2018 (UTC)

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== [[WP:ACE2015|ArbCom elections are now open!]] ==

Hi, 
You appear to be eligible to vote in the current [[WP:ACE2015|Arbitration Committee election]]. The [[WP:ARBCOM|Arbitration Committee]] is the panel of editors responsible for conducting the Wikipedia [[WP:RFAR|arbitration process]]. It has the authority to enact binding solutions for disputes between editors, primarily related to serious behavioural issues that the community has been unable to resolve. This includes the ability to impose [[WP:BAN|site bans]], [[WP:TBAN|topic bans]], editing restrictions, and other measures needed to maintain our editing environment. The [[WP:ARBPOL|arbitration policy]] describes the Committee's roles and responsibilities in greater detail. If you wish to participate, you are welcome to [[WP:ACE2015/C|review the candidates' statements]] and submit your choices on [[Special:SecurePoll/vote/398|the voting page]]. For the Election committee, [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|talk]]) 14:09, 24 November 2015 (UTC)


== Jeffrey Tucker quotes ==

You are welcome to add the quotes you like — Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:HHHIsMyHomeBoy|HHHIsMyHomeBoy]] ([[User talk:HHHIsMyHomeBoy|talk]] • [[Special:Contributions/HHHIsMyHomeBoy|contribs]]) 18:50, 23 May 2016 (UTC) 

== [[WP:ACE2016|ArbCom Elections 2016]]: Voting now open! ==

{{Ivmbox|Hello, LiberatorG. Voting in the '''[[WP:ACE2016|2016 Arbitration Committee elections]]''' is open from Monday, 00:00, 21 November through Sunday, 23:59, 4 December to all unblocked users who have registered an account before Wednesday, 00:00, 28 October 2016 and have made at least 150 mainspace edits before Sunday, 00:00, 1 November 2016.

The [[WP:ARBCOM|Arbitration Committee]] is the panel of editors responsible for conducting the [[Wikipedia:Arbitration|Wikipedia arbitration process]]. It has the authority to impose binding solutions to disputes between editors, primarily for serious conduct disputes the community has been unable to resolve. This includes the authority to impose [[WP:BAN|site bans]], [[WP:TBAN|topic bans]], editing restrictions, and other measures needed to maintain our editing environment. The [[Wikipedia:Arbitration/Policy|arbitration policy]] describes the Committee's roles and responsibilities in greater detail.

If you wish to participate in the 2016 election, please review [[Wikipedia:Arbitration Committee Elections December 2016/Candidates|the candidates' statements]] and submit your choices on '''[[Special:SecurePoll/vote/399|the voting page]]'''. [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|talk]]) 22:08, 21 November 2016 (UTC)
|Scale of justice 2.svg|imagesize=40px}}


== Request for opinion ==

Sir, can I request your opinion on the discussion "Timeline of computer security hacker history" which is located on my talk page?
Please leave your opinion/comments on my talk page instead of here if you have one.
[[User:Mdikici4001|Mdikici4001]] ([[User talk:Mdikici4001|talk]]) 02:49, 18 January 2017 (UTC)

== ArbCom 2017 election voter message ==

{{Ivmbox|Hello, LiberatorG. Voting in the '''[[WP:ACE2017|2017 Arbitration Committee elections]]''' is now open until 23.59 on Sunday, 10 December. All users who registered an account before Saturday, 28 October 2017, made at least 150 mainspace edits before Wednesday, 1 November 2017 and are not currently blocked are eligible to vote. Users with alternate accounts may only vote once.

The [[WP:ARBCOM|Arbitration Committee]] is the panel of editors responsible for conducting the [[Wikipedia:Arbitration|Wikipedia arbitration process]]. It has the authority to impose binding solutions to disputes between editors, primarily for serious conduct disputes the community has been unable to resolve. This includes the authority to impose [[WP:BAN|site bans]], [[WP:TBAN|topic bans]], editing restrictions, and other measures needed to maintain our editing environment. The [[Wikipedia:Arbitration/Policy|arbitration policy]] describes the Committee's roles and responsibilities in greater detail.

If you wish to participate in the 2017 election, please review [[Wikipedia:Arbitration Committee Elections December 2017/Candidates|the candidates]] and submit your choices on the '''[[Special:SecurePoll/vote/400|voting page]]'''. [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|talk]]) 18:42, 3 December 2017 (UTC)
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== Code page 866 ==

Thank you for that. I was, of course, purely thinking in terms of box drawing in existing text files in OEM-437 (hence "OEM box drawing characters"), and that 866 is the only encoding supported by modern browsers that will display them correctly. I had not noticed that I had made an incorrect statement (because, of course, the ''characters'' themselves are included in Unicode and thus UTF-8, just not in the same representations).

I'm somewhat considering adding it back with corrections, but suspect viewing old pre-Unicode text files with box drawing in a web browser is probably too niche a usage case to be worth mentioning. -- [[User:HarJIT|HarJIT]] ([[User talk:HarJIT|talk]]) 21:10, 8 March 2018 (UTC)

:{{reply to|HarJIT}} If you want to share old pre-Unicode text files containing non-ASCII characters, it is generally better to convert them to a Unicode encoding first. That will work regardless of its code page, and you wouldn't have to verify that the only non-ASCII characters in use are the box drawing characters. You could then use the content with a wider variety of modern tools and editors, copy and paste into other Unicode files, etc. -[[User:LiberatorG|LiberatorG]] ([[User talk:LiberatorG#top|talk]]) 16:26, 9 March 2018 (UTC)

Uniform Resource Identifier

2018-06-30T04:05:00Z

Crasshopper: /* Allowed Schemes */

{{Redirect|URI}}
{{good article}}

A '''Uniform Resource Identifier''' ('''URI''') is a [[character string (computer science)|string]] of [[character (computing)|character]]s designed for unambiguous [[identifier|identification]] of [[resource (computer science)|resources]] and extensibility via the URI scheme.

Such identification enables interaction with representations of the resource over a network, typically the [[World Wide Web]], using specific [[Communications protocol|protocols]]. Schemes specifying a concrete [[syntax]] and associated protocols define each URI. The most common form of URI is the Uniform Resource Locator ([[URL]]), frequently referred to informally as a ''web address.'' More rarely seen in usage is the [[Uniform Resource Name|Uniform Resource Name (URN)]], which was designed to complement URLs by providing a mechanism for the identification of resources in particular [[namespace]]s.

==URL and URN==
A [[Uniform Resource Name]] (URN) is a URI that identifies a resource by name in a particular namespace. A URN may be used to talk about a resource without implying its location or how to access it. For example, in the [[International Standard Book Number|International Standard Book Number (ISBN)]] system, ''<nowiki>ISBN</nowiki> 0-486-27557-4'' identifies a specific edition of Shakespeare's play ''[[Romeo and Juliet]]''. The URN for that edition would be ''<nowiki>urn:isbn:0-486-27557-4</nowiki>''. However, it gives no information as to where to find a copy of that book.

A [[Uniform Resource Locator]] (URL) is a URI that specifies the means of acting upon or obtaining the representation of a resource, i.e. specifying both its primary access mechanism and network location. For example, the URL <code><nowiki>http://example.org/wiki/Main_Page</nowiki></code> refers to a resource identified as <code><nowiki>/wiki/Main_Page</nowiki></code> whose representation, in the form of [[HTML]] and related code, is obtainable via the [[Hypertext Transfer Protocol]] (''http:'') from a network host whose [[domain name]] is <code><nowiki>example.org</nowiki></code>.

A URN may be compared to a person's name, while a URL may be compared to their street address. In other words, a URN identifies an item and a URL provides a method for finding it.

Technical publications, especially standards produced by the [[Internet Engineering Task Force|IETF]] and by the [[World Wide Web Consortium|W3C]], normally reflect a view outlined in a [[W3C Recommendation]] of 2001, which acknowledges the precedence of the term URI rather than endorsing any formal subdivision into URL and URN. {{cquote|URL is a useful but informal concept: a URL is a type of URI that identifies a resource via a representation of its primary access mechanism (e.g., its network "location"), rather than by some other attributes it may have.{{sfnp|Joint W3C/IETF URI Planning Interest Group|2001}}}}

As such, a URL is simply a URI that happens to point to a resource over a network.{{efn|A report published in 2002 by a joint W3C/IETF working group aimed to normalize the divergent views held within the IETF and W3C over the relationship between the various 'UR*' terms and standards. While not published as a full standard by either organization, it has become the basis for the above common understanding and has informed many standards since then.}}{{sfnp|Joint W3C/IETF URI Planning Interest Group|2002}} However, in non-technical contexts and in software for the World Wide Web, the term "URL" remains widely used. Additionally, the term "web address" (which has no formal definition) often occurs in non-technical publications as a synonym for a URI that uses the ''http'' or ''https'' schemes. Such assumptions can lead to confusion, for example, in the case of XML namespaces that have a [[#Relation to XML namespaces|visual similarity to resolvable URIs]].

Specifications produced by the [[WHATWG]] prefer ''URL'' over ''URI'', and so newer HTML5 APIs use ''URL'' over ''URI''.<ref>{{cite web |title=URL Standard: 6.3. URL APIs elsewhere |url=https://url.spec.whatwg.org/#url-apis-elsewhere}}</ref> {{cquote|Standardize on the term URL. URI and IRI are just confusing. In practice a single algorithm is used for both so keeping them distinct is not helping anyone. URL also easily wins the search result popularity contest.<ref>{{cite web |title=URL Standard: Goals|url=https://url.spec.whatwg.org/#goals}}</ref>}}

While most URI schemes were originally designed to be used with a particular [[protocol (computing)|protocol]], and often have the same name, they are semantically different from protocols. For example, the scheme ''http'' is generally used for interacting with [[web resource]]s using HTTP, but the scheme ''[[file URI scheme|file]]'' has no protocol.

==Generic syntax==
===Definition===
Each URI begins with a scheme name that refers to a specification for assigning identifiers within that scheme. As such, the URI syntax is a federated and extensible naming system wherein each scheme's specification may further restrict the syntax and semantics of identifiers using that scheme. The URI generic syntax is a superset of the syntax of all URI schemes. It was first defined in [[Request for Comments|Request for Comments (RFC)]] 2396, published in August 1998,{{sfnp|RFC 2396|1998}} and finalized in <nowiki>RFC</nowiki> 3986, published in January 2005.{{sfnp|RFC 3986|2005}}
<section begin=syntax />
The ''URI generic syntax'' consists of a hierarchical sequence of five ''components'':{{sfnp|RFC 3986|2005|loc=§3}}

<pre>
URI = scheme:[//authority]path[?query][#fragment]
</pre>

where the authority component divides into three ''subcomponents'':

<pre>
authority = [userinfo@]host[:port]
</pre>

It comprises:
* A non-empty '''{{visible anchor|scheme}}''' component followed by a colon (<code>:</code>), consisting of a sequence of characters beginning with a letter and followed by any combination of letters, digits, plus (<code>+</code>), period (<code>.</code>), or hyphen (<code>-</code>). Although schemes are case-insensitive, the canonical form is lowercase and documents that specify schemes must do so with lowercase letters. Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>[[skype]]</code>, <code>steam</code>, <code>[[svn]]</code>, <code>cvs</code>, <code>[[telnet]]</code>, <code>[[sms]]</code>, <code>[[smtp]]</code>, <code>[[ldap]]</code>, <code>dav</code>, <code>[[jabber]]</code>, <code>[[xmpp]]</code>, <code>[[udp]]</code>, <code>view-source</code>, <code>[[websockets|ws]]</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>. URI schemes should be registered with the [[Internet Assigned Numbers Authority|Internet Assigned Numbers Authority (IANA)]], although non-registered schemes are used in practice.{{efn|The procedures for registering new URI schemes were originally defined in 1999 by <nowiki>RFC 2717</nowiki>, and are now defined by <nowiki>RFC 7595</nowiki>, published in June 2015.{{sfnp|IETF|2015}}}}
* An optional non-empty '''authority''' component preceded by two slashes (<code>//</code>), comprising:
** An optional '''userinfo''' subcomponent that may consist of a [[User (computing)|user name]] and an optional [[password]] preceded by a colon (<code>:</code>), followed by an at symbol (<code>@</code>). Use of the format <code>username:password</code> in the userinfo subcomponent is deprecated for security reasons. Applications should not render as clear text any data after the first colon (<code>:</code>) found within a userinfo subcomponent unless the data after the colon is the empty string (indicating no password).
** A non-empty '''host''' subcomponent, consisting of either a registered name (including but not limited to a [[hostname]]), or an [[IP address]]. [[IPv4]] addresses must be in [[dot-decimal notation]], and [[IPv6]] addresses must be enclosed in brackets (<code>[]</code>).{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{efn|For URIs relating to resources on the World Wide Web, some web browsers allow {{code|.0}} portions of dot-decimal notation to be dropped or raw integer IP addresses to be used.{{sfnp|Lawrence|2014}}}}
** An optional '''port''' subcomponent preceded by a colon (<code>:</code>).
* A '''path''' component, consisting of a sequence of path segments separated by a slash (<code>/</code>). A path is always defined for a URI, though the defined path may be empty (zero length). A segment may also be empty, resulting in two consecutive slashes (<code>//</code>) in the path component. A path component may resemble or map exactly to a [[Path (computing)|file system path]], but does not always imply a relation to one. If an authority component is present, then the path component must either be empty or begin with a slash (<code>/</code>). If an authority component is absent, then the path cannot begin with an empty segment, that is with two slashes (<code>//</code>), as the following characters would be interpreted as an authority component.{{sfnp|RFC 2396|1998|loc=§3.3}} The final segment of the path may be referred to as a '[[Clean URL#Slug|slug]]'.

{| class="wikitable" style="float: right; font-size: 0.9em; margin-left: 1em"
|-
! Query delimiter
! Example
|-
| Ampersand (<code>&</code>)
| <code>key1=value1&key2=value2</code>
|-
| Semicolon (<code>;</code>){{efn|Historic <nowiki>RFC 1866</nowiki> (obsoleted by <nowiki>RFC 2854</nowiki>) encourages CGI authors to support ';' in addition to '&'.{{sfnp|RFC 1866|1995|loc=§8.2.1}}}}{{Incomplete short citation|date=August 2016}}
| <code>key1=value1;key2=value2</code>
|}
* An optional '''query''' component preceded by a question mark (<code>?</code>), containing a [[query string]] of non-hierarchical data. Its syntax is not well defined, but by convention is most often a sequence of [[attribute–value pair]]s separated by a [[delimiter]].
* An optional '''fragment''' component preceded by an [[Number sign|hash]] (<code>#</code>). The fragment contains a [[fragment identifier]] providing direction to a secondary resource, such as a section heading in an article identified by the remainder of the URI. When the primary resource is an [[HTML]] document, the fragment is often an [[HTML#Attributes|<code>id</code> attribute]] of a specific element, and web browsers will scroll this element into view.<section end=syntax />

Strings of data [[Octet (computing)|octets]] within a URI are represented as characters. Permitted characters within a URI are the [[ASCII]] characters for the lowercase and uppercase letters of the modern [[English alphabet]], the [[Arabic numerals]], [[hyphen]], [[Full stop|period]], [[underscore]], and [[tilde]].{{sfnp|RFC 3986|2005|loc=§2}} Octets represented by any other character must be [[percent-encoding|percent-encoded]].

Of the ASCII character set, the characters <code>: / ? # [ ] @</code> are reserved for use as delimiters of the generic URI components and must be percent-encoded — for example, <code>%3F</code> for a question mark.{{sfnp|RFC 3986|2005|loc=§2.2}} The characters <code>! $ & ' ( ) * + , ; =</code> are permitted by generic URI syntax to be used unencoded in the user information, host, and path as delimiters.{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{sfnp|RFC 3986|2005|loc=§3.3}} Additionally, <code>:</code> and <code>@</code> may appear unencoded within the path, query, and fragment; and <code>?</code> and <code>/</code> may appear unencoded as data within the query or fragment.{{sfnp|RFC 3986|2005|loc=§3.3}}{{sfnp|RFC 3986|2005|loc=§3.4}}

===Examples===
The following figure displays example URIs and their component parts.

<pre>
userinfo host port
┌───┴──┐ ┌──────┴──────┐ ┌┴┐
https://john.doe@www.example.com:123/forum/questions/?tag=networking&order=newest#top
└─┬─┘ └─────────────┬────────────┘└───────┬───────┘ └────────────┬────────────┘ └┬┘
scheme authority path query fragment

ldap://[2001:db8::7]/c=GB?objectClass?one
└─┬┘ └─────┬─────┘└─┬─┘ └──────┬──────┘
scheme authority path query

mailto:John.Doe@example.com
└──┬─┘ └─────────┬────────┘
scheme path

news:comp.infosystems.www.servers.unix
└─┬┘ └───────────────┬───────────────┘
scheme path

tel:+1-816-555-1212
└┬┘ └──────┬──────┘
scheme path

telnet://192.0.2.16:80/
└──┬─┘ └─────┬─────┘│
scheme authority path

urn:oasis:names:specification:docbook:dtd:xml:4.1.2
└┬┘ └──────────────────────┬──────────────────────┘
scheme path
</pre>

===URI Schemes===

Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>[[skype]]</code>, <code>steam</code>, <code>[[svn]]</code>, <code>cvs</code>, <code>[[telnet]]</code>, <code>[[sms]]</code>, <code>[[smtp]]</code>, <code>[[ldap]]</code>, <code>dav</code>, <code>[[jabber]]</code>, <code>[[xmpp]]</code>, <code>[[udp]]</code>, <code>view-source</code>, <code>[[websockets|ws]]</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>.

Unpopular schemes include <code>xri</code>, <code>tag</code>, <code>lastfm</code>, <code>[[mongodb]]</code>, <code>msword</code>, <code>moz</code>, <code>palm</code>, <code>paparazzi</code>, <code>com-eventbrite-attendee</code>, <code>dina-playsingle</code>, <code>[[adium]]xtra</code>, <code>urn</code>, <code>onenote</code>, <code>rediss</code> (''sic''), <code>ipp</code>, <code>things</code>, <code>stuns</code>, <code>prospero</code>, <code>iotdisco</code>, <code>tip</code>, <code>tool</code>, <code>tv</code>, <code>unreal</code>, <code>submit</code>, <code>market</code>, <code>ham</code>, <code>fish</code>, <code>finger</code>, <code>iris</code>, <code>[[IP over avian carriers]]</code>, and <code>[[Gopher (protocol)|gopher]]</code>.<ref>https://www.iana.org/assignments/uri-schemes/uri-schemes.xhtml</ref>

==URI references==
===Definition===
A ''URI reference'' is either a URI, or a ''relative reference'' when it does not begin with a scheme component followed by a colon (<code>:</code>).{{sfnp|RFC 3986|2005|loc=§4.1}} A path segment that contains a colon character (e.g., <code>foo:bar</code>) cannot be used as the first path segment of a relative reference if its path component does not begin with a slash (<code>/</code>), as it would be mistaken for a scheme component. Such a path segment must be preceded by a dot path segment (e.g., <code>./foo:bar</code>).{{sfnp|RFC 3986|2005|loc=§4.2}}

Web document [[markup language]]s frequently use URI references to point to other resources, such as external documents or specific portions of the same logical document:{{sfnp|RFC 3986|2005|loc=§4.4}}
* in [[HTML]], the value of the <code>src</code> attribute of the <code>img</code> element provides a URI reference, as does the value of the <code>href</code> attribute of the <code>a</code> or <code>link</code> element;
* in [[XML]], the [[system identifier]] appearing after the <code>SYSTEM</code> keyword in a [[Document Type Definition|DTD]] is a fragmentless URI reference;
* in [[XSLT]], the value of the <code>href</code> attribute of the <code>xsl:import</code> element/instruction is a URI reference; likewise the first argument to the <code>document()</code> function.

===Examples===
<pre>
https://example.com/path/resource.txt#fragment
//example.com/path/resource.txt
/path/resource.txt
path/resource.txt
/path/resource.txt
../resource.txt
./resource.txt
resource.txt
#fragment
</pre>

===Suffix references===
As URI usage has become commonplace, traditional media (television, radio, newspapers, billboards, etc.) have increasingly used a suffix of the URI as a reference, consisting of only the authority and path portions of the URI, such as

<pre>
www.w3.org/Addressing/
</pre>

Such references are primarily intended for human interpretation rather than for machines, with the assumption that context-based heuristics are sufficient to complete the URI (e.g., most registered names beginning with <code>www</code> are likely to have a URI prefix of <code>http://</code>). Although there is no standard set of heuristics for disambiguating a URI suffix, many client implementations allow them to be entered by the user and heuristically resolved. Although this practice of using suffix references is common, it should be avoided whenever possible and should never be used in situations where long-term references are expected, as the heuristics will change over time, particularly when a new URI scheme becomes popular, and are often incorrect when used out of context. Furthermore, they can lead to security issues along the lines of those described in <nowiki>RFC</nowiki> 1535. As a URI suffix has the same syntax as a relative reference with a relative path, a suffix reference cannot be used in contexts where a relative reference is expected. As a result, suffix references are limited to places where there is no defined base URI, such as dialog boxes and off-line advertisements.{{sfnp|RFC 3986|2005|loc=§4.5}}

==URI resolution==
===Definition===
An ''absolute URI'' is a URI with no fragment component.

''Resolving'' a URI reference against a ''base URI'' results in a ''target URI''. This implies that the base URI exists and is an absolute URI. The base URI can be obtained, in order of precedence, from:{{sfnp|RFC 3986|2005|loc=§5.1}}

* the reference URI itself if it is a URI;
* the content of the representation;
* the entity encapsulating the representation;
* the URI used for the actual retrieval of the representation;
* the context of the application.

===Examples===
Within a representation with a well defined base URI of

<pre>
http://a/b/c/d;p?q
</pre>

a relative reference is resolved to its target URI as follows:{{sfnp|RFC 3986|2005|loc=§5.4}}

<pre>
"g:h" -> "g:h"
"g" -> "http://a/b/c/g"
"./g" -> "http://a/b/c/g"
"g/" -> "http://a/b/c/g/"
"/g" -> "http://a/g"
"//g" -> "http://g"
"?y" -> "http://a/b/c/d;p?y"
"g?y" -> "http://a/b/c/g?y"
"#s" -> "http://a/b/c/d;p?q#s"
"g#s" -> "http://a/b/c/g#s"
"g?y#s" -> "http://a/b/c/g?y#s"
";x" -> "http://a/b/c/;x"
"g;x" -> "http://a/b/c/g;x"
"g;x?y#s" -> "http://a/b/c/g;x?y#s"
"" -> "http://a/b/c/d;p?q"
"." -> "http://a/b/c/"
"./" -> "http://a/b/c/"
".." -> "http://a/b/"
"../" -> "http://a/b/"
"../g" -> "http://a/b/g"
"../.." -> "http://a/"
"../../" -> "http://a/"
"../../g" -> "http://a/g"
</pre>

==History==
===Naming, addressing, and identifying resources===
URIs and URLs have a shared history. In 1994, [[Tim Berners-Lee|Tim Berners-Lee's]] proposals for [[hypertext]]{{sfnp|Palmer|2001}} implicitly introduced the idea of a URL as a short string representing a resource that is the target of a [[hyperlink]]. At the time, people referred to it as a "hypertext name"{{sfnp|W3C|1992}} or "document name".

Over the next three and a half years, as the World Wide Web's core technologies of HTML, HTTP, and web browsers developed, a need to distinguish a string that provided an address for a resource from a string that merely named a resource emerged. Although not yet formally defined, the term ''Uniform Resource Locator'' came to represent the former, and the more contentious ''Uniform Resource Name'' came to represent the latter.

During the debate over defining URLs and URNs it became evident that the two concepts embodied by the terms were merely aspects of the fundamental, overarching notion of resource ''identification''. In June 1994, the IETF published Berners-Lee's <nowiki>RFC 1630</nowiki>: the first Request for Comments that acknowledged the existence of URLs and URNs, and, more importantly, defined a formal syntax for ''Universal Resource Identifiers'' — URL-like strings whose precise syntaxes and semantics depended on their schemes. In addition, this RFC attempted to summarize the syntaxes of URL schemes in use at the time. It also acknowledged, but did not standardize, the existence of relative URLs and fragment identifiers.

===Refinement of specifications===
In December 1994, <nowiki>RFC 1738</nowiki> formally defined relative and absolute URLs, refined the general URL syntax, defined how to resolve relative URLs to absolute form, and better enumerated the URL schemes then in use. The agreed definition and syntax of URNs had to wait until the publication of <nowiki>RFC 2141</nowiki> in May 1997.

The publication of <nowiki>RFC 2396</nowiki> in August 1998 saw the URI syntax become a separate specification{{sfnp|RFC 2396|1998}} and most of the parts of RFCs 1630 and 1738 relating to URIs and URLs in general were revised and expanded by the [[IETF]]. The new RFC changed the meaning of "U" in "URI" to "Uniform" from "Universal".

In December 1999, <nowiki>RFC 2732</nowiki> provided a minor update to <nowiki>RFC 2396</nowiki>, allowing URIs to accommodate [[IPv6]] addresses. A number of shortcomings discovered in the two specifications led to a community effort, coordinated by <nowiki>RFC 2396</nowiki> co-author [[Roy Fielding]], that culminated in the publication of <nowiki>RFC 3986</nowiki> in January 2005. While obsoleting the prior standard, it did not render the details of existing URL schemes obsolete; <nowiki>RFC 1738</nowiki> continues to govern such schemes except where otherwise superseded. <nowiki>RFC 2616</nowiki> for example, refines the <code>http</code> scheme. Simultaneously, the IETF published the content of <nowiki>RFC 3986</nowiki> as the full standard STD 66, reflecting the establishment of the URI generic syntax as an official Internet protocol.

In 2001, the W3C's Technical Architecture Group (TAG) published a guide to [[best practices]] and canonical URIs for publishing multiple versions of a given resource.{{sfnp|W3C|2001}} For example, content might differ by language or by size to adjust for capacity or settings of the device used to access that content.

In August 2002, <nowiki>RFC 3305</nowiki> pointed out that the term "URL" had, despite widespread public use, faded into near obsolescence, and serves only as a reminder that some URIs act as addresses by having schemes implying network accessibility, regardless of any such actual use. As URI-based standards such as [[Resource Description Framework]] make evident, resource identification need not suggest the retrieval of resource representations over the Internet, nor need they imply network-based resources at all.

The [[Semantic Web]] uses the HTTP URI scheme to identify both documents and concepts in the real world, a distinction which has caused confusion as to how to distinguish the two. The TAG published an e-mail in 2005 on how to solve the problem, which became known as the ''httpRange-14 resolution''.{{sfnp|Fielding|2005}} The W3C subsequently published an Interest Group Note titled ''Cool URIs for the Semantic Web'',{{sfnp|W3C|2008}} which explained the use of [[content negotiation]] and the [[HTTP 303]] response code for redirections in more detail.

==Relation to XML namespaces==
In [[XML]], a [[XML namespace|namespace]] is an abstract domain to which a collection of element and attribute names can be assigned. The namespace name is a character string which must adhere to the generic URI syntax.{{sfnp|Morrison|2006}} However, the name is generally not considered to be a URI,{{sfnp|Harold|2004}} because the URI specification bases the decision not only on lexical components, but also on their intended use. A namespace name does not necessarily imply any of the semantics of URI schemes; for example, a namespace name beginning with ''http:'' may have no connotation to the use of the [[HTTP]].

Originally, the namespace name could match the syntax of any non-empty URI reference, but the use of relative URI references was deprecated by the W3C.{{sfnp|W3C|2009}} A separate W3C specification for namespaces in XML 1.1 permits [[Internationalized resource identifier|internationalized resource identifier (IRI)]] references to serve as the basis for namespace names in addition to URI references.{{sfnp|W3C|2006}}

==See also==
* [[CURIE]] – defines a generic, abbreviated syntax for expressing URIs
* [[Dereferenceable Uniform Resource Identifier]] – a resource retrieval mechanism that uses any of the internet protocols (e.g. HTTP) to obtain a copy or representation of the resource it identifies
* [[Extensible Resource Identifier]] – a scheme and resolution protocol for abstract identifiers compatible with URIs
* [[Internationalized Resource Identifier]] (IRI) – a generalization of URIs allowing the use of Unicode
* [[Persistent uniform resource locator]] (PURL) – a URI that is used to redirect to the location of the requested web resource
* [[Uniform Naming Convention]] – a common syntax used by Microsoft to describe the location of a network resource, such as a shared file, directory, or printer
* [[Resource Directory Description Language]] – a descriptive language to provide machine- and human-readable information about a particular namespace and about the XML documents that use it
* [[Universally unique identifier|UUID]]

== Notes ==
{{Notelist}}

== References ==
=== Citations ===
{{Reflist|25em}}

=== Cited works ===
{{refbegin|32em}}
* {{cite web |first=Roy T.|last=Fielding|authorlink=Roy Fielding |title = [httpRange-14] Resolved |url = http://lists.w3.org/Archives/Public/www-tag/2005Jun/0039.html|date=18 June 2005 |accessdate=24 July 2009|ref=harv}}
* {{cite book |first=Elliotte Rusty|last=Harold |authorlink=Elliotte Rusty Harold |year=2004 |title=XML 1.1 Bible |edition=Third |publisher=[[Wiley Publishing]] |page=291 |isbn=0-7645-4986-3 |ref=harv }}
* {{cite web |url = http://www.w3.org/TR/uri-clarification/ |author=Joint W3C/IETF URI Planning Interest Group|title=URIs, URLs, and URNs: Clarifications and Recommendations 1.0|date=21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2001}}}}
* {{cite web |url = https://tools.ietf.org/html/rfc3305|title=Report from the Joint W3C/IETF URI Planning Interest Group: Uniform Resource Identifiers (URIs), URLs, and Uniform Resource Names (URNs): Clarifications and Recommendations|editor1-first=M.|editor1-last=Mealling |editor2-first=R.|editor2-last=Denenberg|publisher=[[World Wide Web Consortium]]|date=August 2002 |accessdate=13 September 2015 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2002}}}}
* {{cite web |url=https://tools.ietf.org/html/rfc7595|title=Guidelines and Registration Procedures for URI Schemes|editor-first=D.|editor-last=Thaler|author1-first=T.|author1-last=Hansen|author2-first=T.|author2-last=Hardie|publisher=[[Internet Engineering Task Force]]|date=June 2015|issn=2070-1721|ref={{SfnRef|IETF|2015}}}}
* {{cite book |last=Morrison|first=Michael|year=2006|title=Sams Teach Yourself XML|publisher=[[Sams Publishing]]|chapter=Hour 5: ''Putting Namespaces to Use''|page=91 |ref=harv }}
* {{cite web |first=Sean B.|last=Palmer |title=The Early History of HTML |url = http://infomesh.net/html/history/early/ |year=2001 |accessdate=2009-04-30 |ref=harv }}
* {{cite web |author = URI Planning Interest Group, W3C/IETF |title = URIs, URLs, and URNs: Clarifications and Recommendations 1.0 |url = http://www.w3.org/TR/uri-clarification/ |date = 21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|URI Planning Interest Group|2009}}}}
* {{cite web |url = http://www.w3.org/History/19921103-hypertext/hypertext/WWW/Addressing/Addressing.html|title=W3 Naming Schemes |publisher=[[World Wide Web Consortium]]|year=1992|accessdate=2009-07-24|ref={{SfnRef|W3C|1992}}}}
* {{cite web |url = http://www.w3.org/2001/tag/doc/alternatives-discovery.html |title=On Linking Alternative Representations To Enable Discovery And Publishing|publisher=[[World Wide Web Consortium]]|year=2006|orig-year=2001|accessdate=2012-04-03|ref={{SfnRef|W3C|2001}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use|title=Namespaces in XML 1.1 (Second Edition)|date=16 August 2006 |at = 2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray|editor2-first=Dave |editor2-last=Hollander|editor3-first=Andrew|editor3-last=Layman|editor4-first=Richard|editor4-last=Tobin|publisher=[[World Wide Web Consortium]] |accessdate=31 August 2015|ref={{SfnRef|W3C|2006}}}}
* {{cite web |url = http://www.w3.org/TR/cooluris/|title=Cool URIs for the Semantic Web|editor1-first=Leo|editor1-last=Sauermann|editor2-first=Richard|editor2-last=Cyganiak|author1-first=Danny|author1-last=Ayers|author2-first=Max|author2-last=Völkel|publisher=[[World Wide Web Consortium]]|date=3 December 2008|accessdate=2012-04-03|ref={{SfnRef|W3C|2008}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use |title=Namespaces in XML 1.0 (Third Edition)|date=8 December 2009 |at=2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray |editor2-first=Dave |editor2-last=Hollander |editor3-first=Andrew|editor3-last=Layman |editor4-first=Richard|editor4-last=Tobin |editor5-first=Henry S. |editor5-last=Thompson |publisher=[[World Wide Web Consortium]]|accessdate=31 August 2015 |ref={{SfnRef|W3C|2009}} }}
* {{cite web |url = https://tools.ietf.org/html/rfc1866#section-8.2.1 |title=Hypertext Markup Language - 2.0|author1-first=Tim|author1-last=Berners-Lee |author1-link=Tim Berners-Lee|author2-first=Dan|author2-last=Connolly|publisher=[[Internet Engineering Task Force]] |date=November 1995 |accessdate=13 September 2015}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc2396 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=2396 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=August 1998 |accessdate=31 August 2015 |ref={{SfnRef|RFC 2396|1998}}}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc3986 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=3986 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=January 2005 |accessdate=31 August 2015 |ref={{SfnRef|RFC 3986|2005}}}}
* {{cite web |last1=Lawrence |first1=Eric |title=Browser Arcana: IP Literals in URLs |url = http://blogs.msdn.com/b/ieinternals/archive/2014/03/06/browser-arcana-ipv4-ipv6-literal-urls-dotted-va-dotless.aspx |website=IEInternals |publisher=[[Microsoft]] |date=6 March 2014 |accessdate=2016-04-25 |ref = harv }}
{{refend}}

==External links==
* [http://www.iana.org/assignments/uri-schemes.html URI Schemes] – [[Internet Assigned Numbers Authority|IANA]]-maintained registry of URI Schemes
* [http://www.w3.org/wiki/UriSchemes URI schemes on the W3C wiki]
* [http://www.w3.org/TR/webarch/#identification Architecture of the World Wide Web, Volume One, §2: Identification] – by W3C
* [http://www.w3.org/TR/uri-clarification/ W3C URI Clarification]

{{Semantic Web|state=collapsed}}
{{URI scheme}}
{{Hypermedia}}

{{Authority control}}

[[Category:Application layer protocols]]
[[Category:Internet protocols]]
[[Category:Internet Standards]]
[[Category:Semantic Web| ]]
[[Category:Uniform Resource Locator]]

Uniform Resource Identifier

2018-06-30T04:04:25Z

Crasshopper: /* Generic syntax */ clear the new text out of the way

{{Redirect|URI}}
{{good article}}

A '''Uniform Resource Identifier''' ('''URI''') is a [[character string (computer science)|string]] of [[character (computing)|character]]s designed for unambiguous [[identifier|identification]] of [[resource (computer science)|resources]] and extensibility via the URI scheme.

Such identification enables interaction with representations of the resource over a network, typically the [[World Wide Web]], using specific [[Communications protocol|protocols]]. Schemes specifying a concrete [[syntax]] and associated protocols define each URI. The most common form of URI is the Uniform Resource Locator ([[URL]]), frequently referred to informally as a ''web address.'' More rarely seen in usage is the [[Uniform Resource Name|Uniform Resource Name (URN)]], which was designed to complement URLs by providing a mechanism for the identification of resources in particular [[namespace]]s.

==URL and URN==
A [[Uniform Resource Name]] (URN) is a URI that identifies a resource by name in a particular namespace. A URN may be used to talk about a resource without implying its location or how to access it. For example, in the [[International Standard Book Number|International Standard Book Number (ISBN)]] system, ''<nowiki>ISBN</nowiki> 0-486-27557-4'' identifies a specific edition of Shakespeare's play ''[[Romeo and Juliet]]''. The URN for that edition would be ''<nowiki>urn:isbn:0-486-27557-4</nowiki>''. However, it gives no information as to where to find a copy of that book.

A [[Uniform Resource Locator]] (URL) is a URI that specifies the means of acting upon or obtaining the representation of a resource, i.e. specifying both its primary access mechanism and network location. For example, the URL <code><nowiki>http://example.org/wiki/Main_Page</nowiki></code> refers to a resource identified as <code><nowiki>/wiki/Main_Page</nowiki></code> whose representation, in the form of [[HTML]] and related code, is obtainable via the [[Hypertext Transfer Protocol]] (''http:'') from a network host whose [[domain name]] is <code><nowiki>example.org</nowiki></code>.

A URN may be compared to a person's name, while a URL may be compared to their street address. In other words, a URN identifies an item and a URL provides a method for finding it.

Technical publications, especially standards produced by the [[Internet Engineering Task Force|IETF]] and by the [[World Wide Web Consortium|W3C]], normally reflect a view outlined in a [[W3C Recommendation]] of 2001, which acknowledges the precedence of the term URI rather than endorsing any formal subdivision into URL and URN. {{cquote|URL is a useful but informal concept: a URL is a type of URI that identifies a resource via a representation of its primary access mechanism (e.g., its network "location"), rather than by some other attributes it may have.{{sfnp|Joint W3C/IETF URI Planning Interest Group|2001}}}}

As such, a URL is simply a URI that happens to point to a resource over a network.{{efn|A report published in 2002 by a joint W3C/IETF working group aimed to normalize the divergent views held within the IETF and W3C over the relationship between the various 'UR*' terms and standards. While not published as a full standard by either organization, it has become the basis for the above common understanding and has informed many standards since then.}}{{sfnp|Joint W3C/IETF URI Planning Interest Group|2002}} However, in non-technical contexts and in software for the World Wide Web, the term "URL" remains widely used. Additionally, the term "web address" (which has no formal definition) often occurs in non-technical publications as a synonym for a URI that uses the ''http'' or ''https'' schemes. Such assumptions can lead to confusion, for example, in the case of XML namespaces that have a [[#Relation to XML namespaces|visual similarity to resolvable URIs]].

Specifications produced by the [[WHATWG]] prefer ''URL'' over ''URI'', and so newer HTML5 APIs use ''URL'' over ''URI''.<ref>{{cite web |title=URL Standard: 6.3. URL APIs elsewhere |url=https://url.spec.whatwg.org/#url-apis-elsewhere}}</ref> {{cquote|Standardize on the term URL. URI and IRI are just confusing. In practice a single algorithm is used for both so keeping them distinct is not helping anyone. URL also easily wins the search result popularity contest.<ref>{{cite web |title=URL Standard: Goals|url=https://url.spec.whatwg.org/#goals}}</ref>}}

While most URI schemes were originally designed to be used with a particular [[protocol (computing)|protocol]], and often have the same name, they are semantically different from protocols. For example, the scheme ''http'' is generally used for interacting with [[web resource]]s using HTTP, but the scheme ''[[file URI scheme|file]]'' has no protocol.

==Generic syntax==
===Definition===
Each URI begins with a scheme name that refers to a specification for assigning identifiers within that scheme. As such, the URI syntax is a federated and extensible naming system wherein each scheme's specification may further restrict the syntax and semantics of identifiers using that scheme. The URI generic syntax is a superset of the syntax of all URI schemes. It was first defined in [[Request for Comments|Request for Comments (RFC)]] 2396, published in August 1998,{{sfnp|RFC 2396|1998}} and finalized in <nowiki>RFC</nowiki> 3986, published in January 2005.{{sfnp|RFC 3986|2005}}
<section begin=syntax />
The ''URI generic syntax'' consists of a hierarchical sequence of five ''components'':{{sfnp|RFC 3986|2005|loc=§3}}

<pre>
URI = scheme:[//authority]path[?query][#fragment]
</pre>

where the authority component divides into three ''subcomponents'':

<pre>
authority = [userinfo@]host[:port]
</pre>

It comprises:
* A non-empty '''{{visible anchor|scheme}}''' component followed by a colon (<code>:</code>), consisting of a sequence of characters beginning with a letter and followed by any combination of letters, digits, plus (<code>+</code>), period (<code>.</code>), or hyphen (<code>-</code>). Although schemes are case-insensitive, the canonical form is lowercase and documents that specify schemes must do so with lowercase letters. Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>[[skype]]</code>, <code>steam</code>, <code>[[svn]]</code>, <code>cvs</code>, <code>[[telnet]]</code>, <code>[[sms]]</code>, <code>[[smtp]]</code>, <code>[[ldap]]</code>, <code>dav</code>, <code>[[jabber]]</code>, <code>[[xmpp]]</code>, <code>[[udp]]</code>, <code>view-source</code>, <code>[[websockets|ws]]</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>. URI schemes should be registered with the [[Internet Assigned Numbers Authority|Internet Assigned Numbers Authority (IANA)]], although non-registered schemes are used in practice.{{efn|The procedures for registering new URI schemes were originally defined in 1999 by <nowiki>RFC 2717</nowiki>, and are now defined by <nowiki>RFC 7595</nowiki>, published in June 2015.{{sfnp|IETF|2015}}}}
* An optional non-empty '''authority''' component preceded by two slashes (<code>//</code>), comprising:
** An optional '''userinfo''' subcomponent that may consist of a [[User (computing)|user name]] and an optional [[password]] preceded by a colon (<code>:</code>), followed by an at symbol (<code>@</code>). Use of the format <code>username:password</code> in the userinfo subcomponent is deprecated for security reasons. Applications should not render as clear text any data after the first colon (<code>:</code>) found within a userinfo subcomponent unless the data after the colon is the empty string (indicating no password).
** A non-empty '''host''' subcomponent, consisting of either a registered name (including but not limited to a [[hostname]]), or an [[IP address]]. [[IPv4]] addresses must be in [[dot-decimal notation]], and [[IPv6]] addresses must be enclosed in brackets (<code>[]</code>).{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{efn|For URIs relating to resources on the World Wide Web, some web browsers allow {{code|.0}} portions of dot-decimal notation to be dropped or raw integer IP addresses to be used.{{sfnp|Lawrence|2014}}}}
** An optional '''port''' subcomponent preceded by a colon (<code>:</code>).
* A '''path''' component, consisting of a sequence of path segments separated by a slash (<code>/</code>). A path is always defined for a URI, though the defined path may be empty (zero length). A segment may also be empty, resulting in two consecutive slashes (<code>//</code>) in the path component. A path component may resemble or map exactly to a [[Path (computing)|file system path]], but does not always imply a relation to one. If an authority component is present, then the path component must either be empty or begin with a slash (<code>/</code>). If an authority component is absent, then the path cannot begin with an empty segment, that is with two slashes (<code>//</code>), as the following characters would be interpreted as an authority component.{{sfnp|RFC 2396|1998|loc=§3.3}} The final segment of the path may be referred to as a '[[Clean URL#Slug|slug]]'.

{| class="wikitable" style="float: right; font-size: 0.9em; margin-left: 1em"
|-
! Query delimiter
! Example
|-
| Ampersand (<code>&</code>)
| <code>key1=value1&key2=value2</code>
|-
| Semicolon (<code>;</code>){{efn|Historic <nowiki>RFC 1866</nowiki> (obsoleted by <nowiki>RFC 2854</nowiki>) encourages CGI authors to support ';' in addition to '&'.{{sfnp|RFC 1866|1995|loc=§8.2.1}}}}{{Incomplete short citation|date=August 2016}}
| <code>key1=value1;key2=value2</code>
|}
* An optional '''query''' component preceded by a question mark (<code>?</code>), containing a [[query string]] of non-hierarchical data. Its syntax is not well defined, but by convention is most often a sequence of [[attribute–value pair]]s separated by a [[delimiter]].
* An optional '''fragment''' component preceded by an [[Number sign|hash]] (<code>#</code>). The fragment contains a [[fragment identifier]] providing direction to a secondary resource, such as a section heading in an article identified by the remainder of the URI. When the primary resource is an [[HTML]] document, the fragment is often an [[HTML#Attributes|<code>id</code> attribute]] of a specific element, and web browsers will scroll this element into view.<section end=syntax />

Strings of data [[Octet (computing)|octets]] within a URI are represented as characters. Permitted characters within a URI are the [[ASCII]] characters for the lowercase and uppercase letters of the modern [[English alphabet]], the [[Arabic numerals]], [[hyphen]], [[Full stop|period]], [[underscore]], and [[tilde]].{{sfnp|RFC 3986|2005|loc=§2}} Octets represented by any other character must be [[percent-encoding|percent-encoded]].

Of the ASCII character set, the characters <code>: / ? # [ ] @</code> are reserved for use as delimiters of the generic URI components and must be percent-encoded — for example, <code>%3F</code> for a question mark.{{sfnp|RFC 3986|2005|loc=§2.2}} The characters <code>! $ & ' ( ) * + , ; =</code> are permitted by generic URI syntax to be used unencoded in the user information, host, and path as delimiters.{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{sfnp|RFC 3986|2005|loc=§3.3}} Additionally, <code>:</code> and <code>@</code> may appear unencoded within the path, query, and fragment; and <code>?</code> and <code>/</code> may appear unencoded as data within the query or fragment.{{sfnp|RFC 3986|2005|loc=§3.3}}{{sfnp|RFC 3986|2005|loc=§3.4}}

===Examples===
The following figure displays example URIs and their component parts.

<pre>
userinfo host port
┌───┴──┐ ┌──────┴──────┐ ┌┴┐
https://john.doe@www.example.com:123/forum/questions/?tag=networking&order=newest#top
└─┬─┘ └─────────────┬────────────┘└───────┬───────┘ └────────────┬────────────┘ └┬┘
scheme authority path query fragment

ldap://[2001:db8::7]/c=GB?objectClass?one
└─┬┘ └─────┬─────┘└─┬─┘ └──────┬──────┘
scheme authority path query

mailto:John.Doe@example.com
└──┬─┘ └─────────┬────────┘
scheme path

news:comp.infosystems.www.servers.unix
└─┬┘ └───────────────┬───────────────┘
scheme path

tel:+1-816-555-1212
└┬┘ └──────┬──────┘
scheme path

telnet://192.0.2.16:80/
└──┬─┘ └─────┬─────┘│
scheme authority path

urn:oasis:names:specification:docbook:dtd:xml:4.1.2
└┬┘ └──────────────────────┬──────────────────────┘
scheme path
</pre>

===Allowed Schemes===

Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>[[skype]]</code>, <code>steam</code>, <code>[[svn]]</code>, <code>cvs</code>, <code>[[telnet]]</code>, <code>[[sms]]</code>, <code>[[smtp]]</code>, <code>[[ldap]]</code>, <code>dav</code>, <code>[[jabber]]</code>, <code>[[xmpp]]</code>, <code>[[udp]]</code>, <code>view-source</code>, <code>[[websockets|ws]]</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>.

Unpopular schemes include <code>xri</code>, <code>tag</code>, <code>lastfm</code>, <code>[[mongodb]]</code>, <code>msword</code>, <code>moz</code>, <code>palm</code>, <code>paparazzi</code>, <code>com-eventbrite-attendee</code>, <code>dina-playsingle</code>, <code>[[adium]]xtra</code>, <code>urn</code>, <code>onenote</code>, <code>rediss</code> (''sic''), <code>ipp</code>, <code>things</code>, <code>stuns</code>, <code>prospero</code>, <code>iotdisco</code>, <code>tip</code>, <code>tool</code>, <code>tv</code>, <code>unreal</code>, <code>submit</code>, <code>market</code>, <code>ham</code>, <code>fish</code>, <code>finger</code>, <code>iris</code>, <code>[[IP over avian carriers]]</code>, and <code>[[Gopher (protocol)|gopher]]</code>.<ref>https://www.iana.org/assignments/uri-schemes/uri-schemes.xhtml</ref>

==URI references==
===Definition===
A ''URI reference'' is either a URI, or a ''relative reference'' when it does not begin with a scheme component followed by a colon (<code>:</code>).{{sfnp|RFC 3986|2005|loc=§4.1}} A path segment that contains a colon character (e.g., <code>foo:bar</code>) cannot be used as the first path segment of a relative reference if its path component does not begin with a slash (<code>/</code>), as it would be mistaken for a scheme component. Such a path segment must be preceded by a dot path segment (e.g., <code>./foo:bar</code>).{{sfnp|RFC 3986|2005|loc=§4.2}}

Web document [[markup language]]s frequently use URI references to point to other resources, such as external documents or specific portions of the same logical document:{{sfnp|RFC 3986|2005|loc=§4.4}}
* in [[HTML]], the value of the <code>src</code> attribute of the <code>img</code> element provides a URI reference, as does the value of the <code>href</code> attribute of the <code>a</code> or <code>link</code> element;
* in [[XML]], the [[system identifier]] appearing after the <code>SYSTEM</code> keyword in a [[Document Type Definition|DTD]] is a fragmentless URI reference;
* in [[XSLT]], the value of the <code>href</code> attribute of the <code>xsl:import</code> element/instruction is a URI reference; likewise the first argument to the <code>document()</code> function.

===Examples===
<pre>
https://example.com/path/resource.txt#fragment
//example.com/path/resource.txt
/path/resource.txt
path/resource.txt
/path/resource.txt
../resource.txt
./resource.txt
resource.txt
#fragment
</pre>

===Suffix references===
As URI usage has become commonplace, traditional media (television, radio, newspapers, billboards, etc.) have increasingly used a suffix of the URI as a reference, consisting of only the authority and path portions of the URI, such as

<pre>
www.w3.org/Addressing/
</pre>

Such references are primarily intended for human interpretation rather than for machines, with the assumption that context-based heuristics are sufficient to complete the URI (e.g., most registered names beginning with <code>www</code> are likely to have a URI prefix of <code>http://</code>). Although there is no standard set of heuristics for disambiguating a URI suffix, many client implementations allow them to be entered by the user and heuristically resolved. Although this practice of using suffix references is common, it should be avoided whenever possible and should never be used in situations where long-term references are expected, as the heuristics will change over time, particularly when a new URI scheme becomes popular, and are often incorrect when used out of context. Furthermore, they can lead to security issues along the lines of those described in <nowiki>RFC</nowiki> 1535. As a URI suffix has the same syntax as a relative reference with a relative path, a suffix reference cannot be used in contexts where a relative reference is expected. As a result, suffix references are limited to places where there is no defined base URI, such as dialog boxes and off-line advertisements.{{sfnp|RFC 3986|2005|loc=§4.5}}

==URI resolution==
===Definition===
An ''absolute URI'' is a URI with no fragment component.

''Resolving'' a URI reference against a ''base URI'' results in a ''target URI''. This implies that the base URI exists and is an absolute URI. The base URI can be obtained, in order of precedence, from:{{sfnp|RFC 3986|2005|loc=§5.1}}

* the reference URI itself if it is a URI;
* the content of the representation;
* the entity encapsulating the representation;
* the URI used for the actual retrieval of the representation;
* the context of the application.

===Examples===
Within a representation with a well defined base URI of

<pre>
http://a/b/c/d;p?q
</pre>

a relative reference is resolved to its target URI as follows:{{sfnp|RFC 3986|2005|loc=§5.4}}

<pre>
"g:h" -> "g:h"
"g" -> "http://a/b/c/g"
"./g" -> "http://a/b/c/g"
"g/" -> "http://a/b/c/g/"
"/g" -> "http://a/g"
"//g" -> "http://g"
"?y" -> "http://a/b/c/d;p?y"
"g?y" -> "http://a/b/c/g?y"
"#s" -> "http://a/b/c/d;p?q#s"
"g#s" -> "http://a/b/c/g#s"
"g?y#s" -> "http://a/b/c/g?y#s"
";x" -> "http://a/b/c/;x"
"g;x" -> "http://a/b/c/g;x"
"g;x?y#s" -> "http://a/b/c/g;x?y#s"
"" -> "http://a/b/c/d;p?q"
"." -> "http://a/b/c/"
"./" -> "http://a/b/c/"
".." -> "http://a/b/"
"../" -> "http://a/b/"
"../g" -> "http://a/b/g"
"../.." -> "http://a/"
"../../" -> "http://a/"
"../../g" -> "http://a/g"
</pre>

==History==
===Naming, addressing, and identifying resources===
URIs and URLs have a shared history. In 1994, [[Tim Berners-Lee|Tim Berners-Lee's]] proposals for [[hypertext]]{{sfnp|Palmer|2001}} implicitly introduced the idea of a URL as a short string representing a resource that is the target of a [[hyperlink]]. At the time, people referred to it as a "hypertext name"{{sfnp|W3C|1992}} or "document name".

Over the next three and a half years, as the World Wide Web's core technologies of HTML, HTTP, and web browsers developed, a need to distinguish a string that provided an address for a resource from a string that merely named a resource emerged. Although not yet formally defined, the term ''Uniform Resource Locator'' came to represent the former, and the more contentious ''Uniform Resource Name'' came to represent the latter.

During the debate over defining URLs and URNs it became evident that the two concepts embodied by the terms were merely aspects of the fundamental, overarching notion of resource ''identification''. In June 1994, the IETF published Berners-Lee's <nowiki>RFC 1630</nowiki>: the first Request for Comments that acknowledged the existence of URLs and URNs, and, more importantly, defined a formal syntax for ''Universal Resource Identifiers'' — URL-like strings whose precise syntaxes and semantics depended on their schemes. In addition, this RFC attempted to summarize the syntaxes of URL schemes in use at the time. It also acknowledged, but did not standardize, the existence of relative URLs and fragment identifiers.

===Refinement of specifications===
In December 1994, <nowiki>RFC 1738</nowiki> formally defined relative and absolute URLs, refined the general URL syntax, defined how to resolve relative URLs to absolute form, and better enumerated the URL schemes then in use. The agreed definition and syntax of URNs had to wait until the publication of <nowiki>RFC 2141</nowiki> in May 1997.

The publication of <nowiki>RFC 2396</nowiki> in August 1998 saw the URI syntax become a separate specification{{sfnp|RFC 2396|1998}} and most of the parts of RFCs 1630 and 1738 relating to URIs and URLs in general were revised and expanded by the [[IETF]]. The new RFC changed the meaning of "U" in "URI" to "Uniform" from "Universal".

In December 1999, <nowiki>RFC 2732</nowiki> provided a minor update to <nowiki>RFC 2396</nowiki>, allowing URIs to accommodate [[IPv6]] addresses. A number of shortcomings discovered in the two specifications led to a community effort, coordinated by <nowiki>RFC 2396</nowiki> co-author [[Roy Fielding]], that culminated in the publication of <nowiki>RFC 3986</nowiki> in January 2005. While obsoleting the prior standard, it did not render the details of existing URL schemes obsolete; <nowiki>RFC 1738</nowiki> continues to govern such schemes except where otherwise superseded. <nowiki>RFC 2616</nowiki> for example, refines the <code>http</code> scheme. Simultaneously, the IETF published the content of <nowiki>RFC 3986</nowiki> as the full standard STD 66, reflecting the establishment of the URI generic syntax as an official Internet protocol.

In 2001, the W3C's Technical Architecture Group (TAG) published a guide to [[best practices]] and canonical URIs for publishing multiple versions of a given resource.{{sfnp|W3C|2001}} For example, content might differ by language or by size to adjust for capacity or settings of the device used to access that content.

In August 2002, <nowiki>RFC 3305</nowiki> pointed out that the term "URL" had, despite widespread public use, faded into near obsolescence, and serves only as a reminder that some URIs act as addresses by having schemes implying network accessibility, regardless of any such actual use. As URI-based standards such as [[Resource Description Framework]] make evident, resource identification need not suggest the retrieval of resource representations over the Internet, nor need they imply network-based resources at all.

The [[Semantic Web]] uses the HTTP URI scheme to identify both documents and concepts in the real world, a distinction which has caused confusion as to how to distinguish the two. The TAG published an e-mail in 2005 on how to solve the problem, which became known as the ''httpRange-14 resolution''.{{sfnp|Fielding|2005}} The W3C subsequently published an Interest Group Note titled ''Cool URIs for the Semantic Web'',{{sfnp|W3C|2008}} which explained the use of [[content negotiation]] and the [[HTTP 303]] response code for redirections in more detail.

==Relation to XML namespaces==
In [[XML]], a [[XML namespace|namespace]] is an abstract domain to which a collection of element and attribute names can be assigned. The namespace name is a character string which must adhere to the generic URI syntax.{{sfnp|Morrison|2006}} However, the name is generally not considered to be a URI,{{sfnp|Harold|2004}} because the URI specification bases the decision not only on lexical components, but also on their intended use. A namespace name does not necessarily imply any of the semantics of URI schemes; for example, a namespace name beginning with ''http:'' may have no connotation to the use of the [[HTTP]].

Originally, the namespace name could match the syntax of any non-empty URI reference, but the use of relative URI references was deprecated by the W3C.{{sfnp|W3C|2009}} A separate W3C specification for namespaces in XML 1.1 permits [[Internationalized resource identifier|internationalized resource identifier (IRI)]] references to serve as the basis for namespace names in addition to URI references.{{sfnp|W3C|2006}}

==See also==
* [[CURIE]] – defines a generic, abbreviated syntax for expressing URIs
* [[Dereferenceable Uniform Resource Identifier]] – a resource retrieval mechanism that uses any of the internet protocols (e.g. HTTP) to obtain a copy or representation of the resource it identifies
* [[Extensible Resource Identifier]] – a scheme and resolution protocol for abstract identifiers compatible with URIs
* [[Internationalized Resource Identifier]] (IRI) – a generalization of URIs allowing the use of Unicode
* [[Persistent uniform resource locator]] (PURL) – a URI that is used to redirect to the location of the requested web resource
* [[Uniform Naming Convention]] – a common syntax used by Microsoft to describe the location of a network resource, such as a shared file, directory, or printer
* [[Resource Directory Description Language]] – a descriptive language to provide machine- and human-readable information about a particular namespace and about the XML documents that use it
* [[Universally unique identifier|UUID]]

== Notes ==
{{Notelist}}

== References ==
=== Citations ===
{{Reflist|25em}}

=== Cited works ===
{{refbegin|32em}}
* {{cite web |first=Roy T.|last=Fielding|authorlink=Roy Fielding |title = [httpRange-14] Resolved |url = http://lists.w3.org/Archives/Public/www-tag/2005Jun/0039.html|date=18 June 2005 |accessdate=24 July 2009|ref=harv}}
* {{cite book |first=Elliotte Rusty|last=Harold |authorlink=Elliotte Rusty Harold |year=2004 |title=XML 1.1 Bible |edition=Third |publisher=[[Wiley Publishing]] |page=291 |isbn=0-7645-4986-3 |ref=harv }}
* {{cite web |url = http://www.w3.org/TR/uri-clarification/ |author=Joint W3C/IETF URI Planning Interest Group|title=URIs, URLs, and URNs: Clarifications and Recommendations 1.0|date=21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2001}}}}
* {{cite web |url = https://tools.ietf.org/html/rfc3305|title=Report from the Joint W3C/IETF URI Planning Interest Group: Uniform Resource Identifiers (URIs), URLs, and Uniform Resource Names (URNs): Clarifications and Recommendations|editor1-first=M.|editor1-last=Mealling |editor2-first=R.|editor2-last=Denenberg|publisher=[[World Wide Web Consortium]]|date=August 2002 |accessdate=13 September 2015 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2002}}}}
* {{cite web |url=https://tools.ietf.org/html/rfc7595|title=Guidelines and Registration Procedures for URI Schemes|editor-first=D.|editor-last=Thaler|author1-first=T.|author1-last=Hansen|author2-first=T.|author2-last=Hardie|publisher=[[Internet Engineering Task Force]]|date=June 2015|issn=2070-1721|ref={{SfnRef|IETF|2015}}}}
* {{cite book |last=Morrison|first=Michael|year=2006|title=Sams Teach Yourself XML|publisher=[[Sams Publishing]]|chapter=Hour 5: ''Putting Namespaces to Use''|page=91 |ref=harv }}
* {{cite web |first=Sean B.|last=Palmer |title=The Early History of HTML |url = http://infomesh.net/html/history/early/ |year=2001 |accessdate=2009-04-30 |ref=harv }}
* {{cite web |author = URI Planning Interest Group, W3C/IETF |title = URIs, URLs, and URNs: Clarifications and Recommendations 1.0 |url = http://www.w3.org/TR/uri-clarification/ |date = 21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|URI Planning Interest Group|2009}}}}
* {{cite web |url = http://www.w3.org/History/19921103-hypertext/hypertext/WWW/Addressing/Addressing.html|title=W3 Naming Schemes |publisher=[[World Wide Web Consortium]]|year=1992|accessdate=2009-07-24|ref={{SfnRef|W3C|1992}}}}
* {{cite web |url = http://www.w3.org/2001/tag/doc/alternatives-discovery.html |title=On Linking Alternative Representations To Enable Discovery And Publishing|publisher=[[World Wide Web Consortium]]|year=2006|orig-year=2001|accessdate=2012-04-03|ref={{SfnRef|W3C|2001}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use|title=Namespaces in XML 1.1 (Second Edition)|date=16 August 2006 |at = 2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray|editor2-first=Dave |editor2-last=Hollander|editor3-first=Andrew|editor3-last=Layman|editor4-first=Richard|editor4-last=Tobin|publisher=[[World Wide Web Consortium]] |accessdate=31 August 2015|ref={{SfnRef|W3C|2006}}}}
* {{cite web |url = http://www.w3.org/TR/cooluris/|title=Cool URIs for the Semantic Web|editor1-first=Leo|editor1-last=Sauermann|editor2-first=Richard|editor2-last=Cyganiak|author1-first=Danny|author1-last=Ayers|author2-first=Max|author2-last=Völkel|publisher=[[World Wide Web Consortium]]|date=3 December 2008|accessdate=2012-04-03|ref={{SfnRef|W3C|2008}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use |title=Namespaces in XML 1.0 (Third Edition)|date=8 December 2009 |at=2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray |editor2-first=Dave |editor2-last=Hollander |editor3-first=Andrew|editor3-last=Layman |editor4-first=Richard|editor4-last=Tobin |editor5-first=Henry S. |editor5-last=Thompson |publisher=[[World Wide Web Consortium]]|accessdate=31 August 2015 |ref={{SfnRef|W3C|2009}} }}
* {{cite web |url = https://tools.ietf.org/html/rfc1866#section-8.2.1 |title=Hypertext Markup Language - 2.0|author1-first=Tim|author1-last=Berners-Lee |author1-link=Tim Berners-Lee|author2-first=Dan|author2-last=Connolly|publisher=[[Internet Engineering Task Force]] |date=November 1995 |accessdate=13 September 2015}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc2396 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=2396 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=August 1998 |accessdate=31 August 2015 |ref={{SfnRef|RFC 2396|1998}}}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc3986 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=3986 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=January 2005 |accessdate=31 August 2015 |ref={{SfnRef|RFC 3986|2005}}}}
* {{cite web |last1=Lawrence |first1=Eric |title=Browser Arcana: IP Literals in URLs |url = http://blogs.msdn.com/b/ieinternals/archive/2014/03/06/browser-arcana-ipv4-ipv6-literal-urls-dotted-va-dotless.aspx |website=IEInternals |publisher=[[Microsoft]] |date=6 March 2014 |accessdate=2016-04-25 |ref = harv }}
{{refend}}

==External links==
* [http://www.iana.org/assignments/uri-schemes.html URI Schemes] – [[Internet Assigned Numbers Authority|IANA]]-maintained registry of URI Schemes
* [http://www.w3.org/wiki/UriSchemes URI schemes on the W3C wiki]
* [http://www.w3.org/TR/webarch/#identification Architecture of the World Wide Web, Volume One, §2: Identification] – by W3C
* [http://www.w3.org/TR/uri-clarification/ W3C URI Clarification]

{{Semantic Web|state=collapsed}}
{{URI scheme}}
{{Hypermedia}}

{{Authority control}}

[[Category:Application layer protocols]]
[[Category:Internet protocols]]
[[Category:Internet Standards]]
[[Category:Semantic Web| ]]
[[Category:Uniform Resource Locator]]

Uniform Resource Identifier

2018-06-29T19:45:14Z

Crasshopper: /* Definition */ formatting

{{Redirect|URI}}
{{good article}}

A '''Uniform Resource Identifier''' ('''URI''') is a [[character string (computer science)|string]] of [[character (computing)|character]]s designed for unambiguous [[identifier|identification]] of [[resource (computer science)|resources]] and extensibility via the URI scheme.

Such identification enables interaction with representations of the resource over a network, typically the [[World Wide Web]], using specific [[Communications protocol|protocols]]. Schemes specifying a concrete [[syntax]] and associated protocols define each URI. The most common form of URI is the Uniform Resource Locator ([[URL]]), frequently referred to informally as a ''web address.'' More rarely seen in usage is the [[Uniform Resource Name|Uniform Resource Name (URN)]], which was designed to complement URLs by providing a mechanism for the identification of resources in particular [[namespace]]s.

==URL and URN==
A [[Uniform Resource Name]] (URN) is a URI that identifies a resource by name in a particular namespace. A URN may be used to talk about a resource without implying its location or how to access it. For example, in the [[International Standard Book Number|International Standard Book Number (ISBN)]] system, ''<nowiki>ISBN</nowiki> 0-486-27557-4'' identifies a specific edition of Shakespeare's play ''[[Romeo and Juliet]]''. The URN for that edition would be ''<nowiki>urn:isbn:0-486-27557-4</nowiki>''. However, it gives no information as to where to find a copy of that book.

A [[Uniform Resource Locator]] (URL) is a URI that specifies the means of acting upon or obtaining the representation of a resource, i.e. specifying both its primary access mechanism and network location. For example, the URL <code><nowiki>http://example.org/wiki/Main_Page</nowiki></code> refers to a resource identified as <code><nowiki>/wiki/Main_Page</nowiki></code> whose representation, in the form of [[HTML]] and related code, is obtainable via the [[Hypertext Transfer Protocol]] (''http:'') from a network host whose [[domain name]] is <code><nowiki>example.org</nowiki></code>.

A URN may be compared to a person's name, while a URL may be compared to their street address. In other words, a URN identifies an item and a URL provides a method for finding it.

Technical publications, especially standards produced by the [[Internet Engineering Task Force|IETF]] and by the [[World Wide Web Consortium|W3C]], normally reflect a view outlined in a [[W3C Recommendation]] of 2001, which acknowledges the precedence of the term URI rather than endorsing any formal subdivision into URL and URN. {{cquote|URL is a useful but informal concept: a URL is a type of URI that identifies a resource via a representation of its primary access mechanism (e.g., its network "location"), rather than by some other attributes it may have.{{sfnp|Joint W3C/IETF URI Planning Interest Group|2001}}}}

As such, a URL is simply a URI that happens to point to a resource over a network.{{efn|A report published in 2002 by a joint W3C/IETF working group aimed to normalize the divergent views held within the IETF and W3C over the relationship between the various 'UR*' terms and standards. While not published as a full standard by either organization, it has become the basis for the above common understanding and has informed many standards since then.}}{{sfnp|Joint W3C/IETF URI Planning Interest Group|2002}} However, in non-technical contexts and in software for the World Wide Web, the term "URL" remains widely used. Additionally, the term "web address" (which has no formal definition) often occurs in non-technical publications as a synonym for a URI that uses the ''http'' or ''https'' schemes. Such assumptions can lead to confusion, for example, in the case of XML namespaces that have a [[#Relation to XML namespaces|visual similarity to resolvable URIs]].

Specifications produced by the [[WHATWG]] prefer URL over URI, and so newer HTML5 APIs use URL over URI.<ref>{{cite web |title=URL Standard: 6.3. URL APIs elsewhere |url=https://url.spec.whatwg.org/#url-apis-elsewhere}}</ref> {{cquote|Standardize on the term URL. URI and IRI are just confusing. In practice a single algorithm is used for both so keeping them distinct is not helping anyone. URL also easily wins the search result popularity contest.<ref>{{cite web |title=URL Standard: Goals|url=https://url.spec.whatwg.org/#goals}}</ref>}}

While most URI schemes were originally designed to be used with a particular [[protocol (computing)|protocol]], and often have the same name, they are semantically different from protocols. For example, the scheme ''http'' is generally used for interacting with [[web resource]]s using HTTP, but the scheme ''[[file URI scheme|file]]'' has no protocol.

==Generic syntax==
===Definition===
Each URI begins with a scheme name that refers to a specification for assigning identifiers within that scheme. As such, the URI syntax is a federated and extensible naming system wherein each scheme's specification may further restrict the syntax and semantics of identifiers using that scheme. The URI generic syntax is a superset of the syntax of all URI schemes. It was first defined in [[Request for Comments|Request for Comments (RFC)]] 2396, published in August 1998,{{sfnp|RFC 2396|1998}} and finalized in <nowiki>RFC</nowiki> 3986, published in January 2005.{{sfnp|RFC 3986|2005}}
<section begin=syntax />
The ''URI generic syntax'' consists of a hierarchical sequence of five ''components'':{{sfnp|RFC 3986|2005|loc=§3}}

<pre>
URI = scheme:[//authority]path[?query][#fragment]
</pre>

where the authority component divides into three ''subcomponents'':

<pre>
authority = [userinfo@]host[:port]
</pre>

It comprises:
* A non-empty '''{{visible anchor|scheme}}''' component followed by a colon (<code>:</code>), consisting of a sequence of characters beginning with a letter and followed by any combination of letters, digits, plus (<code>+</code>), period (<code>.</code>), or hyphen (<code>-</code>). Although schemes are case-insensitive, the canonical form is lowercase and documents that specify schemes must do so with lowercase letters. Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>[[skype]]</code>, <code>steam</code>, <code>[[svn]]</code>, <code>cvs</code>, <code>[[telnet]]</code>, <code>[[sms]]</code>, <code>[[smtp]]</code>, <code>[[ldap]]</code>, <code>dav</code>, <code>[[jabber]]</code>, <code>[[xmpp]]</code>, <code>[[udp]]</code>, <code>view-source</code>, <code>[[websockets|ws]]</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>. Unpopular schemes include <code>xri</code>, <code>tag</code>, <code>lastfm</code>, <code>[[mongodb]]</code>, <code>msword</code>, <code>moz</code>, <code>palm</code>, <code>paparazzi</code>, <code>com-eventbrite-attendee</code>, <code>dina-playsingle</code>, <code>[[adium]]xtra</code>, <code>urn</code>, <code>onenote</code>, <code>rediss</code> (''sic''), <code>ipp</code>, <code>things</code>, <code>stuns</code>, <code>prospero</code>, <code>iotdisco</code>, <code>tip</code>, <code>tool</code>, <code>tv</code>, <code>unreal</code>, <code>submit</code>, <code>market</code>, <code>ham</code>, <code>fish</code>, <code>finger</code>, <code>iris</code>, <code>[[IP over avian carriers]]</code>, and <code>[[Gopher (protocol)|gopher]]</code>.<ref>https://www.iana.org/assignments/uri-schemes/uri-schemes.xhtml</ref> URI schemes should be registered with the [[Internet Assigned Numbers Authority|Internet Assigned Numbers Authority (IANA)]], although non-registered schemes are used in practice.{{efn|The procedures for registering new URI schemes were originally defined in 1999 by <nowiki>RFC 2717</nowiki>, and are now defined by <nowiki>RFC 7595</nowiki>, published in June 2015.{{sfnp|IETF|2015}}}}
* An optional non-empty '''authority''' component preceded by two slashes (<code>//</code>), comprising:
** An optional '''userinfo''' subcomponent that may consist of a [[User (computing)|user name]] and an optional [[password]] preceded by a colon (<code>:</code>), followed by an at symbol (<code>@</code>). Use of the format <code>username:password</code> in the userinfo subcomponent is deprecated for security reasons. Applications should not render as clear text any data after the first colon (<code>:</code>) found within a userinfo subcomponent unless the data after the colon is the empty string (indicating no password).
** A non-empty '''host''' subcomponent, consisting of either a registered name (including but not limited to a [[hostname]]), or an [[IP address]]. [[IPv4]] addresses must be in [[dot-decimal notation]], and [[IPv6]] addresses must be enclosed in brackets (<code>[]</code>).{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{efn|For URIs relating to resources on the World Wide Web, some web browsers allow {{code|.0}} portions of dot-decimal notation to be dropped or raw integer IP addresses to be used.{{sfnp|Lawrence|2014}}}}
** An optional '''port''' subcomponent preceded by a colon (<code>:</code>).
* A '''path''' component, consisting of a sequence of path segments separated by a slash (<code>/</code>). A path is always defined for a URI, though the defined path may be empty (zero length). A segment may also be empty, resulting in two consecutive slashes (<code>//</code>) in the path component. A path component may resemble or map exactly to a [[Path (computing)|file system path]], but does not always imply a relation to one. If an authority component is present, then the path component must either be empty or begin with a slash (<code>/</code>). If an authority component is absent, then the path cannot begin with an empty segment, that is with two slashes (<code>//</code>), as the following characters would be interpreted as an authority component.{{sfnp|RFC 2396|1998|loc=§3.3}} The final segment of the path may be referred to as a '[[Clean URL#Slug|slug]]'.

{| class="wikitable" style="float: right; font-size: 0.9em; margin-left: 1em"
|-
! Query delimiter
! Example
|-
| Ampersand (<code>&</code>)
| <code>key1=value1&key2=value2</code>
|-
| Semicolon (<code>;</code>){{efn|Historic <nowiki>RFC 1866</nowiki> (obsoleted by <nowiki>RFC 2854</nowiki>) encourages CGI authors to support ';' in addition to '&'.{{sfnp|RFC 1866|1995|loc=§8.2.1}}}}{{Incomplete short citation|date=August 2016}}
| <code>key1=value1;key2=value2</code>
|}
* An optional '''query''' component preceded by a question mark (<code>?</code>), containing a [[query string]] of non-hierarchical data. Its syntax is not well defined, but by convention is most often a sequence of [[attribute–value pair]]s separated by a [[delimiter]].
* An optional '''fragment''' component preceded by an [[Number sign|hash]] (<code>#</code>). The fragment contains a [[fragment identifier]] providing direction to a secondary resource, such as a section heading in an article identified by the remainder of the URI. When the primary resource is an [[HTML]] document, the fragment is often an [[HTML#Attributes|<code>id</code> attribute]] of a specific element, and web browsers will scroll this element into view.<section end=syntax />

Strings of data [[Octet (computing)|octets]] within a URI are represented as characters. Permitted characters within a URI are the [[ASCII]] characters for the lowercase and uppercase letters of the modern [[English alphabet]], the [[Arabic numerals]], [[hyphen]], [[Full stop|period]], [[underscore]], and [[tilde]].{{sfnp|RFC 3986|2005|loc=§2}} Octets represented by any other character must be [[percent-encoding|percent-encoded]].

Of the ASCII character set, the characters <code>: / ? # [ ] @</code> are reserved for use as delimiters of the generic URI components and must be percent-encoded — for example, <code>%3F</code> for a question mark.{{sfnp|RFC 3986|2005|loc=§2.2}} The characters <code>! $ & ' ( ) * + , ; =</code> are permitted by generic URI syntax to be used unencoded in the user information, host, and path as delimiters.{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{sfnp|RFC 3986|2005|loc=§3.3}} Additionally, <code>:</code> and <code>@</code> may appear unencoded within the path, query, and fragment; and <code>?</code> and <code>/</code> may appear unencoded as data within the query or fragment.{{sfnp|RFC 3986|2005|loc=§3.3}}{{sfnp|RFC 3986|2005|loc=§3.4}}

===Examples===
The following figure displays example URIs and their component parts.

<pre>
userinfo host port
┌───┴──┐ ┌──────┴──────┐ ┌┴┐
https://john.doe@www.example.com:123/forum/questions/?tag=networking&order=newest#top
└─┬─┘ └─────────────┬────────────┘└───────┬───────┘ └────────────┬────────────┘ └┬┘
scheme authority path query fragment

ldap://[2001:db8::7]/c=GB?objectClass?one
└─┬┘ └─────┬─────┘└─┬─┘ └──────┬──────┘
scheme authority path query

mailto:John.Doe@example.com
└──┬─┘ └─────────┬────────┘
scheme path

news:comp.infosystems.www.servers.unix
└─┬┘ └───────────────┬───────────────┘
scheme path

tel:+1-816-555-1212
└┬┘ └──────┬──────┘
scheme path

telnet://192.0.2.16:80/
└──┬─┘ └─────┬─────┘│
scheme authority path

urn:oasis:names:specification:docbook:dtd:xml:4.1.2
└┬┘ └──────────────────────┬──────────────────────┘
scheme path
</pre>

==URI references==
===Definition===
A ''URI reference'' is either a URI, or a ''relative reference'' when it does not begin with a scheme component followed by a colon (<code>:</code>).{{sfnp|RFC 3986|2005|loc=§4.1}} A path segment that contains a colon character (e.g., <code>foo:bar</code>) cannot be used as the first path segment of a relative reference if its path component does not begin with a slash (<code>/</code>), as it would be mistaken for a scheme component. Such a path segment must be preceded by a dot path segment (e.g., <code>./foo:bar</code>).{{sfnp|RFC 3986|2005|loc=§4.2}}

Web document [[markup language]]s frequently use URI references to point to other resources, such as external documents or specific portions of the same logical document:{{sfnp|RFC 3986|2005|loc=§4.4}}
* in [[HTML]], the value of the <code>src</code> attribute of the <code>img</code> element provides a URI reference, as does the value of the <code>href</code> attribute of the <code>a</code> or <code>link</code> element;
* in [[XML]], the [[system identifier]] appearing after the <code>SYSTEM</code> keyword in a [[Document Type Definition|DTD]] is a fragmentless URI reference;
* in [[XSLT]], the value of the <code>href</code> attribute of the <code>xsl:import</code> element/instruction is a URI reference; likewise the first argument to the <code>document()</code> function.

===Examples===
<pre>
https://example.com/path/resource.txt#fragment
//example.com/path/resource.txt
/path/resource.txt
path/resource.txt
/path/resource.txt
../resource.txt
./resource.txt
resource.txt
#fragment
</pre>

===Suffix references===
As URI usage has become commonplace, traditional media (television, radio, newspapers, billboards, etc.) have increasingly used a suffix of the URI as a reference, consisting of only the authority and path portions of the URI, such as

<pre>
www.w3.org/Addressing/
</pre>

Such references are primarily intended for human interpretation rather than for machines, with the assumption that context-based heuristics are sufficient to complete the URI (e.g., most registered names beginning with <code>www</code> are likely to have a URI prefix of <code>http://</code>). Although there is no standard set of heuristics for disambiguating a URI suffix, many client implementations allow them to be entered by the user and heuristically resolved. Although this practice of using suffix references is common, it should be avoided whenever possible and should never be used in situations where long-term references are expected, as the heuristics will change over time, particularly when a new URI scheme becomes popular, and are often incorrect when used out of context. Furthermore, they can lead to security issues along the lines of those described in <nowiki>RFC</nowiki> 1535. As a URI suffix has the same syntax as a relative reference with a relative path, a suffix reference cannot be used in contexts where a relative reference is expected. As a result, suffix references are limited to places where there is no defined base URI, such as dialog boxes and off-line advertisements.{{sfnp|RFC 3986|2005|loc=§4.5}}

==URI resolution==
===Definition===
An ''absolute URI'' is a URI with no fragment component.

''Resolving'' a URI reference against a ''base URI'' results in a ''target URI''. This implies that the base URI exists and is an absolute URI. The base URI can be obtained, in order of precedence, from:{{sfnp|RFC 3986|2005|loc=§5.1}}

* the reference URI itself if it is a URI;
* the content of the representation;
* the entity encapsulating the representation;
* the URI used for the actual retrieval of the representation;
* the context of the application.

===Examples===
Within a representation with a well defined base URI of

<pre>
http://a/b/c/d;p?q
</pre>

a relative reference is resolved to its target URI as follows:{{sfnp|RFC 3986|2005|loc=§5.4}}

<pre>
"g:h" -> "g:h"
"g" -> "http://a/b/c/g"
"./g" -> "http://a/b/c/g"
"g/" -> "http://a/b/c/g/"
"/g" -> "http://a/g"
"//g" -> "http://g"
"?y" -> "http://a/b/c/d;p?y"
"g?y" -> "http://a/b/c/g?y"
"#s" -> "http://a/b/c/d;p?q#s"
"g#s" -> "http://a/b/c/g#s"
"g?y#s" -> "http://a/b/c/g?y#s"
";x" -> "http://a/b/c/;x"
"g;x" -> "http://a/b/c/g;x"
"g;x?y#s" -> "http://a/b/c/g;x?y#s"
"" -> "http://a/b/c/d;p?q"
"." -> "http://a/b/c/"
"./" -> "http://a/b/c/"
".." -> "http://a/b/"
"../" -> "http://a/b/"
"../g" -> "http://a/b/g"
"../.." -> "http://a/"
"../../" -> "http://a/"
"../../g" -> "http://a/g"
</pre>

==History==
===Naming, addressing, and identifying resources===
URIs and URLs have a shared history. In 1994, [[Tim Berners-Lee|Tim Berners-Lee's]] proposals for [[hypertext]]{{sfnp|Palmer|2001}} implicitly introduced the idea of a URL as a short string representing a resource that is the target of a [[hyperlink]]. At the time, people referred to it as a "hypertext name"{{sfnp|W3C|1992}} or "document name".

Over the next three and a half years, as the World Wide Web's core technologies of HTML, HTTP, and web browsers developed, a need to distinguish a string that provided an address for a resource from a string that merely named a resource emerged. Although not yet formally defined, the term ''Uniform Resource Locator'' came to represent the former, and the more contentious ''Uniform Resource Name'' came to represent the latter.

During the debate over defining URLs and URNs it became evident that the two concepts embodied by the terms were merely aspects of the fundamental, overarching notion of resource ''identification''. In June 1994, the IETF published Berners-Lee's <nowiki>RFC 1630</nowiki>: the first Request for Comments that acknowledged the existence of URLs and URNs, and, more importantly, defined a formal syntax for ''Universal Resource Identifiers'' — URL-like strings whose precise syntaxes and semantics depended on their schemes. In addition, this RFC attempted to summarize the syntaxes of URL schemes in use at the time. It also acknowledged, but did not standardize, the existence of relative URLs and fragment identifiers.

===Refinement of specifications===
In December 1994, <nowiki>RFC 1738</nowiki> formally defined relative and absolute URLs, refined the general URL syntax, defined how to resolve relative URLs to absolute form, and better enumerated the URL schemes then in use. The agreed definition and syntax of URNs had to wait until the publication of <nowiki>RFC 2141</nowiki> in May 1997.

The publication of <nowiki>RFC 2396</nowiki> in August 1998 saw the URI syntax become a separate specification{{sfnp|RFC 2396|1998}} and most of the parts of RFCs 1630 and 1738 relating to URIs and URLs in general were revised and expanded by the [[IETF]]. The new RFC changed the meaning of "U" in "URI" to "Uniform" from "Universal".

In December 1999, <nowiki>RFC 2732</nowiki> provided a minor update to <nowiki>RFC 2396</nowiki>, allowing URIs to accommodate [[IPv6]] addresses. A number of shortcomings discovered in the two specifications led to a community effort, coordinated by <nowiki>RFC 2396</nowiki> co-author [[Roy Fielding]], that culminated in the publication of <nowiki>RFC 3986</nowiki> in January 2005. While obsoleting the prior standard, it did not render the details of existing URL schemes obsolete; <nowiki>RFC 1738</nowiki> continues to govern such schemes except where otherwise superseded. <nowiki>RFC 2616</nowiki> for example, refines the <code>http</code> scheme. Simultaneously, the IETF published the content of <nowiki>RFC 3986</nowiki> as the full standard STD 66, reflecting the establishment of the URI generic syntax as an official Internet protocol.

In 2001, the W3C's Technical Architecture Group (TAG) published a guide to [[best practices]] and canonical URIs for publishing multiple versions of a given resource.{{sfnp|W3C|2001}} For example, content might differ by language or by size to adjust for capacity or settings of the device used to access that content.

In August 2002, <nowiki>RFC 3305</nowiki> pointed out that the term "URL" had, despite widespread public use, faded into near obsolescence, and serves only as a reminder that some URIs act as addresses by having schemes implying network accessibility, regardless of any such actual use. As URI-based standards such as [[Resource Description Framework]] make evident, resource identification need not suggest the retrieval of resource representations over the Internet, nor need they imply network-based resources at all.

The [[Semantic Web]] uses the HTTP URI scheme to identify both documents and concepts in the real world, a distinction which has caused confusion as to how to distinguish the two. The TAG published an e-mail in 2005 on how to solve the problem, which became known as the ''httpRange-14 resolution''.{{sfnp|Fielding|2005}} The W3C subsequently published an Interest Group Note titled ''Cool URIs for the Semantic Web'',{{sfnp|W3C|2008}} which explained the use of [[content negotiation]] and the [[HTTP 303]] response code for redirections in more detail.

==Relation to XML namespaces==
In [[XML]], a [[XML namespace|namespace]] is an abstract domain to which a collection of element and attribute names can be assigned. The namespace name is a character string which must adhere to the generic URI syntax.{{sfnp|Morrison|2006}} However, the name is generally not considered to be a URI,{{sfnp|Harold|2004}} because the URI specification bases the decision not only on lexical components, but also on their intended use. A namespace name does not necessarily imply any of the semantics of URI schemes; for example, a namespace name beginning with ''http:'' may have no connotation to the use of the [[HTTP]].

Originally, the namespace name could match the syntax of any non-empty URI reference, but the use of relative URI references was deprecated by the W3C.{{sfnp|W3C|2009}} A separate W3C specification for namespaces in XML 1.1 permits [[Internationalized resource identifier|internationalized resource identifier (IRI)]] references to serve as the basis for namespace names in addition to URI references.{{sfnp|W3C|2006}}

==See also==
* [[CURIE]] – defines a generic, abbreviated syntax for expressing URIs
* [[Dereferenceable Uniform Resource Identifier]] – a resource retrieval mechanism that uses any of the internet protocols (e.g. HTTP) to obtain a copy or representation of the resource it identifies
* [[Extensible Resource Identifier]] – a scheme and resolution protocol for abstract identifiers compatible with URIs
* [[Internationalized Resource Identifier]] (IRI) – a generalization of URIs allowing the use of Unicode
* [[Persistent uniform resource locator]] (PURL) – a URI that is used to redirect to the location of the requested web resource
* [[Uniform Naming Convention]] – a common syntax used by Microsoft to describe the location of a network resource, such as a shared file, directory, or printer
* [[Resource Directory Description Language]] – a descriptive language to provide machine- and human-readable information about a particular namespace and about the XML documents that use it
* [[Universally unique identifier|UUID]]

== Notes ==
{{Notelist}}

== References ==
=== Citations ===
{{Reflist|25em}}

=== Cited works ===
{{refbegin|32em}}
* {{cite web |first=Roy T.|last=Fielding|authorlink=Roy Fielding |title = [httpRange-14] Resolved |url = http://lists.w3.org/Archives/Public/www-tag/2005Jun/0039.html|date=18 June 2005 |accessdate=24 July 2009|ref=harv}}
* {{cite book |first=Elliotte Rusty|last=Harold |authorlink=Elliotte Rusty Harold |year=2004 |title=XML 1.1 Bible |edition=Third |publisher=[[Wiley Publishing]] |page=291 |isbn=0-7645-4986-3 |ref=harv }}
* {{cite web |url = http://www.w3.org/TR/uri-clarification/ |author=Joint W3C/IETF URI Planning Interest Group|title=URIs, URLs, and URNs: Clarifications and Recommendations 1.0|date=21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2001}}}}
* {{cite web |url = https://tools.ietf.org/html/rfc3305|title=Report from the Joint W3C/IETF URI Planning Interest Group: Uniform Resource Identifiers (URIs), URLs, and Uniform Resource Names (URNs): Clarifications and Recommendations|editor1-first=M.|editor1-last=Mealling |editor2-first=R.|editor2-last=Denenberg|publisher=[[World Wide Web Consortium]]|date=August 2002 |accessdate=13 September 2015 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2002}}}}
* {{cite web |url=https://tools.ietf.org/html/rfc7595|title=Guidelines and Registration Procedures for URI Schemes|editor-first=D.|editor-last=Thaler|author1-first=T.|author1-last=Hansen|author2-first=T.|author2-last=Hardie|publisher=[[Internet Engineering Task Force]]|date=June 2015|issn=2070-1721|ref={{SfnRef|IETF|2015}}}}
* {{cite book |last=Morrison|first=Michael|year=2006|title=Sams Teach Yourself XML|publisher=[[Sams Publishing]]|chapter=Hour 5: ''Putting Namespaces to Use''|page=91 |ref=harv }}
* {{cite web |first=Sean B.|last=Palmer |title=The Early History of HTML |url = http://infomesh.net/html/history/early/ |year=2001 |accessdate=2009-04-30 |ref=harv }}
* {{cite web |author = URI Planning Interest Group, W3C/IETF |title = URIs, URLs, and URNs: Clarifications and Recommendations 1.0 |url = http://www.w3.org/TR/uri-clarification/ |date = 21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|URI Planning Interest Group|2009}}}}
* {{cite web |url = http://www.w3.org/History/19921103-hypertext/hypertext/WWW/Addressing/Addressing.html|title=W3 Naming Schemes |publisher=[[World Wide Web Consortium]]|year=1992|accessdate=2009-07-24|ref={{SfnRef|W3C|1992}}}}
* {{cite web |url = http://www.w3.org/2001/tag/doc/alternatives-discovery.html |title=On Linking Alternative Representations To Enable Discovery And Publishing|publisher=[[World Wide Web Consortium]]|year=2006|orig-year=2001|accessdate=2012-04-03|ref={{SfnRef|W3C|2001}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use|title=Namespaces in XML 1.1 (Second Edition)|date=16 August 2006 |at = 2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray|editor2-first=Dave |editor2-last=Hollander|editor3-first=Andrew|editor3-last=Layman|editor4-first=Richard|editor4-last=Tobin|publisher=[[World Wide Web Consortium]] |accessdate=31 August 2015|ref={{SfnRef|W3C|2006}}}}
* {{cite web |url = http://www.w3.org/TR/cooluris/|title=Cool URIs for the Semantic Web|editor1-first=Leo|editor1-last=Sauermann|editor2-first=Richard|editor2-last=Cyganiak|author1-first=Danny|author1-last=Ayers|author2-first=Max|author2-last=Völkel|publisher=[[World Wide Web Consortium]]|date=3 December 2008|accessdate=2012-04-03|ref={{SfnRef|W3C|2008}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use |title=Namespaces in XML 1.0 (Third Edition)|date=8 December 2009 |at=2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray |editor2-first=Dave |editor2-last=Hollander |editor3-first=Andrew|editor3-last=Layman |editor4-first=Richard|editor4-last=Tobin |editor5-first=Henry S. |editor5-last=Thompson |publisher=[[World Wide Web Consortium]]|accessdate=31 August 2015 |ref={{SfnRef|W3C|2009}} }}
* {{cite web |url = https://tools.ietf.org/html/rfc1866#section-8.2.1 |title=Hypertext Markup Language - 2.0|author1-first=Tim|author1-last=Berners-Lee |author1-link=Tim Berners-Lee|author2-first=Dan|author2-last=Connolly|publisher=[[Internet Engineering Task Force]] |date=November 1995 |accessdate=13 September 2015}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc2396 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=2396 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=August 1998 |accessdate=31 August 2015 |ref={{SfnRef|RFC 2396|1998}}}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc3986 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=3986 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=January 2005 |accessdate=31 August 2015 |ref={{SfnRef|RFC 3986|2005}}}}
* {{cite web |last1=Lawrence |first1=Eric |title=Browser Arcana: IP Literals in URLs |url = http://blogs.msdn.com/b/ieinternals/archive/2014/03/06/browser-arcana-ipv4-ipv6-literal-urls-dotted-va-dotless.aspx |website=IEInternals |publisher=[[Microsoft]] |date=6 March 2014 |accessdate=2016-04-25 |ref = harv }}
{{refend}}

==External links==
* [http://www.iana.org/assignments/uri-schemes.html URI Schemes] – [[Internet Assigned Numbers Authority|IANA]]-maintained registry of URI Schemes
* [http://www.w3.org/wiki/UriSchemes URI schemes on the W3C wiki]
* [http://www.w3.org/TR/webarch/#identification Architecture of the World Wide Web, Volume One, §2: Identification] – by W3C
* [http://www.w3.org/TR/uri-clarification/ W3C URI Clarification]

{{Semantic Web|state=collapsed}}
{{URI scheme}}
{{Hypermedia}}

{{Authority control}}

[[Category:Application layer protocols]]
[[Category:Internet protocols]]
[[Category:Internet Standards]]
[[Category:Semantic Web| ]]
[[Category:Uniform Resource Locator]]

Uniform Resource Identifier

2018-06-29T19:44:05Z

Crasshopper: /* Definition */ link some of them in

{{Redirect|URI}}
{{good article}}

A '''Uniform Resource Identifier''' ('''URI''') is a [[character string (computer science)|string]] of [[character (computing)|character]]s designed for unambiguous [[identifier|identification]] of [[resource (computer science)|resources]] and extensibility via the URI scheme.

Such identification enables interaction with representations of the resource over a network, typically the [[World Wide Web]], using specific [[Communications protocol|protocols]]. Schemes specifying a concrete [[syntax]] and associated protocols define each URI. The most common form of URI is the Uniform Resource Locator ([[URL]]), frequently referred to informally as a ''web address.'' More rarely seen in usage is the [[Uniform Resource Name|Uniform Resource Name (URN)]], which was designed to complement URLs by providing a mechanism for the identification of resources in particular [[namespace]]s.

==URL and URN==
A [[Uniform Resource Name]] (URN) is a URI that identifies a resource by name in a particular namespace. A URN may be used to talk about a resource without implying its location or how to access it. For example, in the [[International Standard Book Number|International Standard Book Number (ISBN)]] system, ''<nowiki>ISBN</nowiki> 0-486-27557-4'' identifies a specific edition of Shakespeare's play ''[[Romeo and Juliet]]''. The URN for that edition would be ''<nowiki>urn:isbn:0-486-27557-4</nowiki>''. However, it gives no information as to where to find a copy of that book.

A [[Uniform Resource Locator]] (URL) is a URI that specifies the means of acting upon or obtaining the representation of a resource, i.e. specifying both its primary access mechanism and network location. For example, the URL <code><nowiki>http://example.org/wiki/Main_Page</nowiki></code> refers to a resource identified as <code><nowiki>/wiki/Main_Page</nowiki></code> whose representation, in the form of [[HTML]] and related code, is obtainable via the [[Hypertext Transfer Protocol]] (''http:'') from a network host whose [[domain name]] is <code><nowiki>example.org</nowiki></code>.

A URN may be compared to a person's name, while a URL may be compared to their street address. In other words, a URN identifies an item and a URL provides a method for finding it.

Technical publications, especially standards produced by the [[Internet Engineering Task Force|IETF]] and by the [[World Wide Web Consortium|W3C]], normally reflect a view outlined in a [[W3C Recommendation]] of 2001, which acknowledges the precedence of the term URI rather than endorsing any formal subdivision into URL and URN. {{cquote|URL is a useful but informal concept: a URL is a type of URI that identifies a resource via a representation of its primary access mechanism (e.g., its network "location"), rather than by some other attributes it may have.{{sfnp|Joint W3C/IETF URI Planning Interest Group|2001}}}}

As such, a URL is simply a URI that happens to point to a resource over a network.{{efn|A report published in 2002 by a joint W3C/IETF working group aimed to normalize the divergent views held within the IETF and W3C over the relationship between the various 'UR*' terms and standards. While not published as a full standard by either organization, it has become the basis for the above common understanding and has informed many standards since then.}}{{sfnp|Joint W3C/IETF URI Planning Interest Group|2002}} However, in non-technical contexts and in software for the World Wide Web, the term "URL" remains widely used. Additionally, the term "web address" (which has no formal definition) often occurs in non-technical publications as a synonym for a URI that uses the ''http'' or ''https'' schemes. Such assumptions can lead to confusion, for example, in the case of XML namespaces that have a [[#Relation to XML namespaces|visual similarity to resolvable URIs]].

Specifications produced by the [[WHATWG]] prefer URL over URI, and so newer HTML5 APIs use URL over URI.<ref>{{cite web |title=URL Standard: 6.3. URL APIs elsewhere |url=https://url.spec.whatwg.org/#url-apis-elsewhere}}</ref> {{cquote|Standardize on the term URL. URI and IRI are just confusing. In practice a single algorithm is used for both so keeping them distinct is not helping anyone. URL also easily wins the search result popularity contest.<ref>{{cite web |title=URL Standard: Goals|url=https://url.spec.whatwg.org/#goals}}</ref>}}

While most URI schemes were originally designed to be used with a particular [[protocol (computing)|protocol]], and often have the same name, they are semantically different from protocols. For example, the scheme ''http'' is generally used for interacting with [[web resource]]s using HTTP, but the scheme ''[[file URI scheme|file]]'' has no protocol.

==Generic syntax==
===Definition===
Each URI begins with a scheme name that refers to a specification for assigning identifiers within that scheme. As such, the URI syntax is a federated and extensible naming system wherein each scheme's specification may further restrict the syntax and semantics of identifiers using that scheme. The URI generic syntax is a superset of the syntax of all URI schemes. It was first defined in [[Request for Comments|Request for Comments (RFC)]] 2396, published in August 1998,{{sfnp|RFC 2396|1998}} and finalized in <nowiki>RFC</nowiki> 3986, published in January 2005.{{sfnp|RFC 3986|2005}}
<section begin=syntax />
The ''URI generic syntax'' consists of a hierarchical sequence of five ''components'':{{sfnp|RFC 3986|2005|loc=§3}}

<pre>
URI = scheme:[//authority]path[?query][#fragment]
</pre>

where the authority component divides into three ''subcomponents'':

<pre>
authority = [userinfo@]host[:port]
</pre>

It comprises:
* A non-empty '''{{visible anchor|scheme}}''' component followed by a colon (<code>:</code>), consisting of a sequence of characters beginning with a letter and followed by any combination of letters, digits, plus (<code>+</code>), period (<code>.</code>), or hyphen (<code>-</code>). Although schemes are case-insensitive, the canonical form is lowercase and documents that specify schemes must do so with lowercase letters. Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>[[skype]]</code>, <code>steam</code>, <code>[[svn]]</code>, <code>cvs</code>, <code>[[telnet]]</code>, <code>[[sms]]</code>, <code>[[smtp]]</code>, <code>[[ldap]]</code>, <code>dav</code>, <code>[[jabber]]</code>, <code>[[xmpp]]</code>, <code>[[udp]]</code>, <code>view-source</code>, <code>[[websockets|ws]]</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>. Unpopular schemes include <code>xri</code>, <code>tag</code>, <code>lastfm</code>, <code>[[mongodb]]</code>, <code>msword</code>, <code>moz</code>, <code>palm</code>, <code>paparazzi</code>, <code>com-eventbrite-attendee</code>, <code>dina-playsingle</code>, <code>[[adium]]xtra</code>, <code>urn</code>, <code>onenote</code>, <code>rediss</code> (''sic''), <code>ipp</code>, <code>things</code>, <code>stuns</code>, <code>prospero</code>, <code>iotdisco</code>, <code>tip</code>, <code>tool</code>, <code>tv</code>, <code>unreal</code>, <code>submit</code>, <code>market</code>, <code>ham</code>, <code>fish</code>, <code>finger</code>, <code>iris</code>, <code>[[IP over avian carriers]]</code>, and <code>[[Gopher (protocol)|gopher]].<ref>https://www.iana.org/assignments/uri-schemes/uri-schemes.xhtml</ref> URI schemes should be registered with the [[Internet Assigned Numbers Authority|Internet Assigned Numbers Authority (IANA)]], although non-registered schemes are used in practice.{{efn|The procedures for registering new URI schemes were originally defined in 1999 by <nowiki>RFC 2717</nowiki>, and are now defined by <nowiki>RFC 7595</nowiki>, published in June 2015.{{sfnp|IETF|2015}}}}
* An optional non-empty '''authority''' component preceded by two slashes (<code>//</code>), comprising:
** An optional '''userinfo''' subcomponent that may consist of a [[User (computing)|user name]] and an optional [[password]] preceded by a colon (<code>:</code>), followed by an at symbol (<code>@</code>). Use of the format <code>username:password</code> in the userinfo subcomponent is deprecated for security reasons. Applications should not render as clear text any data after the first colon (<code>:</code>) found within a userinfo subcomponent unless the data after the colon is the empty string (indicating no password).
** A non-empty '''host''' subcomponent, consisting of either a registered name (including but not limited to a [[hostname]]), or an [[IP address]]. [[IPv4]] addresses must be in [[dot-decimal notation]], and [[IPv6]] addresses must be enclosed in brackets (<code>[]</code>).{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{efn|For URIs relating to resources on the World Wide Web, some web browsers allow {{code|.0}} portions of dot-decimal notation to be dropped or raw integer IP addresses to be used.{{sfnp|Lawrence|2014}}}}
** An optional '''port''' subcomponent preceded by a colon (<code>:</code>).
* A '''path''' component, consisting of a sequence of path segments separated by a slash (<code>/</code>). A path is always defined for a URI, though the defined path may be empty (zero length). A segment may also be empty, resulting in two consecutive slashes (<code>//</code>) in the path component. A path component may resemble or map exactly to a [[Path (computing)|file system path]], but does not always imply a relation to one. If an authority component is present, then the path component must either be empty or begin with a slash (<code>/</code>). If an authority component is absent, then the path cannot begin with an empty segment, that is with two slashes (<code>//</code>), as the following characters would be interpreted as an authority component.{{sfnp|RFC 2396|1998|loc=§3.3}} The final segment of the path may be referred to as a '[[Clean URL#Slug|slug]]'.

{| class="wikitable" style="float: right; font-size: 0.9em; margin-left: 1em"
|-
! Query delimiter
! Example
|-
| Ampersand (<code>&</code>)
| <code>key1=value1&key2=value2</code>
|-
| Semicolon (<code>;</code>){{efn|Historic <nowiki>RFC 1866</nowiki> (obsoleted by <nowiki>RFC 2854</nowiki>) encourages CGI authors to support ';' in addition to '&'.{{sfnp|RFC 1866|1995|loc=§8.2.1}}}}{{Incomplete short citation|date=August 2016}}
| <code>key1=value1;key2=value2</code>
|}
* An optional '''query''' component preceded by a question mark (<code>?</code>), containing a [[query string]] of non-hierarchical data. Its syntax is not well defined, but by convention is most often a sequence of [[attribute–value pair]]s separated by a [[delimiter]].
* An optional '''fragment''' component preceded by an [[Number sign|hash]] (<code>#</code>). The fragment contains a [[fragment identifier]] providing direction to a secondary resource, such as a section heading in an article identified by the remainder of the URI. When the primary resource is an [[HTML]] document, the fragment is often an [[HTML#Attributes|<code>id</code> attribute]] of a specific element, and web browsers will scroll this element into view.<section end=syntax />

Strings of data [[Octet (computing)|octets]] within a URI are represented as characters. Permitted characters within a URI are the [[ASCII]] characters for the lowercase and uppercase letters of the modern [[English alphabet]], the [[Arabic numerals]], [[hyphen]], [[Full stop|period]], [[underscore]], and [[tilde]].{{sfnp|RFC 3986|2005|loc=§2}} Octets represented by any other character must be [[percent-encoding|percent-encoded]].

Of the ASCII character set, the characters <code>: / ? # [ ] @</code> are reserved for use as delimiters of the generic URI components and must be percent-encoded — for example, <code>%3F</code> for a question mark.{{sfnp|RFC 3986|2005|loc=§2.2}} The characters <code>! $ & ' ( ) * + , ; =</code> are permitted by generic URI syntax to be used unencoded in the user information, host, and path as delimiters.{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{sfnp|RFC 3986|2005|loc=§3.3}} Additionally, <code>:</code> and <code>@</code> may appear unencoded within the path, query, and fragment; and <code>?</code> and <code>/</code> may appear unencoded as data within the query or fragment.{{sfnp|RFC 3986|2005|loc=§3.3}}{{sfnp|RFC 3986|2005|loc=§3.4}}

===Examples===
The following figure displays example URIs and their component parts.

<pre>
userinfo host port
┌───┴──┐ ┌──────┴──────┐ ┌┴┐
https://john.doe@www.example.com:123/forum/questions/?tag=networking&order=newest#top
└─┬─┘ └─────────────┬────────────┘└───────┬───────┘ └────────────┬────────────┘ └┬┘
scheme authority path query fragment

ldap://[2001:db8::7]/c=GB?objectClass?one
└─┬┘ └─────┬─────┘└─┬─┘ └──────┬──────┘
scheme authority path query

mailto:John.Doe@example.com
└──┬─┘ └─────────┬────────┘
scheme path

news:comp.infosystems.www.servers.unix
└─┬┘ └───────────────┬───────────────┘
scheme path

tel:+1-816-555-1212
└┬┘ └──────┬──────┘
scheme path

telnet://192.0.2.16:80/
└──┬─┘ └─────┬─────┘│
scheme authority path

urn:oasis:names:specification:docbook:dtd:xml:4.1.2
└┬┘ └──────────────────────┬──────────────────────┘
scheme path
</pre>

==URI references==
===Definition===
A ''URI reference'' is either a URI, or a ''relative reference'' when it does not begin with a scheme component followed by a colon (<code>:</code>).{{sfnp|RFC 3986|2005|loc=§4.1}} A path segment that contains a colon character (e.g., <code>foo:bar</code>) cannot be used as the first path segment of a relative reference if its path component does not begin with a slash (<code>/</code>), as it would be mistaken for a scheme component. Such a path segment must be preceded by a dot path segment (e.g., <code>./foo:bar</code>).{{sfnp|RFC 3986|2005|loc=§4.2}}

Web document [[markup language]]s frequently use URI references to point to other resources, such as external documents or specific portions of the same logical document:{{sfnp|RFC 3986|2005|loc=§4.4}}
* in [[HTML]], the value of the <code>src</code> attribute of the <code>img</code> element provides a URI reference, as does the value of the <code>href</code> attribute of the <code>a</code> or <code>link</code> element;
* in [[XML]], the [[system identifier]] appearing after the <code>SYSTEM</code> keyword in a [[Document Type Definition|DTD]] is a fragmentless URI reference;
* in [[XSLT]], the value of the <code>href</code> attribute of the <code>xsl:import</code> element/instruction is a URI reference; likewise the first argument to the <code>document()</code> function.

===Examples===
<pre>
https://example.com/path/resource.txt#fragment
//example.com/path/resource.txt
/path/resource.txt
path/resource.txt
/path/resource.txt
../resource.txt
./resource.txt
resource.txt
#fragment
</pre>

===Suffix references===
As URI usage has become commonplace, traditional media (television, radio, newspapers, billboards, etc.) have increasingly used a suffix of the URI as a reference, consisting of only the authority and path portions of the URI, such as

<pre>
www.w3.org/Addressing/
</pre>

Such references are primarily intended for human interpretation rather than for machines, with the assumption that context-based heuristics are sufficient to complete the URI (e.g., most registered names beginning with <code>www</code> are likely to have a URI prefix of <code>http://</code>). Although there is no standard set of heuristics for disambiguating a URI suffix, many client implementations allow them to be entered by the user and heuristically resolved. Although this practice of using suffix references is common, it should be avoided whenever possible and should never be used in situations where long-term references are expected, as the heuristics will change over time, particularly when a new URI scheme becomes popular, and are often incorrect when used out of context. Furthermore, they can lead to security issues along the lines of those described in <nowiki>RFC</nowiki> 1535. As a URI suffix has the same syntax as a relative reference with a relative path, a suffix reference cannot be used in contexts where a relative reference is expected. As a result, suffix references are limited to places where there is no defined base URI, such as dialog boxes and off-line advertisements.{{sfnp|RFC 3986|2005|loc=§4.5}}

==URI resolution==
===Definition===
An ''absolute URI'' is a URI with no fragment component.

''Resolving'' a URI reference against a ''base URI'' results in a ''target URI''. This implies that the base URI exists and is an absolute URI. The base URI can be obtained, in order of precedence, from:{{sfnp|RFC 3986|2005|loc=§5.1}}

* the reference URI itself if it is a URI;
* the content of the representation;
* the entity encapsulating the representation;
* the URI used for the actual retrieval of the representation;
* the context of the application.

===Examples===
Within a representation with a well defined base URI of

<pre>
http://a/b/c/d;p?q
</pre>

a relative reference is resolved to its target URI as follows:{{sfnp|RFC 3986|2005|loc=§5.4}}

<pre>
"g:h" -> "g:h"
"g" -> "http://a/b/c/g"
"./g" -> "http://a/b/c/g"
"g/" -> "http://a/b/c/g/"
"/g" -> "http://a/g"
"//g" -> "http://g"
"?y" -> "http://a/b/c/d;p?y"
"g?y" -> "http://a/b/c/g?y"
"#s" -> "http://a/b/c/d;p?q#s"
"g#s" -> "http://a/b/c/g#s"
"g?y#s" -> "http://a/b/c/g?y#s"
";x" -> "http://a/b/c/;x"
"g;x" -> "http://a/b/c/g;x"
"g;x?y#s" -> "http://a/b/c/g;x?y#s"
"" -> "http://a/b/c/d;p?q"
"." -> "http://a/b/c/"
"./" -> "http://a/b/c/"
".." -> "http://a/b/"
"../" -> "http://a/b/"
"../g" -> "http://a/b/g"
"../.." -> "http://a/"
"../../" -> "http://a/"
"../../g" -> "http://a/g"
</pre>

==History==
===Naming, addressing, and identifying resources===
URIs and URLs have a shared history. In 1994, [[Tim Berners-Lee|Tim Berners-Lee's]] proposals for [[hypertext]]{{sfnp|Palmer|2001}} implicitly introduced the idea of a URL as a short string representing a resource that is the target of a [[hyperlink]]. At the time, people referred to it as a "hypertext name"{{sfnp|W3C|1992}} or "document name".

Over the next three and a half years, as the World Wide Web's core technologies of HTML, HTTP, and web browsers developed, a need to distinguish a string that provided an address for a resource from a string that merely named a resource emerged. Although not yet formally defined, the term ''Uniform Resource Locator'' came to represent the former, and the more contentious ''Uniform Resource Name'' came to represent the latter.

During the debate over defining URLs and URNs it became evident that the two concepts embodied by the terms were merely aspects of the fundamental, overarching notion of resource ''identification''. In June 1994, the IETF published Berners-Lee's <nowiki>RFC 1630</nowiki>: the first Request for Comments that acknowledged the existence of URLs and URNs, and, more importantly, defined a formal syntax for ''Universal Resource Identifiers'' — URL-like strings whose precise syntaxes and semantics depended on their schemes. In addition, this RFC attempted to summarize the syntaxes of URL schemes in use at the time. It also acknowledged, but did not standardize, the existence of relative URLs and fragment identifiers.

===Refinement of specifications===
In December 1994, <nowiki>RFC 1738</nowiki> formally defined relative and absolute URLs, refined the general URL syntax, defined how to resolve relative URLs to absolute form, and better enumerated the URL schemes then in use. The agreed definition and syntax of URNs had to wait until the publication of <nowiki>RFC 2141</nowiki> in May 1997.

The publication of <nowiki>RFC 2396</nowiki> in August 1998 saw the URI syntax become a separate specification{{sfnp|RFC 2396|1998}} and most of the parts of RFCs 1630 and 1738 relating to URIs and URLs in general were revised and expanded by the [[IETF]]. The new RFC changed the meaning of "U" in "URI" to "Uniform" from "Universal".

In December 1999, <nowiki>RFC 2732</nowiki> provided a minor update to <nowiki>RFC 2396</nowiki>, allowing URIs to accommodate [[IPv6]] addresses. A number of shortcomings discovered in the two specifications led to a community effort, coordinated by <nowiki>RFC 2396</nowiki> co-author [[Roy Fielding]], that culminated in the publication of <nowiki>RFC 3986</nowiki> in January 2005. While obsoleting the prior standard, it did not render the details of existing URL schemes obsolete; <nowiki>RFC 1738</nowiki> continues to govern such schemes except where otherwise superseded. <nowiki>RFC 2616</nowiki> for example, refines the <code>http</code> scheme. Simultaneously, the IETF published the content of <nowiki>RFC 3986</nowiki> as the full standard STD 66, reflecting the establishment of the URI generic syntax as an official Internet protocol.

In 2001, the W3C's Technical Architecture Group (TAG) published a guide to [[best practices]] and canonical URIs for publishing multiple versions of a given resource.{{sfnp|W3C|2001}} For example, content might differ by language or by size to adjust for capacity or settings of the device used to access that content.

In August 2002, <nowiki>RFC 3305</nowiki> pointed out that the term "URL" had, despite widespread public use, faded into near obsolescence, and serves only as a reminder that some URIs act as addresses by having schemes implying network accessibility, regardless of any such actual use. As URI-based standards such as [[Resource Description Framework]] make evident, resource identification need not suggest the retrieval of resource representations over the Internet, nor need they imply network-based resources at all.

The [[Semantic Web]] uses the HTTP URI scheme to identify both documents and concepts in the real world, a distinction which has caused confusion as to how to distinguish the two. The TAG published an e-mail in 2005 on how to solve the problem, which became known as the ''httpRange-14 resolution''.{{sfnp|Fielding|2005}} The W3C subsequently published an Interest Group Note titled ''Cool URIs for the Semantic Web'',{{sfnp|W3C|2008}} which explained the use of [[content negotiation]] and the [[HTTP 303]] response code for redirections in more detail.

==Relation to XML namespaces==
In [[XML]], a [[XML namespace|namespace]] is an abstract domain to which a collection of element and attribute names can be assigned. The namespace name is a character string which must adhere to the generic URI syntax.{{sfnp|Morrison|2006}} However, the name is generally not considered to be a URI,{{sfnp|Harold|2004}} because the URI specification bases the decision not only on lexical components, but also on their intended use. A namespace name does not necessarily imply any of the semantics of URI schemes; for example, a namespace name beginning with ''http:'' may have no connotation to the use of the [[HTTP]].

Originally, the namespace name could match the syntax of any non-empty URI reference, but the use of relative URI references was deprecated by the W3C.{{sfnp|W3C|2009}} A separate W3C specification for namespaces in XML 1.1 permits [[Internationalized resource identifier|internationalized resource identifier (IRI)]] references to serve as the basis for namespace names in addition to URI references.{{sfnp|W3C|2006}}

==See also==
* [[CURIE]] – defines a generic, abbreviated syntax for expressing URIs
* [[Dereferenceable Uniform Resource Identifier]] – a resource retrieval mechanism that uses any of the internet protocols (e.g. HTTP) to obtain a copy or representation of the resource it identifies
* [[Extensible Resource Identifier]] – a scheme and resolution protocol for abstract identifiers compatible with URIs
* [[Internationalized Resource Identifier]] (IRI) – a generalization of URIs allowing the use of Unicode
* [[Persistent uniform resource locator]] (PURL) – a URI that is used to redirect to the location of the requested web resource
* [[Uniform Naming Convention]] – a common syntax used by Microsoft to describe the location of a network resource, such as a shared file, directory, or printer
* [[Resource Directory Description Language]] – a descriptive language to provide machine- and human-readable information about a particular namespace and about the XML documents that use it
* [[Universally unique identifier|UUID]]

== Notes ==
{{Notelist}}

== References ==
=== Citations ===
{{Reflist|25em}}

=== Cited works ===
{{refbegin|32em}}
* {{cite web |first=Roy T.|last=Fielding|authorlink=Roy Fielding |title = [httpRange-14] Resolved |url = http://lists.w3.org/Archives/Public/www-tag/2005Jun/0039.html|date=18 June 2005 |accessdate=24 July 2009|ref=harv}}
* {{cite book |first=Elliotte Rusty|last=Harold |authorlink=Elliotte Rusty Harold |year=2004 |title=XML 1.1 Bible |edition=Third |publisher=[[Wiley Publishing]] |page=291 |isbn=0-7645-4986-3 |ref=harv }}
* {{cite web |url = http://www.w3.org/TR/uri-clarification/ |author=Joint W3C/IETF URI Planning Interest Group|title=URIs, URLs, and URNs: Clarifications and Recommendations 1.0|date=21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2001}}}}
* {{cite web |url = https://tools.ietf.org/html/rfc3305|title=Report from the Joint W3C/IETF URI Planning Interest Group: Uniform Resource Identifiers (URIs), URLs, and Uniform Resource Names (URNs): Clarifications and Recommendations|editor1-first=M.|editor1-last=Mealling |editor2-first=R.|editor2-last=Denenberg|publisher=[[World Wide Web Consortium]]|date=August 2002 |accessdate=13 September 2015 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2002}}}}
* {{cite web |url=https://tools.ietf.org/html/rfc7595|title=Guidelines and Registration Procedures for URI Schemes|editor-first=D.|editor-last=Thaler|author1-first=T.|author1-last=Hansen|author2-first=T.|author2-last=Hardie|publisher=[[Internet Engineering Task Force]]|date=June 2015|issn=2070-1721|ref={{SfnRef|IETF|2015}}}}
* {{cite book |last=Morrison|first=Michael|year=2006|title=Sams Teach Yourself XML|publisher=[[Sams Publishing]]|chapter=Hour 5: ''Putting Namespaces to Use''|page=91 |ref=harv }}
* {{cite web |first=Sean B.|last=Palmer |title=The Early History of HTML |url = http://infomesh.net/html/history/early/ |year=2001 |accessdate=2009-04-30 |ref=harv }}
* {{cite web |author = URI Planning Interest Group, W3C/IETF |title = URIs, URLs, and URNs: Clarifications and Recommendations 1.0 |url = http://www.w3.org/TR/uri-clarification/ |date = 21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|URI Planning Interest Group|2009}}}}
* {{cite web |url = http://www.w3.org/History/19921103-hypertext/hypertext/WWW/Addressing/Addressing.html|title=W3 Naming Schemes |publisher=[[World Wide Web Consortium]]|year=1992|accessdate=2009-07-24|ref={{SfnRef|W3C|1992}}}}
* {{cite web |url = http://www.w3.org/2001/tag/doc/alternatives-discovery.html |title=On Linking Alternative Representations To Enable Discovery And Publishing|publisher=[[World Wide Web Consortium]]|year=2006|orig-year=2001|accessdate=2012-04-03|ref={{SfnRef|W3C|2001}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use|title=Namespaces in XML 1.1 (Second Edition)|date=16 August 2006 |at = 2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray|editor2-first=Dave |editor2-last=Hollander|editor3-first=Andrew|editor3-last=Layman|editor4-first=Richard|editor4-last=Tobin|publisher=[[World Wide Web Consortium]] |accessdate=31 August 2015|ref={{SfnRef|W3C|2006}}}}
* {{cite web |url = http://www.w3.org/TR/cooluris/|title=Cool URIs for the Semantic Web|editor1-first=Leo|editor1-last=Sauermann|editor2-first=Richard|editor2-last=Cyganiak|author1-first=Danny|author1-last=Ayers|author2-first=Max|author2-last=Völkel|publisher=[[World Wide Web Consortium]]|date=3 December 2008|accessdate=2012-04-03|ref={{SfnRef|W3C|2008}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use |title=Namespaces in XML 1.0 (Third Edition)|date=8 December 2009 |at=2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray |editor2-first=Dave |editor2-last=Hollander |editor3-first=Andrew|editor3-last=Layman |editor4-first=Richard|editor4-last=Tobin |editor5-first=Henry S. |editor5-last=Thompson |publisher=[[World Wide Web Consortium]]|accessdate=31 August 2015 |ref={{SfnRef|W3C|2009}} }}
* {{cite web |url = https://tools.ietf.org/html/rfc1866#section-8.2.1 |title=Hypertext Markup Language - 2.0|author1-first=Tim|author1-last=Berners-Lee |author1-link=Tim Berners-Lee|author2-first=Dan|author2-last=Connolly|publisher=[[Internet Engineering Task Force]] |date=November 1995 |accessdate=13 September 2015}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc2396 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=2396 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=August 1998 |accessdate=31 August 2015 |ref={{SfnRef|RFC 2396|1998}}}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc3986 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=3986 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=January 2005 |accessdate=31 August 2015 |ref={{SfnRef|RFC 3986|2005}}}}
* {{cite web |last1=Lawrence |first1=Eric |title=Browser Arcana: IP Literals in URLs |url = http://blogs.msdn.com/b/ieinternals/archive/2014/03/06/browser-arcana-ipv4-ipv6-literal-urls-dotted-va-dotless.aspx |website=IEInternals |publisher=[[Microsoft]] |date=6 March 2014 |accessdate=2016-04-25 |ref = harv }}
{{refend}}

==External links==
* [http://www.iana.org/assignments/uri-schemes.html URI Schemes] – [[Internet Assigned Numbers Authority|IANA]]-maintained registry of URI Schemes
* [http://www.w3.org/wiki/UriSchemes URI schemes on the W3C wiki]
* [http://www.w3.org/TR/webarch/#identification Architecture of the World Wide Web, Volume One, §2: Identification] – by W3C
* [http://www.w3.org/TR/uri-clarification/ W3C URI Clarification]

{{Semantic Web|state=collapsed}}
{{URI scheme}}
{{Hypermedia}}

{{Authority control}}

[[Category:Application layer protocols]]
[[Category:Internet protocols]]
[[Category:Internet Standards]]
[[Category:Semantic Web| ]]
[[Category:Uniform Resource Locator]]

Uniform Resource Identifier

2018-06-29T19:37:29Z

Crasshopper: /* Definition */ typoes

{{Redirect|URI}}
{{good article}}

A '''Uniform Resource Identifier''' ('''URI''') is a [[character string (computer science)|string]] of [[character (computing)|character]]s designed for unambiguous [[identifier|identification]] of [[resource (computer science)|resources]] and extensibility via the URI scheme.

Such identification enables interaction with representations of the resource over a network, typically the [[World Wide Web]], using specific [[Communications protocol|protocols]]. Schemes specifying a concrete [[syntax]] and associated protocols define each URI. The most common form of URI is the Uniform Resource Locator ([[URL]]), frequently referred to informally as a ''web address.'' More rarely seen in usage is the [[Uniform Resource Name|Uniform Resource Name (URN)]], which was designed to complement URLs by providing a mechanism for the identification of resources in particular [[namespace]]s.

==URL and URN==
A [[Uniform Resource Name]] (URN) is a URI that identifies a resource by name in a particular namespace. A URN may be used to talk about a resource without implying its location or how to access it. For example, in the [[International Standard Book Number|International Standard Book Number (ISBN)]] system, ''<nowiki>ISBN</nowiki> 0-486-27557-4'' identifies a specific edition of Shakespeare's play ''[[Romeo and Juliet]]''. The URN for that edition would be ''<nowiki>urn:isbn:0-486-27557-4</nowiki>''. However, it gives no information as to where to find a copy of that book.

A [[Uniform Resource Locator]] (URL) is a URI that specifies the means of acting upon or obtaining the representation of a resource, i.e. specifying both its primary access mechanism and network location. For example, the URL <code><nowiki>http://example.org/wiki/Main_Page</nowiki></code> refers to a resource identified as <code><nowiki>/wiki/Main_Page</nowiki></code> whose representation, in the form of [[HTML]] and related code, is obtainable via the [[Hypertext Transfer Protocol]] (''http:'') from a network host whose [[domain name]] is <code><nowiki>example.org</nowiki></code>.

A URN may be compared to a person's name, while a URL may be compared to their street address. In other words, a URN identifies an item and a URL provides a method for finding it.

Technical publications, especially standards produced by the [[Internet Engineering Task Force|IETF]] and by the [[World Wide Web Consortium|W3C]], normally reflect a view outlined in a [[W3C Recommendation]] of 2001, which acknowledges the precedence of the term URI rather than endorsing any formal subdivision into URL and URN. {{cquote|URL is a useful but informal concept: a URL is a type of URI that identifies a resource via a representation of its primary access mechanism (e.g., its network "location"), rather than by some other attributes it may have.{{sfnp|Joint W3C/IETF URI Planning Interest Group|2001}}}}

As such, a URL is simply a URI that happens to point to a resource over a network.{{efn|A report published in 2002 by a joint W3C/IETF working group aimed to normalize the divergent views held within the IETF and W3C over the relationship between the various 'UR*' terms and standards. While not published as a full standard by either organization, it has become the basis for the above common understanding and has informed many standards since then.}}{{sfnp|Joint W3C/IETF URI Planning Interest Group|2002}} However, in non-technical contexts and in software for the World Wide Web, the term "URL" remains widely used. Additionally, the term "web address" (which has no formal definition) often occurs in non-technical publications as a synonym for a URI that uses the ''http'' or ''https'' schemes. Such assumptions can lead to confusion, for example, in the case of XML namespaces that have a [[#Relation to XML namespaces|visual similarity to resolvable URIs]].

Specifications produced by the [[WHATWG]] prefer URL over URI, and so newer HTML5 APIs use URL over URI.<ref>{{cite web |title=URL Standard: 6.3. URL APIs elsewhere |url=https://url.spec.whatwg.org/#url-apis-elsewhere}}</ref> {{cquote|Standardize on the term URL. URI and IRI are just confusing. In practice a single algorithm is used for both so keeping them distinct is not helping anyone. URL also easily wins the search result popularity contest.<ref>{{cite web |title=URL Standard: Goals|url=https://url.spec.whatwg.org/#goals}}</ref>}}

While most URI schemes were originally designed to be used with a particular [[protocol (computing)|protocol]], and often have the same name, they are semantically different from protocols. For example, the scheme ''http'' is generally used for interacting with [[web resource]]s using HTTP, but the scheme ''[[file URI scheme|file]]'' has no protocol.

==Generic syntax==
===Definition===
Each URI begins with a scheme name that refers to a specification for assigning identifiers within that scheme. As such, the URI syntax is a federated and extensible naming system wherein each scheme's specification may further restrict the syntax and semantics of identifiers using that scheme. The URI generic syntax is a superset of the syntax of all URI schemes. It was first defined in [[Request for Comments|Request for Comments (RFC)]] 2396, published in August 1998,{{sfnp|RFC 2396|1998}} and finalized in <nowiki>RFC</nowiki> 3986, published in January 2005.{{sfnp|RFC 3986|2005}}
<section begin=syntax />
The ''URI generic syntax'' consists of a hierarchical sequence of five ''components'':{{sfnp|RFC 3986|2005|loc=§3}}

<pre>
URI = scheme:[//authority]path[?query][#fragment]
</pre>

where the authority component divides into three ''subcomponents'':

<pre>
authority = [userinfo@]host[:port]
</pre>

It comprises:
* A non-empty '''{{visible anchor|scheme}}''' component followed by a colon (<code>:</code>), consisting of a sequence of characters beginning with a letter and followed by any combination of letters, digits, plus (<code>+</code>), period (<code>.</code>), or hyphen (<code>-</code>). Although schemes are case-insensitive, the canonical form is lowercase and documents that specify schemes must do so with lowercase letters. Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>skype</code>, <code>steam</code>, <code>svn</code>, <code>telnet</code>, <code>sms</code>, <code>smtp</code>, <code>ldap</code>, <code>jabber</code>, <code>xmpp</code>, <code>udp</code>, <code>view-source</code>, <code>ws</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>. Unpopular schemes include <code>xri</code>, <code>tag</code>, <code>lastfm</code>, <code>mongodb</code>, <code>msword</code>, <code>moz</code>, <code>palm</code>, <code>paparazzi</code>, <code>urn</code>, <code>onenote</code>, <code>rediss</code> (''sic''), <code>ipp</code>, <code>things</code>, <code>stuns</code>, <code>prospero</code>, <code>iotdisco</code>, <code>tip</code>, <code>tool</code>, <code>tv</code>, <code>unreal</code>, <code>submit</code>, <code>market</code>, <code>ham</code>, <code>fish</code>, <code>finger</code>, <code>iris</code>, <code>[[IP over avian carriers]]</code>, and <code>[[Gopher (protocol)|gopher]].<ref>https://www.iana.org/assignments/uri-schemes/uri-schemes.xhtml</ref> URI schemes should be registered with the [[Internet Assigned Numbers Authority|Internet Assigned Numbers Authority (IANA)]], although non-registered schemes are used in practice.{{efn|The procedures for registering new URI schemes were originally defined in 1999 by <nowiki>RFC 2717</nowiki>, and are now defined by <nowiki>RFC 7595</nowiki>, published in June 2015.{{sfnp|IETF|2015}}}}
* An optional non-empty '''authority''' component preceded by two slashes (<code>//</code>), comprising:
** An optional '''userinfo''' subcomponent that may consist of a [[User (computing)|user name]] and an optional [[password]] preceded by a colon (<code>:</code>), followed by an at symbol (<code>@</code>). Use of the format <code>username:password</code> in the userinfo subcomponent is deprecated for security reasons. Applications should not render as clear text any data after the first colon (<code>:</code>) found within a userinfo subcomponent unless the data after the colon is the empty string (indicating no password).
** A non-empty '''host''' subcomponent, consisting of either a registered name (including but not limited to a [[hostname]]), or an [[IP address]]. [[IPv4]] addresses must be in [[dot-decimal notation]], and [[IPv6]] addresses must be enclosed in brackets (<code>[]</code>).{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{efn|For URIs relating to resources on the World Wide Web, some web browsers allow {{code|.0}} portions of dot-decimal notation to be dropped or raw integer IP addresses to be used.{{sfnp|Lawrence|2014}}}}
** An optional '''port''' subcomponent preceded by a colon (<code>:</code>).
* A '''path''' component, consisting of a sequence of path segments separated by a slash (<code>/</code>). A path is always defined for a URI, though the defined path may be empty (zero length). A segment may also be empty, resulting in two consecutive slashes (<code>//</code>) in the path component. A path component may resemble or map exactly to a [[Path (computing)|file system path]], but does not always imply a relation to one. If an authority component is present, then the path component must either be empty or begin with a slash (<code>/</code>). If an authority component is absent, then the path cannot begin with an empty segment, that is with two slashes (<code>//</code>), as the following characters would be interpreted as an authority component.{{sfnp|RFC 2396|1998|loc=§3.3}} The final segment of the path may be referred to as a '[[Clean URL#Slug|slug]]'.

{| class="wikitable" style="float: right; font-size: 0.9em; margin-left: 1em"
|-
! Query delimiter
! Example
|-
| Ampersand (<code>&</code>)
| <code>key1=value1&key2=value2</code>
|-
| Semicolon (<code>;</code>){{efn|Historic <nowiki>RFC 1866</nowiki> (obsoleted by <nowiki>RFC 2854</nowiki>) encourages CGI authors to support ';' in addition to '&'.{{sfnp|RFC 1866|1995|loc=§8.2.1}}}}{{Incomplete short citation|date=August 2016}}
| <code>key1=value1;key2=value2</code>
|}
* An optional '''query''' component preceded by a question mark (<code>?</code>), containing a [[query string]] of non-hierarchical data. Its syntax is not well defined, but by convention is most often a sequence of [[attribute–value pair]]s separated by a [[delimiter]].
* An optional '''fragment''' component preceded by an [[Number sign|hash]] (<code>#</code>). The fragment contains a [[fragment identifier]] providing direction to a secondary resource, such as a section heading in an article identified by the remainder of the URI. When the primary resource is an [[HTML]] document, the fragment is often an [[HTML#Attributes|<code>id</code> attribute]] of a specific element, and web browsers will scroll this element into view.<section end=syntax />

Strings of data [[Octet (computing)|octets]] within a URI are represented as characters. Permitted characters within a URI are the [[ASCII]] characters for the lowercase and uppercase letters of the modern [[English alphabet]], the [[Arabic numerals]], [[hyphen]], [[Full stop|period]], [[underscore]], and [[tilde]].{{sfnp|RFC 3986|2005|loc=§2}} Octets represented by any other character must be [[percent-encoding|percent-encoded]].

Of the ASCII character set, the characters <code>: / ? # [ ] @</code> are reserved for use as delimiters of the generic URI components and must be percent-encoded — for example, <code>%3F</code> for a question mark.{{sfnp|RFC 3986|2005|loc=§2.2}} The characters <code>! $ & ' ( ) * + , ; =</code> are permitted by generic URI syntax to be used unencoded in the user information, host, and path as delimiters.{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{sfnp|RFC 3986|2005|loc=§3.3}} Additionally, <code>:</code> and <code>@</code> may appear unencoded within the path, query, and fragment; and <code>?</code> and <code>/</code> may appear unencoded as data within the query or fragment.{{sfnp|RFC 3986|2005|loc=§3.3}}{{sfnp|RFC 3986|2005|loc=§3.4}}

===Examples===
The following figure displays example URIs and their component parts.

<pre>
userinfo host port
┌───┴──┐ ┌──────┴──────┐ ┌┴┐
https://john.doe@www.example.com:123/forum/questions/?tag=networking&order=newest#top
└─┬─┘ └─────────────┬────────────┘└───────┬───────┘ └────────────┬────────────┘ └┬┘
scheme authority path query fragment

ldap://[2001:db8::7]/c=GB?objectClass?one
└─┬┘ └─────┬─────┘└─┬─┘ └──────┬──────┘
scheme authority path query

mailto:John.Doe@example.com
└──┬─┘ └─────────┬────────┘
scheme path

news:comp.infosystems.www.servers.unix
└─┬┘ └───────────────┬───────────────┘
scheme path

tel:+1-816-555-1212
└┬┘ └──────┬──────┘
scheme path

telnet://192.0.2.16:80/
└──┬─┘ └─────┬─────┘│
scheme authority path

urn:oasis:names:specification:docbook:dtd:xml:4.1.2
└┬┘ └──────────────────────┬──────────────────────┘
scheme path
</pre>

==URI references==
===Definition===
A ''URI reference'' is either a URI, or a ''relative reference'' when it does not begin with a scheme component followed by a colon (<code>:</code>).{{sfnp|RFC 3986|2005|loc=§4.1}} A path segment that contains a colon character (e.g., <code>foo:bar</code>) cannot be used as the first path segment of a relative reference if its path component does not begin with a slash (<code>/</code>), as it would be mistaken for a scheme component. Such a path segment must be preceded by a dot path segment (e.g., <code>./foo:bar</code>).{{sfnp|RFC 3986|2005|loc=§4.2}}

Web document [[markup language]]s frequently use URI references to point to other resources, such as external documents or specific portions of the same logical document:{{sfnp|RFC 3986|2005|loc=§4.4}}
* in [[HTML]], the value of the <code>src</code> attribute of the <code>img</code> element provides a URI reference, as does the value of the <code>href</code> attribute of the <code>a</code> or <code>link</code> element;
* in [[XML]], the [[system identifier]] appearing after the <code>SYSTEM</code> keyword in a [[Document Type Definition|DTD]] is a fragmentless URI reference;
* in [[XSLT]], the value of the <code>href</code> attribute of the <code>xsl:import</code> element/instruction is a URI reference; likewise the first argument to the <code>document()</code> function.

===Examples===
<pre>
https://example.com/path/resource.txt#fragment
//example.com/path/resource.txt
/path/resource.txt
path/resource.txt
/path/resource.txt
../resource.txt
./resource.txt
resource.txt
#fragment
</pre>

===Suffix references===
As URI usage has become commonplace, traditional media (television, radio, newspapers, billboards, etc.) have increasingly used a suffix of the URI as a reference, consisting of only the authority and path portions of the URI, such as

<pre>
www.w3.org/Addressing/
</pre>

Such references are primarily intended for human interpretation rather than for machines, with the assumption that context-based heuristics are sufficient to complete the URI (e.g., most registered names beginning with <code>www</code> are likely to have a URI prefix of <code>http://</code>). Although there is no standard set of heuristics for disambiguating a URI suffix, many client implementations allow them to be entered by the user and heuristically resolved. Although this practice of using suffix references is common, it should be avoided whenever possible and should never be used in situations where long-term references are expected, as the heuristics will change over time, particularly when a new URI scheme becomes popular, and are often incorrect when used out of context. Furthermore, they can lead to security issues along the lines of those described in <nowiki>RFC</nowiki> 1535. As a URI suffix has the same syntax as a relative reference with a relative path, a suffix reference cannot be used in contexts where a relative reference is expected. As a result, suffix references are limited to places where there is no defined base URI, such as dialog boxes and off-line advertisements.{{sfnp|RFC 3986|2005|loc=§4.5}}

==URI resolution==
===Definition===
An ''absolute URI'' is a URI with no fragment component.

''Resolving'' a URI reference against a ''base URI'' results in a ''target URI''. This implies that the base URI exists and is an absolute URI. The base URI can be obtained, in order of precedence, from:{{sfnp|RFC 3986|2005|loc=§5.1}}

* the reference URI itself if it is a URI;
* the content of the representation;
* the entity encapsulating the representation;
* the URI used for the actual retrieval of the representation;
* the context of the application.

===Examples===
Within a representation with a well defined base URI of

<pre>
http://a/b/c/d;p?q
</pre>

a relative reference is resolved to its target URI as follows:{{sfnp|RFC 3986|2005|loc=§5.4}}

<pre>
"g:h" -> "g:h"
"g" -> "http://a/b/c/g"
"./g" -> "http://a/b/c/g"
"g/" -> "http://a/b/c/g/"
"/g" -> "http://a/g"
"//g" -> "http://g"
"?y" -> "http://a/b/c/d;p?y"
"g?y" -> "http://a/b/c/g?y"
"#s" -> "http://a/b/c/d;p?q#s"
"g#s" -> "http://a/b/c/g#s"
"g?y#s" -> "http://a/b/c/g?y#s"
";x" -> "http://a/b/c/;x"
"g;x" -> "http://a/b/c/g;x"
"g;x?y#s" -> "http://a/b/c/g;x?y#s"
"" -> "http://a/b/c/d;p?q"
"." -> "http://a/b/c/"
"./" -> "http://a/b/c/"
".." -> "http://a/b/"
"../" -> "http://a/b/"
"../g" -> "http://a/b/g"
"../.." -> "http://a/"
"../../" -> "http://a/"
"../../g" -> "http://a/g"
</pre>

==History==
===Naming, addressing, and identifying resources===
URIs and URLs have a shared history. In 1994, [[Tim Berners-Lee|Tim Berners-Lee's]] proposals for [[hypertext]]{{sfnp|Palmer|2001}} implicitly introduced the idea of a URL as a short string representing a resource that is the target of a [[hyperlink]]. At the time, people referred to it as a "hypertext name"{{sfnp|W3C|1992}} or "document name".

Over the next three and a half years, as the World Wide Web's core technologies of HTML, HTTP, and web browsers developed, a need to distinguish a string that provided an address for a resource from a string that merely named a resource emerged. Although not yet formally defined, the term ''Uniform Resource Locator'' came to represent the former, and the more contentious ''Uniform Resource Name'' came to represent the latter.

During the debate over defining URLs and URNs it became evident that the two concepts embodied by the terms were merely aspects of the fundamental, overarching notion of resource ''identification''. In June 1994, the IETF published Berners-Lee's <nowiki>RFC 1630</nowiki>: the first Request for Comments that acknowledged the existence of URLs and URNs, and, more importantly, defined a formal syntax for ''Universal Resource Identifiers'' — URL-like strings whose precise syntaxes and semantics depended on their schemes. In addition, this RFC attempted to summarize the syntaxes of URL schemes in use at the time. It also acknowledged, but did not standardize, the existence of relative URLs and fragment identifiers.

===Refinement of specifications===
In December 1994, <nowiki>RFC 1738</nowiki> formally defined relative and absolute URLs, refined the general URL syntax, defined how to resolve relative URLs to absolute form, and better enumerated the URL schemes then in use. The agreed definition and syntax of URNs had to wait until the publication of <nowiki>RFC 2141</nowiki> in May 1997.

The publication of <nowiki>RFC 2396</nowiki> in August 1998 saw the URI syntax become a separate specification{{sfnp|RFC 2396|1998}} and most of the parts of RFCs 1630 and 1738 relating to URIs and URLs in general were revised and expanded by the [[IETF]]. The new RFC changed the meaning of "U" in "URI" to "Uniform" from "Universal".

In December 1999, <nowiki>RFC 2732</nowiki> provided a minor update to <nowiki>RFC 2396</nowiki>, allowing URIs to accommodate [[IPv6]] addresses. A number of shortcomings discovered in the two specifications led to a community effort, coordinated by <nowiki>RFC 2396</nowiki> co-author [[Roy Fielding]], that culminated in the publication of <nowiki>RFC 3986</nowiki> in January 2005. While obsoleting the prior standard, it did not render the details of existing URL schemes obsolete; <nowiki>RFC 1738</nowiki> continues to govern such schemes except where otherwise superseded. <nowiki>RFC 2616</nowiki> for example, refines the <code>http</code> scheme. Simultaneously, the IETF published the content of <nowiki>RFC 3986</nowiki> as the full standard STD 66, reflecting the establishment of the URI generic syntax as an official Internet protocol.

In 2001, the W3C's Technical Architecture Group (TAG) published a guide to [[best practices]] and canonical URIs for publishing multiple versions of a given resource.{{sfnp|W3C|2001}} For example, content might differ by language or by size to adjust for capacity or settings of the device used to access that content.

In August 2002, <nowiki>RFC 3305</nowiki> pointed out that the term "URL" had, despite widespread public use, faded into near obsolescence, and serves only as a reminder that some URIs act as addresses by having schemes implying network accessibility, regardless of any such actual use. As URI-based standards such as [[Resource Description Framework]] make evident, resource identification need not suggest the retrieval of resource representations over the Internet, nor need they imply network-based resources at all.

The [[Semantic Web]] uses the HTTP URI scheme to identify both documents and concepts in the real world, a distinction which has caused confusion as to how to distinguish the two. The TAG published an e-mail in 2005 on how to solve the problem, which became known as the ''httpRange-14 resolution''.{{sfnp|Fielding|2005}} The W3C subsequently published an Interest Group Note titled ''Cool URIs for the Semantic Web'',{{sfnp|W3C|2008}} which explained the use of [[content negotiation]] and the [[HTTP 303]] response code for redirections in more detail.

==Relation to XML namespaces==
In [[XML]], a [[XML namespace|namespace]] is an abstract domain to which a collection of element and attribute names can be assigned. The namespace name is a character string which must adhere to the generic URI syntax.{{sfnp|Morrison|2006}} However, the name is generally not considered to be a URI,{{sfnp|Harold|2004}} because the URI specification bases the decision not only on lexical components, but also on their intended use. A namespace name does not necessarily imply any of the semantics of URI schemes; for example, a namespace name beginning with ''http:'' may have no connotation to the use of the [[HTTP]].

Originally, the namespace name could match the syntax of any non-empty URI reference, but the use of relative URI references was deprecated by the W3C.{{sfnp|W3C|2009}} A separate W3C specification for namespaces in XML 1.1 permits [[Internationalized resource identifier|internationalized resource identifier (IRI)]] references to serve as the basis for namespace names in addition to URI references.{{sfnp|W3C|2006}}

==See also==
* [[CURIE]] – defines a generic, abbreviated syntax for expressing URIs
* [[Dereferenceable Uniform Resource Identifier]] – a resource retrieval mechanism that uses any of the internet protocols (e.g. HTTP) to obtain a copy or representation of the resource it identifies
* [[Extensible Resource Identifier]] – a scheme and resolution protocol for abstract identifiers compatible with URIs
* [[Internationalized Resource Identifier]] (IRI) – a generalization of URIs allowing the use of Unicode
* [[Persistent uniform resource locator]] (PURL) – a URI that is used to redirect to the location of the requested web resource
* [[Uniform Naming Convention]] – a common syntax used by Microsoft to describe the location of a network resource, such as a shared file, directory, or printer
* [[Resource Directory Description Language]] – a descriptive language to provide machine- and human-readable information about a particular namespace and about the XML documents that use it
* [[Universally unique identifier|UUID]]

== Notes ==
{{Notelist}}

== References ==
=== Citations ===
{{Reflist|25em}}

=== Cited works ===
{{refbegin|32em}}
* {{cite web |first=Roy T.|last=Fielding|authorlink=Roy Fielding |title = [httpRange-14] Resolved |url = http://lists.w3.org/Archives/Public/www-tag/2005Jun/0039.html|date=18 June 2005 |accessdate=24 July 2009|ref=harv}}
* {{cite book |first=Elliotte Rusty|last=Harold |authorlink=Elliotte Rusty Harold |year=2004 |title=XML 1.1 Bible |edition=Third |publisher=[[Wiley Publishing]] |page=291 |isbn=0-7645-4986-3 |ref=harv }}
* {{cite web |url = http://www.w3.org/TR/uri-clarification/ |author=Joint W3C/IETF URI Planning Interest Group|title=URIs, URLs, and URNs: Clarifications and Recommendations 1.0|date=21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2001}}}}
* {{cite web |url = https://tools.ietf.org/html/rfc3305|title=Report from the Joint W3C/IETF URI Planning Interest Group: Uniform Resource Identifiers (URIs), URLs, and Uniform Resource Names (URNs): Clarifications and Recommendations|editor1-first=M.|editor1-last=Mealling |editor2-first=R.|editor2-last=Denenberg|publisher=[[World Wide Web Consortium]]|date=August 2002 |accessdate=13 September 2015 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2002}}}}
* {{cite web |url=https://tools.ietf.org/html/rfc7595|title=Guidelines and Registration Procedures for URI Schemes|editor-first=D.|editor-last=Thaler|author1-first=T.|author1-last=Hansen|author2-first=T.|author2-last=Hardie|publisher=[[Internet Engineering Task Force]]|date=June 2015|issn=2070-1721|ref={{SfnRef|IETF|2015}}}}
* {{cite book |last=Morrison|first=Michael|year=2006|title=Sams Teach Yourself XML|publisher=[[Sams Publishing]]|chapter=Hour 5: ''Putting Namespaces to Use''|page=91 |ref=harv }}
* {{cite web |first=Sean B.|last=Palmer |title=The Early History of HTML |url = http://infomesh.net/html/history/early/ |year=2001 |accessdate=2009-04-30 |ref=harv }}
* {{cite web |author = URI Planning Interest Group, W3C/IETF |title = URIs, URLs, and URNs: Clarifications and Recommendations 1.0 |url = http://www.w3.org/TR/uri-clarification/ |date = 21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|URI Planning Interest Group|2009}}}}
* {{cite web |url = http://www.w3.org/History/19921103-hypertext/hypertext/WWW/Addressing/Addressing.html|title=W3 Naming Schemes |publisher=[[World Wide Web Consortium]]|year=1992|accessdate=2009-07-24|ref={{SfnRef|W3C|1992}}}}
* {{cite web |url = http://www.w3.org/2001/tag/doc/alternatives-discovery.html |title=On Linking Alternative Representations To Enable Discovery And Publishing|publisher=[[World Wide Web Consortium]]|year=2006|orig-year=2001|accessdate=2012-04-03|ref={{SfnRef|W3C|2001}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use|title=Namespaces in XML 1.1 (Second Edition)|date=16 August 2006 |at = 2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray|editor2-first=Dave |editor2-last=Hollander|editor3-first=Andrew|editor3-last=Layman|editor4-first=Richard|editor4-last=Tobin|publisher=[[World Wide Web Consortium]] |accessdate=31 August 2015|ref={{SfnRef|W3C|2006}}}}
* {{cite web |url = http://www.w3.org/TR/cooluris/|title=Cool URIs for the Semantic Web|editor1-first=Leo|editor1-last=Sauermann|editor2-first=Richard|editor2-last=Cyganiak|author1-first=Danny|author1-last=Ayers|author2-first=Max|author2-last=Völkel|publisher=[[World Wide Web Consortium]]|date=3 December 2008|accessdate=2012-04-03|ref={{SfnRef|W3C|2008}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use |title=Namespaces in XML 1.0 (Third Edition)|date=8 December 2009 |at=2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray |editor2-first=Dave |editor2-last=Hollander |editor3-first=Andrew|editor3-last=Layman |editor4-first=Richard|editor4-last=Tobin |editor5-first=Henry S. |editor5-last=Thompson |publisher=[[World Wide Web Consortium]]|accessdate=31 August 2015 |ref={{SfnRef|W3C|2009}} }}
* {{cite web |url = https://tools.ietf.org/html/rfc1866#section-8.2.1 |title=Hypertext Markup Language - 2.0|author1-first=Tim|author1-last=Berners-Lee |author1-link=Tim Berners-Lee|author2-first=Dan|author2-last=Connolly|publisher=[[Internet Engineering Task Force]] |date=November 1995 |accessdate=13 September 2015}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc2396 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=2396 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=August 1998 |accessdate=31 August 2015 |ref={{SfnRef|RFC 2396|1998}}}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc3986 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=3986 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=January 2005 |accessdate=31 August 2015 |ref={{SfnRef|RFC 3986|2005}}}}
* {{cite web |last1=Lawrence |first1=Eric |title=Browser Arcana: IP Literals in URLs |url = http://blogs.msdn.com/b/ieinternals/archive/2014/03/06/browser-arcana-ipv4-ipv6-literal-urls-dotted-va-dotless.aspx |website=IEInternals |publisher=[[Microsoft]] |date=6 March 2014 |accessdate=2016-04-25 |ref = harv }}
{{refend}}

==External links==
* [http://www.iana.org/assignments/uri-schemes.html URI Schemes] – [[Internet Assigned Numbers Authority|IANA]]-maintained registry of URI Schemes
* [http://www.w3.org/wiki/UriSchemes URI schemes on the W3C wiki]
* [http://www.w3.org/TR/webarch/#identification Architecture of the World Wide Web, Volume One, §2: Identification] – by W3C
* [http://www.w3.org/TR/uri-clarification/ W3C URI Clarification]

{{Semantic Web|state=collapsed}}
{{URI scheme}}
{{Hypermedia}}

{{Authority control}}

[[Category:Application layer protocols]]
[[Category:Internet protocols]]
[[Category:Internet Standards]]
[[Category:Semantic Web| ]]
[[Category:Uniform Resource Locator]]

Uniform Resource Identifier

2018-06-29T19:36:05Z

Crasshopper: /* Definition */ added more examples of URI schemes

{{Redirect|URI}}
{{good article}}

A '''Uniform Resource Identifier''' ('''URI''') is a [[character string (computer science)|string]] of [[character (computing)|character]]s designed for unambiguous [[identifier|identification]] of [[resource (computer science)|resources]] and extensibility via the URI scheme.

Such identification enables interaction with representations of the resource over a network, typically the [[World Wide Web]], using specific [[Communications protocol|protocols]]. Schemes specifying a concrete [[syntax]] and associated protocols define each URI. The most common form of URI is the Uniform Resource Locator ([[URL]]), frequently referred to informally as a ''web address.'' More rarely seen in usage is the [[Uniform Resource Name|Uniform Resource Name (URN)]], which was designed to complement URLs by providing a mechanism for the identification of resources in particular [[namespace]]s.

==URL and URN==
A [[Uniform Resource Name]] (URN) is a URI that identifies a resource by name in a particular namespace. A URN may be used to talk about a resource without implying its location or how to access it. For example, in the [[International Standard Book Number|International Standard Book Number (ISBN)]] system, ''<nowiki>ISBN</nowiki> 0-486-27557-4'' identifies a specific edition of Shakespeare's play ''[[Romeo and Juliet]]''. The URN for that edition would be ''<nowiki>urn:isbn:0-486-27557-4</nowiki>''. However, it gives no information as to where to find a copy of that book.

A [[Uniform Resource Locator]] (URL) is a URI that specifies the means of acting upon or obtaining the representation of a resource, i.e. specifying both its primary access mechanism and network location. For example, the URL <code><nowiki>http://example.org/wiki/Main_Page</nowiki></code> refers to a resource identified as <code><nowiki>/wiki/Main_Page</nowiki></code> whose representation, in the form of [[HTML]] and related code, is obtainable via the [[Hypertext Transfer Protocol]] (''http:'') from a network host whose [[domain name]] is <code><nowiki>example.org</nowiki></code>.

A URN may be compared to a person's name, while a URL may be compared to their street address. In other words, a URN identifies an item and a URL provides a method for finding it.

Technical publications, especially standards produced by the [[Internet Engineering Task Force|IETF]] and by the [[World Wide Web Consortium|W3C]], normally reflect a view outlined in a [[W3C Recommendation]] of 2001, which acknowledges the precedence of the term URI rather than endorsing any formal subdivision into URL and URN. {{cquote|URL is a useful but informal concept: a URL is a type of URI that identifies a resource via a representation of its primary access mechanism (e.g., its network "location"), rather than by some other attributes it may have.{{sfnp|Joint W3C/IETF URI Planning Interest Group|2001}}}}

As such, a URL is simply a URI that happens to point to a resource over a network.{{efn|A report published in 2002 by a joint W3C/IETF working group aimed to normalize the divergent views held within the IETF and W3C over the relationship between the various 'UR*' terms and standards. While not published as a full standard by either organization, it has become the basis for the above common understanding and has informed many standards since then.}}{{sfnp|Joint W3C/IETF URI Planning Interest Group|2002}} However, in non-technical contexts and in software for the World Wide Web, the term "URL" remains widely used. Additionally, the term "web address" (which has no formal definition) often occurs in non-technical publications as a synonym for a URI that uses the ''http'' or ''https'' schemes. Such assumptions can lead to confusion, for example, in the case of XML namespaces that have a [[#Relation to XML namespaces|visual similarity to resolvable URIs]].

Specifications produced by the [[WHATWG]] prefer URL over URI, and so newer HTML5 APIs use URL over URI.<ref>{{cite web |title=URL Standard: 6.3. URL APIs elsewhere |url=https://url.spec.whatwg.org/#url-apis-elsewhere}}</ref> {{cquote|Standardize on the term URL. URI and IRI are just confusing. In practice a single algorithm is used for both so keeping them distinct is not helping anyone. URL also easily wins the search result popularity contest.<ref>{{cite web |title=URL Standard: Goals|url=https://url.spec.whatwg.org/#goals}}</ref>}}

While most URI schemes were originally designed to be used with a particular [[protocol (computing)|protocol]], and often have the same name, they are semantically different from protocols. For example, the scheme ''http'' is generally used for interacting with [[web resource]]s using HTTP, but the scheme ''[[file URI scheme|file]]'' has no protocol.

==Generic syntax==
===Definition===
Each URI begins with a scheme name that refers to a specification for assigning identifiers within that scheme. As such, the URI syntax is a federated and extensible naming system wherein each scheme's specification may further restrict the syntax and semantics of identifiers using that scheme. The URI generic syntax is a superset of the syntax of all URI schemes. It was first defined in [[Request for Comments|Request for Comments (RFC)]] 2396, published in August 1998,{{sfnp|RFC 2396|1998}} and finalized in <nowiki>RFC</nowiki> 3986, published in January 2005.{{sfnp|RFC 3986|2005}}
<section begin=syntax />
The ''URI generic syntax'' consists of a hierarchical sequence of five ''components'':{{sfnp|RFC 3986|2005|loc=§3}}

<pre>
URI = scheme:[//authority]path[?query][#fragment]
</pre>

where the authority component divides into three ''subcomponents'':

<pre>
authority = [userinfo@]host[:port]
</pre>

It comprises:
* A non-empty '''{{visible anchor|scheme}}''' component followed by a colon (<code>:</code>), consisting of a sequence of characters beginning with a letter and followed by any combination of letters, digits, plus (<code>+</code>), period (<code>.</code>), or hyphen (<code>-</code>). Although schemes are case-insensitive, the canonical form is lowercase and documents that specify schemes must do so with lowercase letters. Examples of popular schemes include <code>[[Hypertext Transfer Protocol|http]]</code>, <code>[[HTTP Secure|https]]</code>, <code>[[File Transfer Protocol|ftp]]</code>, <code>[[Mailto|mailto]]</code>, <code>[[File URI scheme|file]]</code>, <code>chrome</code>, <code>skype</code>, <code>steam</code>, <code>svn</code>, <code>telnet</code>, <code>sms</code>, <code>smtp</code>, <code>ldap</code>, <code>jabber</code>, <code>xmpp</code>, <code>udp</code>, <code>view-source</code>, <code>ws</code>, <code>about</code>, <code>[[Data URI scheme|data]]</code>, and <code>[[Internet Relay Chat#URI scheme|irc]]</code>. Unpopular schemes include <code>xri</code>, <code>tag</code>, <code>lastfm</code>, <code>mongodb</code>, <code>msword</code>, <code>moz</code>, <code>palm</code>, <code>paparazzi</code>, <code>urn</code>, <code>onenote</code>, <code>rediss</code> (_sic_), <code>ipp</code>, <code>things</code>, <code>stuns</code>, <code>prospero</code>, <code>iotdisco</code>, <code>tip</code>, <code>tool</code>, <code>tv</code>, <code>unreal</code>, <code>submit</code>, <code>market</code>, <code>ham</code>, <code>fish</code>, <code>finger</code>, <code>iris</code>, <code>[[IP over avian carriers]]</code>, and <code>[[Gopher (protocol)|gopher]].<cite>https://www.iana.org/assignments/uri-schemes/uri-schemes.xhtml</cite> URI schemes should be registered with the [[Internet Assigned Numbers Authority|Internet Assigned Numbers Authority (IANA)]], although non-registered schemes are used in practice.{{efn|The procedures for registering new URI schemes were originally defined in 1999 by <nowiki>RFC 2717</nowiki>, and are now defined by <nowiki>RFC 7595</nowiki>, published in June 2015.{{sfnp|IETF|2015}}}}
* An optional non-empty '''authority''' component preceded by two slashes (<code>//</code>), comprising:
** An optional '''userinfo''' subcomponent that may consist of a [[User (computing)|user name]] and an optional [[password]] preceded by a colon (<code>:</code>), followed by an at symbol (<code>@</code>). Use of the format <code>username:password</code> in the userinfo subcomponent is deprecated for security reasons. Applications should not render as clear text any data after the first colon (<code>:</code>) found within a userinfo subcomponent unless the data after the colon is the empty string (indicating no password).
** A non-empty '''host''' subcomponent, consisting of either a registered name (including but not limited to a [[hostname]]), or an [[IP address]]. [[IPv4]] addresses must be in [[dot-decimal notation]], and [[IPv6]] addresses must be enclosed in brackets (<code>[]</code>).{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{efn|For URIs relating to resources on the World Wide Web, some web browsers allow {{code|.0}} portions of dot-decimal notation to be dropped or raw integer IP addresses to be used.{{sfnp|Lawrence|2014}}}}
** An optional '''port''' subcomponent preceded by a colon (<code>:</code>).
* A '''path''' component, consisting of a sequence of path segments separated by a slash (<code>/</code>). A path is always defined for a URI, though the defined path may be empty (zero length). A segment may also be empty, resulting in two consecutive slashes (<code>//</code>) in the path component. A path component may resemble or map exactly to a [[Path (computing)|file system path]], but does not always imply a relation to one. If an authority component is present, then the path component must either be empty or begin with a slash (<code>/</code>). If an authority component is absent, then the path cannot begin with an empty segment, that is with two slashes (<code>//</code>), as the following characters would be interpreted as an authority component.{{sfnp|RFC 2396|1998|loc=§3.3}} The final segment of the path may be referred to as a '[[Clean URL#Slug|slug]]'.

{| class="wikitable" style="float: right; font-size: 0.9em; margin-left: 1em"
|-
! Query delimiter
! Example
|-
| Ampersand (<code>&</code>)
| <code>key1=value1&key2=value2</code>
|-
| Semicolon (<code>;</code>){{efn|Historic <nowiki>RFC 1866</nowiki> (obsoleted by <nowiki>RFC 2854</nowiki>) encourages CGI authors to support ';' in addition to '&'.{{sfnp|RFC 1866|1995|loc=§8.2.1}}}}{{Incomplete short citation|date=August 2016}}
| <code>key1=value1;key2=value2</code>
|}
* An optional '''query''' component preceded by a question mark (<code>?</code>), containing a [[query string]] of non-hierarchical data. Its syntax is not well defined, but by convention is most often a sequence of [[attribute–value pair]]s separated by a [[delimiter]].
* An optional '''fragment''' component preceded by an [[Number sign|hash]] (<code>#</code>). The fragment contains a [[fragment identifier]] providing direction to a secondary resource, such as a section heading in an article identified by the remainder of the URI. When the primary resource is an [[HTML]] document, the fragment is often an [[HTML#Attributes|<code>id</code> attribute]] of a specific element, and web browsers will scroll this element into view.<section end=syntax />

Strings of data [[Octet (computing)|octets]] within a URI are represented as characters. Permitted characters within a URI are the [[ASCII]] characters for the lowercase and uppercase letters of the modern [[English alphabet]], the [[Arabic numerals]], [[hyphen]], [[Full stop|period]], [[underscore]], and [[tilde]].{{sfnp|RFC 3986|2005|loc=§2}} Octets represented by any other character must be [[percent-encoding|percent-encoded]].

Of the ASCII character set, the characters <code>: / ? # [ ] @</code> are reserved for use as delimiters of the generic URI components and must be percent-encoded — for example, <code>%3F</code> for a question mark.{{sfnp|RFC 3986|2005|loc=§2.2}} The characters <code>! $ & ' ( ) * + , ; =</code> are permitted by generic URI syntax to be used unencoded in the user information, host, and path as delimiters.{{sfnp|RFC 3986|2005|loc=§3.2.2}}{{sfnp|RFC 3986|2005|loc=§3.3}} Additionally, <code>:</code> and <code>@</code> may appear unencoded within the path, query, and fragment; and <code>?</code> and <code>/</code> may appear unencoded as data within the query or fragment.{{sfnp|RFC 3986|2005|loc=§3.3}}{{sfnp|RFC 3986|2005|loc=§3.4}}

===Examples===
The following figure displays example URIs and their component parts.

<pre>
userinfo host port
┌───┴──┐ ┌──────┴──────┐ ┌┴┐
https://john.doe@www.example.com:123/forum/questions/?tag=networking&order=newest#top
└─┬─┘ └─────────────┬────────────┘└───────┬───────┘ └────────────┬────────────┘ └┬┘
scheme authority path query fragment

ldap://[2001:db8::7]/c=GB?objectClass?one
└─┬┘ └─────┬─────┘└─┬─┘ └──────┬──────┘
scheme authority path query

mailto:John.Doe@example.com
└──┬─┘ └─────────┬────────┘
scheme path

news:comp.infosystems.www.servers.unix
└─┬┘ └───────────────┬───────────────┘
scheme path

tel:+1-816-555-1212
└┬┘ └──────┬──────┘
scheme path

telnet://192.0.2.16:80/
└──┬─┘ └─────┬─────┘│
scheme authority path

urn:oasis:names:specification:docbook:dtd:xml:4.1.2
└┬┘ └──────────────────────┬──────────────────────┘
scheme path
</pre>

==URI references==
===Definition===
A ''URI reference'' is either a URI, or a ''relative reference'' when it does not begin with a scheme component followed by a colon (<code>:</code>).{{sfnp|RFC 3986|2005|loc=§4.1}} A path segment that contains a colon character (e.g., <code>foo:bar</code>) cannot be used as the first path segment of a relative reference if its path component does not begin with a slash (<code>/</code>), as it would be mistaken for a scheme component. Such a path segment must be preceded by a dot path segment (e.g., <code>./foo:bar</code>).{{sfnp|RFC 3986|2005|loc=§4.2}}

Web document [[markup language]]s frequently use URI references to point to other resources, such as external documents or specific portions of the same logical document:{{sfnp|RFC 3986|2005|loc=§4.4}}
* in [[HTML]], the value of the <code>src</code> attribute of the <code>img</code> element provides a URI reference, as does the value of the <code>href</code> attribute of the <code>a</code> or <code>link</code> element;
* in [[XML]], the [[system identifier]] appearing after the <code>SYSTEM</code> keyword in a [[Document Type Definition|DTD]] is a fragmentless URI reference;
* in [[XSLT]], the value of the <code>href</code> attribute of the <code>xsl:import</code> element/instruction is a URI reference; likewise the first argument to the <code>document()</code> function.

===Examples===
<pre>
https://example.com/path/resource.txt#fragment
//example.com/path/resource.txt
/path/resource.txt
path/resource.txt
/path/resource.txt
../resource.txt
./resource.txt
resource.txt
#fragment
</pre>

===Suffix references===
As URI usage has become commonplace, traditional media (television, radio, newspapers, billboards, etc.) have increasingly used a suffix of the URI as a reference, consisting of only the authority and path portions of the URI, such as

<pre>
www.w3.org/Addressing/
</pre>

Such references are primarily intended for human interpretation rather than for machines, with the assumption that context-based heuristics are sufficient to complete the URI (e.g., most registered names beginning with <code>www</code> are likely to have a URI prefix of <code>http://</code>). Although there is no standard set of heuristics for disambiguating a URI suffix, many client implementations allow them to be entered by the user and heuristically resolved. Although this practice of using suffix references is common, it should be avoided whenever possible and should never be used in situations where long-term references are expected, as the heuristics will change over time, particularly when a new URI scheme becomes popular, and are often incorrect when used out of context. Furthermore, they can lead to security issues along the lines of those described in <nowiki>RFC</nowiki> 1535. As a URI suffix has the same syntax as a relative reference with a relative path, a suffix reference cannot be used in contexts where a relative reference is expected. As a result, suffix references are limited to places where there is no defined base URI, such as dialog boxes and off-line advertisements.{{sfnp|RFC 3986|2005|loc=§4.5}}

==URI resolution==
===Definition===
An ''absolute URI'' is a URI with no fragment component.

''Resolving'' a URI reference against a ''base URI'' results in a ''target URI''. This implies that the base URI exists and is an absolute URI. The base URI can be obtained, in order of precedence, from:{{sfnp|RFC 3986|2005|loc=§5.1}}

* the reference URI itself if it is a URI;
* the content of the representation;
* the entity encapsulating the representation;
* the URI used for the actual retrieval of the representation;
* the context of the application.

===Examples===
Within a representation with a well defined base URI of

<pre>
http://a/b/c/d;p?q
</pre>

a relative reference is resolved to its target URI as follows:{{sfnp|RFC 3986|2005|loc=§5.4}}

<pre>
"g:h" -> "g:h"
"g" -> "http://a/b/c/g"
"./g" -> "http://a/b/c/g"
"g/" -> "http://a/b/c/g/"
"/g" -> "http://a/g"
"//g" -> "http://g"
"?y" -> "http://a/b/c/d;p?y"
"g?y" -> "http://a/b/c/g?y"
"#s" -> "http://a/b/c/d;p?q#s"
"g#s" -> "http://a/b/c/g#s"
"g?y#s" -> "http://a/b/c/g?y#s"
";x" -> "http://a/b/c/;x"
"g;x" -> "http://a/b/c/g;x"
"g;x?y#s" -> "http://a/b/c/g;x?y#s"
"" -> "http://a/b/c/d;p?q"
"." -> "http://a/b/c/"
"./" -> "http://a/b/c/"
".." -> "http://a/b/"
"../" -> "http://a/b/"
"../g" -> "http://a/b/g"
"../.." -> "http://a/"
"../../" -> "http://a/"
"../../g" -> "http://a/g"
</pre>

==History==
===Naming, addressing, and identifying resources===
URIs and URLs have a shared history. In 1994, [[Tim Berners-Lee|Tim Berners-Lee's]] proposals for [[hypertext]]{{sfnp|Palmer|2001}} implicitly introduced the idea of a URL as a short string representing a resource that is the target of a [[hyperlink]]. At the time, people referred to it as a "hypertext name"{{sfnp|W3C|1992}} or "document name".

Over the next three and a half years, as the World Wide Web's core technologies of HTML, HTTP, and web browsers developed, a need to distinguish a string that provided an address for a resource from a string that merely named a resource emerged. Although not yet formally defined, the term ''Uniform Resource Locator'' came to represent the former, and the more contentious ''Uniform Resource Name'' came to represent the latter.

During the debate over defining URLs and URNs it became evident that the two concepts embodied by the terms were merely aspects of the fundamental, overarching notion of resource ''identification''. In June 1994, the IETF published Berners-Lee's <nowiki>RFC 1630</nowiki>: the first Request for Comments that acknowledged the existence of URLs and URNs, and, more importantly, defined a formal syntax for ''Universal Resource Identifiers'' — URL-like strings whose precise syntaxes and semantics depended on their schemes. In addition, this RFC attempted to summarize the syntaxes of URL schemes in use at the time. It also acknowledged, but did not standardize, the existence of relative URLs and fragment identifiers.

===Refinement of specifications===
In December 1994, <nowiki>RFC 1738</nowiki> formally defined relative and absolute URLs, refined the general URL syntax, defined how to resolve relative URLs to absolute form, and better enumerated the URL schemes then in use. The agreed definition and syntax of URNs had to wait until the publication of <nowiki>RFC 2141</nowiki> in May 1997.

The publication of <nowiki>RFC 2396</nowiki> in August 1998 saw the URI syntax become a separate specification{{sfnp|RFC 2396|1998}} and most of the parts of RFCs 1630 and 1738 relating to URIs and URLs in general were revised and expanded by the [[IETF]]. The new RFC changed the meaning of "U" in "URI" to "Uniform" from "Universal".

In December 1999, <nowiki>RFC 2732</nowiki> provided a minor update to <nowiki>RFC 2396</nowiki>, allowing URIs to accommodate [[IPv6]] addresses. A number of shortcomings discovered in the two specifications led to a community effort, coordinated by <nowiki>RFC 2396</nowiki> co-author [[Roy Fielding]], that culminated in the publication of <nowiki>RFC 3986</nowiki> in January 2005. While obsoleting the prior standard, it did not render the details of existing URL schemes obsolete; <nowiki>RFC 1738</nowiki> continues to govern such schemes except where otherwise superseded. <nowiki>RFC 2616</nowiki> for example, refines the <code>http</code> scheme. Simultaneously, the IETF published the content of <nowiki>RFC 3986</nowiki> as the full standard STD 66, reflecting the establishment of the URI generic syntax as an official Internet protocol.

In 2001, the W3C's Technical Architecture Group (TAG) published a guide to [[best practices]] and canonical URIs for publishing multiple versions of a given resource.{{sfnp|W3C|2001}} For example, content might differ by language or by size to adjust for capacity or settings of the device used to access that content.

In August 2002, <nowiki>RFC 3305</nowiki> pointed out that the term "URL" had, despite widespread public use, faded into near obsolescence, and serves only as a reminder that some URIs act as addresses by having schemes implying network accessibility, regardless of any such actual use. As URI-based standards such as [[Resource Description Framework]] make evident, resource identification need not suggest the retrieval of resource representations over the Internet, nor need they imply network-based resources at all.

The [[Semantic Web]] uses the HTTP URI scheme to identify both documents and concepts in the real world, a distinction which has caused confusion as to how to distinguish the two. The TAG published an e-mail in 2005 on how to solve the problem, which became known as the ''httpRange-14 resolution''.{{sfnp|Fielding|2005}} The W3C subsequently published an Interest Group Note titled ''Cool URIs for the Semantic Web'',{{sfnp|W3C|2008}} which explained the use of [[content negotiation]] and the [[HTTP 303]] response code for redirections in more detail.

==Relation to XML namespaces==
In [[XML]], a [[XML namespace|namespace]] is an abstract domain to which a collection of element and attribute names can be assigned. The namespace name is a character string which must adhere to the generic URI syntax.{{sfnp|Morrison|2006}} However, the name is generally not considered to be a URI,{{sfnp|Harold|2004}} because the URI specification bases the decision not only on lexical components, but also on their intended use. A namespace name does not necessarily imply any of the semantics of URI schemes; for example, a namespace name beginning with ''http:'' may have no connotation to the use of the [[HTTP]].

Originally, the namespace name could match the syntax of any non-empty URI reference, but the use of relative URI references was deprecated by the W3C.{{sfnp|W3C|2009}} A separate W3C specification for namespaces in XML 1.1 permits [[Internationalized resource identifier|internationalized resource identifier (IRI)]] references to serve as the basis for namespace names in addition to URI references.{{sfnp|W3C|2006}}

==See also==
* [[CURIE]] – defines a generic, abbreviated syntax for expressing URIs
* [[Dereferenceable Uniform Resource Identifier]] – a resource retrieval mechanism that uses any of the internet protocols (e.g. HTTP) to obtain a copy or representation of the resource it identifies
* [[Extensible Resource Identifier]] – a scheme and resolution protocol for abstract identifiers compatible with URIs
* [[Internationalized Resource Identifier]] (IRI) – a generalization of URIs allowing the use of Unicode
* [[Persistent uniform resource locator]] (PURL) – a URI that is used to redirect to the location of the requested web resource
* [[Uniform Naming Convention]] – a common syntax used by Microsoft to describe the location of a network resource, such as a shared file, directory, or printer
* [[Resource Directory Description Language]] – a descriptive language to provide machine- and human-readable information about a particular namespace and about the XML documents that use it
* [[Universally unique identifier|UUID]]

== Notes ==
{{Notelist}}

== References ==
=== Citations ===
{{Reflist|25em}}

=== Cited works ===
{{refbegin|32em}}
* {{cite web |first=Roy T.|last=Fielding|authorlink=Roy Fielding |title = [httpRange-14] Resolved |url = http://lists.w3.org/Archives/Public/www-tag/2005Jun/0039.html|date=18 June 2005 |accessdate=24 July 2009|ref=harv}}
* {{cite book |first=Elliotte Rusty|last=Harold |authorlink=Elliotte Rusty Harold |year=2004 |title=XML 1.1 Bible |edition=Third |publisher=[[Wiley Publishing]] |page=291 |isbn=0-7645-4986-3 |ref=harv }}
* {{cite web |url = http://www.w3.org/TR/uri-clarification/ |author=Joint W3C/IETF URI Planning Interest Group|title=URIs, URLs, and URNs: Clarifications and Recommendations 1.0|date=21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2001}}}}
* {{cite web |url = https://tools.ietf.org/html/rfc3305|title=Report from the Joint W3C/IETF URI Planning Interest Group: Uniform Resource Identifiers (URIs), URLs, and Uniform Resource Names (URNs): Clarifications and Recommendations|editor1-first=M.|editor1-last=Mealling |editor2-first=R.|editor2-last=Denenberg|publisher=[[World Wide Web Consortium]]|date=August 2002 |accessdate=13 September 2015 |ref={{SfnRef|Joint W3C/IETF URI Planning Interest Group|2002}}}}
* {{cite web |url=https://tools.ietf.org/html/rfc7595|title=Guidelines and Registration Procedures for URI Schemes|editor-first=D.|editor-last=Thaler|author1-first=T.|author1-last=Hansen|author2-first=T.|author2-last=Hardie|publisher=[[Internet Engineering Task Force]]|date=June 2015|issn=2070-1721|ref={{SfnRef|IETF|2015}}}}
* {{cite book |last=Morrison|first=Michael|year=2006|title=Sams Teach Yourself XML|publisher=[[Sams Publishing]]|chapter=Hour 5: ''Putting Namespaces to Use''|page=91 |ref=harv }}
* {{cite web |first=Sean B.|last=Palmer |title=The Early History of HTML |url = http://infomesh.net/html/history/early/ |year=2001 |accessdate=2009-04-30 |ref=harv }}
* {{cite web |author = URI Planning Interest Group, W3C/IETF |title = URIs, URLs, and URNs: Clarifications and Recommendations 1.0 |url = http://www.w3.org/TR/uri-clarification/ |date = 21 September 2001 |accessdate=2009-07-27 |ref={{SfnRef|URI Planning Interest Group|2009}}}}
* {{cite web |url = http://www.w3.org/History/19921103-hypertext/hypertext/WWW/Addressing/Addressing.html|title=W3 Naming Schemes |publisher=[[World Wide Web Consortium]]|year=1992|accessdate=2009-07-24|ref={{SfnRef|W3C|1992}}}}
* {{cite web |url = http://www.w3.org/2001/tag/doc/alternatives-discovery.html |title=On Linking Alternative Representations To Enable Discovery And Publishing|publisher=[[World Wide Web Consortium]]|year=2006|orig-year=2001|accessdate=2012-04-03|ref={{SfnRef|W3C|2001}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use|title=Namespaces in XML 1.1 (Second Edition)|date=16 August 2006 |at = 2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray|editor2-first=Dave |editor2-last=Hollander|editor3-first=Andrew|editor3-last=Layman|editor4-first=Richard|editor4-last=Tobin|publisher=[[World Wide Web Consortium]] |accessdate=31 August 2015|ref={{SfnRef|W3C|2006}}}}
* {{cite web |url = http://www.w3.org/TR/cooluris/|title=Cool URIs for the Semantic Web|editor1-first=Leo|editor1-last=Sauermann|editor2-first=Richard|editor2-last=Cyganiak|author1-first=Danny|author1-last=Ayers|author2-first=Max|author2-last=Völkel|publisher=[[World Wide Web Consortium]]|date=3 December 2008|accessdate=2012-04-03|ref={{SfnRef|W3C|2008}}}}
* {{cite web |url = http://www.w3.org/TR/REC-xml-names/#iri-use |title=Namespaces in XML 1.0 (Third Edition)|date=8 December 2009 |at=2.2 Use of URIs as Namespace Names|editor1-first=Tim|editor1-last=Bray|editor1-link=Tim Bray |editor2-first=Dave |editor2-last=Hollander |editor3-first=Andrew|editor3-last=Layman |editor4-first=Richard|editor4-last=Tobin |editor5-first=Henry S. |editor5-last=Thompson |publisher=[[World Wide Web Consortium]]|accessdate=31 August 2015 |ref={{SfnRef|W3C|2009}} }}
* {{cite web |url = https://tools.ietf.org/html/rfc1866#section-8.2.1 |title=Hypertext Markup Language - 2.0|author1-first=Tim|author1-last=Berners-Lee |author1-link=Tim Berners-Lee|author2-first=Dan|author2-last=Connolly|publisher=[[Internet Engineering Task Force]] |date=November 1995 |accessdate=13 September 2015}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc2396 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=2396 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=August 1998 |accessdate=31 August 2015 |ref={{SfnRef|RFC 2396|1998}}}}
* {{cite IETF |url = http://tools.ietf.org/html/rfc3986 |title=Uniform Resource Identifiers (URI): Generic Syntax |rfc=3986 |first1=Tim |last1=Berners-Lee |authorlink1=Tim Berners-Lee |first2=Roy|last2=Fielding|authorlink2=Roy Fielding |first3=Larry|last3=Masinter|publisher=[[Internet Engineering Task Force]]|date=January 2005 |accessdate=31 August 2015 |ref={{SfnRef|RFC 3986|2005}}}}
* {{cite web |last1=Lawrence |first1=Eric |title=Browser Arcana: IP Literals in URLs |url = http://blogs.msdn.com/b/ieinternals/archive/2014/03/06/browser-arcana-ipv4-ipv6-literal-urls-dotted-va-dotless.aspx |website=IEInternals |publisher=[[Microsoft]] |date=6 March 2014 |accessdate=2016-04-25 |ref = harv }}
{{refend}}

==External links==
* [http://www.iana.org/assignments/uri-schemes.html URI Schemes] – [[Internet Assigned Numbers Authority|IANA]]-maintained registry of URI Schemes
* [http://www.w3.org/wiki/UriSchemes URI schemes on the W3C wiki]
* [http://www.w3.org/TR/webarch/#identification Architecture of the World Wide Web, Volume One, §2: Identification] – by W3C
* [http://www.w3.org/TR/uri-clarification/ W3C URI Clarification]

{{Semantic Web|state=collapsed}}
{{URI scheme}}
{{Hypermedia}}

{{Authority control}}

[[Category:Application layer protocols]]
[[Category:Internet protocols]]
[[Category:Internet Standards]]
[[Category:Semantic Web| ]]
[[Category:Uniform Resource Locator]]

Kuba Kingdom

2018-05-25T16:19:21Z

Crasshopper: /* Kuba art */

{{more footnotes|date=November 2016}}
[[File:Brooklyn Museum 22.1582 Mwaash aMbooy Mask.jpg|thumb|A contemporary ''Mwaash aMbooy'' mask, representing Woot, the mythical founder of the Kuba Kingdom]]
The '''Kuba Kingdom''', also rendered as the '''Kingdom of the Bakuba''', [[Songora]] or '''Bushongo''', was a [[List of kingdoms in pre-colonial Africa|pre-colonial kingdom]] in [[Central Africa]]. The Kuba Kingdom [[Floruit|flourished]] between the 17th and 19th centuries in the region bordered by the [[Sankuru River|Sankuru]], [[Lulua River|Lulua]], and [[Kasai River|Kasai rivers]] in the south-east of the modern-day [[Democratic Republic of the Congo]].

The Kuba Kingdom was a conglomerate of several smaller [[Bushongo language|Bushongo-speaking]] principalities as well as the Kete, [[Coofa]], [[Mbeengi]], and the [[Kuba Twa|Twa Pygmies]]. The original Kuba migrated during the 16th century from the north. Nineteen different ethnic groups are included in the kingdom, which still exists and is presided over by the King (''nyim'').

==History==

[[File:Mulwalwa helmet mask Berlin-Dahlem.jpg|thumb|Helmet mask "mulwalwa", Southern Kuba, 19th or early 20th century]]

===Shyaam a-Mbul===
The kingdom began as a conglomeration of several chiefdoms of various [[ethnic groups]] with no real central authority. In approximately 1625, an individual from outside the area known as Shyaam a-Mbul a Ngoong usurped the position of one of the area rulers and united all the chiefdoms under his leadership. Tradition states that Shyaam a-Mbul was the adopted son of a Kuba queen. He left the Kuba region to find enlightenment in the [[Pende people|Pende]] and [[Kingdom of Kongo|Kongo]] kingdoms to the west. After learning all he could from these states, he returned to Kuba to form the empire's political, social and economic foundations.

===A new government===
{{main|Rulers of Kuba}}
The Kuba government was reorganized toward a merit-based title system, but power still remained firmly in the hands of the aristocracy. The Kuba government was controlled by a king called the ''nyim'' who belonged to the Bushoong clan. The king was responsible to a court council of all the Kuba subgroups, who were represented equally before the king by their elites.

===Growth===
As the kingdom matured, it benefited from advanced techniques adopted from neighboring peoples as well as New World crops introduced from the Americas, such as maize, tobacco, cassava and beans. Kuba became very wealthy, which resulted in great artistic works commissioned by the Kuba nobility. The Kuba kings retained the most fanciful works for court ceremony and were also buried with these artifacts.

===Apex ===
The Kuba Kingdom reached its apex during the mid 19th century. Europeans first reached the area in 1884. Because of the kingdom's relative isolation, it was not as affected by the slave trade as were the [[Kingdom of Kongo|Kongo]] and [[Ndongo]] kingdoms on the coast.

The current reigning monarch, Kot-a-Mbweeky III, has been on the throne since 1969.

==Kuba culture==

===Kuba art===
{{Main|Kuba art}}

''See also: [[Kuba textiles]], [[Kuba divination]], [[Kuba masquerade]], [[Ndop (Kuba)]]''

The Kuba are known for their [[raffia]] embroidered textiles, fiber and beaded hats, carved palm wine cups and cosmetic boxes, but they are most famous for their monumental helmet masks, featuring exquisite geometric patterns, stunning fabrics, seeds, beads and shells. They have been described {{fact}} as a people who cannot bear to leave a surface without ornament.

The boxes, known as Kuba Boxes and called ''ngedi mu ntey'' by the Kuba, are generally used to hold ''tukula'' powder and paste. The boxes are usually in the shape of a square with a faceted lid, a semicircle (sometimes referred to as "half moon"), a rectangle or the shape of a mask. Sometimes they were used for holding razors for cutting raffia, hairpins or ritual objects.

''Tukula'' (called ''twool'' by the Kuba) is a red powder made of ground cam wood. The color red is essential to the Kuba concept of beauty and was therefore used to ornament the face, hair and chest during dances and important ceremonies, as well as to anoint bodies for burial. ''Tukula'' was also mixed with other pigments to dye raffia cloth.

After 1700, King Misha mi-Shyaang a-Mbul introduced wooden sculptures called ''[[Ndop (Kuba)|ndop]]'' figures that were carved to resemble the king and represent his individual reign. These figures always included the king's ''ibol'' or personal symbol, akin to a personal standard.

The carved palm-wine drinking cups and ornately carved boxes are identified with competition between titled court members among the Kuba. With half of all Bushoong men holding titles in the 1880s, competition for influence was sometimes fierce, and it found expression in the elaboration of these essentially commonplace household objects into works of extraordinary beauty.

===Kuba religion and mythos===
The Kuba believed in [[Bumba (god)|Bumba]] the Sky Father who spewed out the sun, moon, stars, and planets. He also created life with the Earth Mother. However these were somewhat distant deities, and the Kuba placed more immediate concern in a supernatural being named Woot, who named the animals and other things.<ref>[http://www.swarthmore.edu/Humanities/pschmid1/kuba.html Swarthmore article]</ref> Woot was the first human and bringer of civilization.<ref name=BMA>{{cite book|last=[[Birmingham Museum of Art]]|title=Birmingham Museum of Art : guide to the collection|year=2010|publisher=Birmingham Museum of Art|location=[Birmingham, Ala]|isbn=978-1-904832-77-5|pages=74|url=http://artsbma.org}}</ref> The Kuba are sometimes known as the "Children of Woot."<ref>[http://www.arcc.ku.edu/Africa/6/KubaCulturePage.html University of Kansas Anthropology site]{{Dead link|date=May 2018}}</ref>

==See also==
*[[William Henry Sheppard]]

==References==
{{Reflist}}

==Further reading==
*{{cite book|last=Vansina|first=Jan|authorlink=Jan Vansina|title=The Children of Woot: A history of the Kuba peoples|date=1978|publisher=University of Wisconsin Press|location=Madison|isbn=9780299074906}}

==External links==
*[http://www.clemson.edu/caah/history/faculty/Miller%20Kuba/Kuba%20Site/kuba_home.html An exhibit of Kuba art held at Clemson University in 2002]{{Dead link|date=May 2018}}
*[http://www.randafricanart.com/sitebuilder/images/Kuba-541x466.jpg map of tribes in the area]
*[https://flickr.com/photos/71909327@N00/sets/72157602085585719/ Photos of Kuba Raffia Cloths]
*[http://www.metmuseum.org/toah/hd/kuba/hd_kuba.htm Kingdoms of the Savanna: The Kuba Kingdom]
*[https://archive.is/20060901175547/http://www.anthro.ku.edu/Central%20African%20Art/website_bwoom/web%20pages/general.htm The Bwoom Mask of the Kuba People]
*[https://web.archive.org/web/20060918162440/http://www.uiowa.edu/~africart/toc/people/Kuba.html Art & Life in Africa]

[[Category:Countries in precolonial Africa]]
[[Category:Former countries in Africa]]
[[Category:Former monarchies of Africa]]
[[Category:Political history of the Democratic Republic of the Congo]]
[[Category:Ethnic groups in the Democratic Republic of the Congo| ]]
[[Category:1625 establishments in Africa]]
[[Category:States and territories established in 1625]]
[[Category:States and territories disestablished in 1900]]

Drawdown (economics)

2018-05-02T13:19:49Z

Crasshopper: /* Further reading */ dead links, and nothing on archive.org

The '''drawdown''' is the measure of the decline from a historical peak in some variable (typically the cumulative profit or total open equity of a financial trading strategy).

Somewhat more formally, if <math>X = (X(t), t \ge 0)</math> is a random process with ''X''(0) = 0, the drawdown at time ''T'', denoted <math>D(T)</math>,
is defined as

<math> D(T)=\max \left\{ 0, \max_{t \in (0,T)} X(t)- X(T) \right\} </math>

The '''maximum drawdown''' (MDD) up to time <math>T</math>
is the maximum of the Drawdown over the history of the variable. More formally,

<math> \text{MDD}(T)=\max_{\tau\in (0,T)}\left[\max_{t \in (0,\tau)} X(t)- X(\tau) \right]</math>

==Finance definitions==
{|class="wikitable floatright" | width="200"
|- style="font-size:75%"
|-align="center"
|colspan="1" | Pseudocode
|-
| rowspan="2" |

The following [[pseudocode]] computes the Drawdown ("DD") and Max Drawdown ("MDD") of the variable "NAV", the Net Asset Value of an investment. Drawdown and Max Drawdown are calculated as percentages:

MDD = 0
peak = -99999
for i = 1 to N step 1
# peak will be the maximum value seen so far (0 to i), only get updated when higher NAV is seen
if (NAV[i] > peak)
peak = NAV[i]
endif
DD[i] = 100.0 * (peak - NAV[i]) / peak
# Same idea as peak variable, MDD keeps track of the maximum drawdown so far. Only get updated when higher DD is seen.
if (DD[i] > MDD)
MDD = DD[i]
endif
endfor
|}

There are two main definitions of a drawdown:

===1. How low it goes (the magnitude)===

:Putting it plainly, a '''drawdown''' is the “pain” period experienced by an investor between a peak (new highs) and subsequent valley (a low point before moving higher).{{cn|date=June 2015}}
:Next, the '''Maximum Drawdown''', or more commonly referred to as Max DD. This is pretty much self explanatory, as the Max DD is the worst (the maximum) peak to valley loss since the investment’s inception.{{cn|date=June 2015}}

In finance, the use of the maximum drawdown as an indicator of risk is particularly popular in the world of [[commodity trading advisor]]s through the widespread use of three performance measures: the [[Calmar ratio]], the [[Sterling ratio]] and the [[Burke ratio]]. These measures can be considered as a modification of the [[Sharpe ratio]] in the sense that the numerator is always the excess of mean returns over the risk-free rate while the standard deviation of returns in the denominator is replaced by some function of the drawdown.

===2. How long it lasts (the duration)===

:The '''Drawdown Duration''' is the length of any peak to peak period, or the time between new equity highs.
:The '''Max Drawdown Duration''' is the worst (the maximum/longest) amount of time an investment has seen between peaks (equity highs).

Many assume Max DD Duration is the length of time between new highs during which the Max DD (magnitude) occurred. But that isn’t always the case. The Max DD duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss (and usually is, because the program needs a long time to recover from the largest loss), but it doesn’t have to be.{{cn|date=June 2015}}

When ''X'' is Brownian motion with drift, the expected behavior of the MDD as a function of
time is known. If ''X'' is represented as
<math>X(t)=\mu t+ \sigma W(t),</math>
where <math>W(t)</math> is a standard [[Wiener process]], then: when <math>\mu > 0</math> the MDD grows logarithmically with time; when <math>\mu = 0</math> the MDD grows as the square root of time; and when
<math>\mu < 0</math> the MDD grows linearly with time (Magdon-Ismail et al. 2004).<ref>http://alumnus.caltech.edu/~amir/drawdown-jrnl.pdf</ref>

==See also==
* [[Risk_return_ratio|Risk Return Ratio]]

==Further reading==
*Burghardt, G., Duncan, R. and L. Liu, "Understanding Drawdowns", working paper, Carr Futures (September 4), 2003
*Chekhlov, A., S. Uryasev and M. Zabarankin, "Drawdown Measure in Portfolio Optimization", International Journal of Theoretical and Applied Finance 8(1), 13-58, 2005. (http://www.ise.ufl.edu/uryasev/files/2011/11/IJTAF_DrawDown_Paper.pdf)
*Eckholdt, H., "Risk Management: Using SAS to Model Portfolio Drawdown, Recovery and Value at Risk" (February), 2004.
*Goldberg, L.R. and O. Mahmoud, "On a Convex Measure of Drawdown Risk", working paper, Center for Risk Management Research, UC Berkeley, 2014. (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2430918)
*Grossman, S. J. and Z. Zhou, "Optimal Investment Strategies for Controlling Drawdowns", Mathematical Finance 3, pp. 241–276, 1993.
*Hamelink, F. and M. Hoesli, "The Maximum Drawdown as a Risk Measure: The Role of Real Estate in the Optimal Portfolio Revisited", working paper (June 24), 2003.
*Hayes, B. T., "Maximum Drawdowns of Hedge Funds with Serial Correlation", Journal of Alternative Investments (vol 8, no 4) (Spring), pp. 26–38, 2006.
*Kim, Daehwan, "Relevance of Maximum Drawdown in the Investment Fund Selection Problem when Utility is Nonadditive", working paper (July), 2010.
*Steiner, Andreas, "Ambiguity in Calculating and Interpreting Maximum Drawdown," working paper (December), 2010.
*Wilkins, K., C. Morales and L. Roman, "Maximum Drawdown Distributions with Volatility Persistence", working paper, 2005.
*Magdon-Ismail, M. and A. Atiya, "Maximum Drawdown", Risk Magazine (October), 2004. (http://alumnus.caltech.edu/~amir/mdd-risk.pdf)
*Magdon-Ismail, M., A. Atiya, A. Pratap, and Y. Abu-Mostafa, "On the Maximum Drawdown of the Brownian Motion", Journal of Applied Probability (vol 41, no 1), 2004. (http://alumnus.caltech.edu/~amir/drawdown-jrnl.pdf)

==References==
{{Reflist}}

[[Category:Business terms]]

Universal property (mathematics)

2018-01-20T17:35:48Z

Crasshopper: ←Redirected page to Universal property

#REDIRECT [[universal property]]

Derived algebraic geometry

2018-01-20T17:35:14Z

Crasshopper:

'''Derived algebraic geometry''' (also called '''spectral algebraic geometry'''<ref>Some authors (e.g., Lurie) use the term "derived algebraic geometry" for the approach based on simplicial commutative rings and the term "spectral algebraic geometry" for the approach based on <math>\textbf{E}_{\infty}</math>-ring spectra. Over a field of characteristic zero, the distinction is usually insignificant.</ref>) is a branch of mathematics that generalizes [[algebraic geometry]] to a situation where [[commutative ring]]s, which provide a local chart, are replaced by [[ring spectra]] in [[algebraic topology]], whose higher homotopy accounts for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's [[scheme theory]] allows the structure sheaf to carry [[nilpotent element]]s. Derived algebraic geometry can be thought of as an extension of this, and provides natural settings for [[intersection theory]] (or [[motivic homotopy theory]]<ref>{{cite arxiv|last=Khan|first=Adeel A.|date=2016-10-21|title=Brave new motivic homotopy theory I|eprint=1610.06871|class=math.AT}}</ref>) of singular algebraic varieties and [[cotangent complex]]es in [[deformation theory]] (cf. F. Francis).

== Introduction ==
Basic objects of study in the field are [[derived scheme]]s and [[derived stack]]s; they generalize, for instance, [[differential graded scheme]]s. The oft-cited example is [[Serre's intersection formula]].<ref>http://mathoverflow.net/questions/12236/serre-intersection-formula-and-derived-algebraic-geometry</ref> In the usual formulation, the formula involves [[Tor functor]] and thus, unless higher Tor vanish, the [[scheme-theoretic intersection]] (i.e., fiber product of immersions) ''does not'' yield the correct [[intersection number]]. In the derived context, one takes the [[derived tensor product]] <math>A \otimes^L B</math>, whose higher homotopy is higher Tor, whose [[étale Spec|Spec]] is not a scheme but a [[derived scheme]]. Hence, the "derived" fiber product yields the correct intersection number. (Currently this is hypothetical; the derived intersection theory has yet to be developed.)

The term "derived" comes from [[derived category]]. It is classic that many operations in algebraic geometry make sense only in the derived category of say [[quasi-coherent sheaf|quasi-coherent sheaves]], rather than the category of such. In the much same way, one usually talks about the [[∞-category]] of derived schemes, etc., as opposed to ordinary category.

== Advantages ==
According to Justin Curry<ref>{{Cite journal|last=Curry|first=Justin|date=2013-03-13|title=Sheaves, Cosheaves and Applications|url=http://arxiv.org/abs/1303.3255|journal=arXiv:1303.3255 [math]|volume=|pages=112|via=}}</ref>:

:The need for a derived perspective can be stated with one picture. In Figure 25 two maps are drawn to the two-sphere S². One is defined on the wedge sum S²∨S¹ and maps the S¹ to a point. The other is defined on the closed disk 𝔻² and maps the boundary circle to a point. If one is only allowed to look at the [[homology]] of the [[fiber (mathematics)|fiber]] for both of these maps, they will not be able to tell them apart. The derived category is the [[universal property (mathematics)|universal]] solution to this problem, as well as many others.

== See also ==
*[[Noncommutative algebraic geometry]]
*[[Simplicial commutative ring]]
*[[Algebra over an operad]]
*[[En-ring]]
*[[Higher Topos Theory]]
*[[∞-topos]]
*[[étale spectrum]]

== Notes ==
{{reflist}}

== References ==
*Ben-Zvi, D., Francis, J., and D. Nadler. ''Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry.''
*Francis, John; [http://www.math.northwestern.edu/~jnkf/writ/thezrev.pdf Derived Algebraic Geometry Over <math>\mathcal{E}_n</math>-Rings]
*{{cite arxiv|title = Derived Algebraic Geometry|eprint= 1401.1044|date = 2014-01-06|first = Bertrand|last = Toën|class= math.AG}}
*{{cite book | last1=Toën | first1=Bertrand | last2=Vezzosi, Gabriele | chapter=From HAG to DAG: derived moduli stacks | zbl=1076.14002 | editor-last=Greenlees | editor-first=J. P. C. | title=Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, September 9–20, 2002 | location=Dordrecht | publisher=Kluwer Academic Publishers | isbn=1-4020-1833-9 | series=NATO Science Series II: Mathematics, Physics and Chemistry | volume=131 | pages=173–216 | year=2004 }}
*{{cite journal | last=Vezzosi | first=Gabriele | title=What is ...a derived stack? | zbl=1228.14004 | journal=Notices Am. Math. Soc. | volume=58 | number=7 | pages=955–958 | year=2011 | url=http://www.ams.org/notices/201107/rtx110700955p.pdf}}

== External links ==
*[http://www.math.harvard.edu/~lurie/ Jacob Lurie's Home Page]
*[http://people.fas.harvard.edu/~amathew/dag.html DAG reading group] at Harvard
*http://ncatlab.org/nlab/show/derived+algebraic+geometry
*[http://www-personal.umich.edu/~erman/DAG.html Michigan Derived Algebraic Geometry RTG Learning Workshop], 2012
*http://mathoverflow.net/questions/217792/derived-algebraic-geometry-how-to-reach-research-level-math/219361#219361
*https://mathoverflow.net/questions/30396/derived-algebraic-geometry-and-chow-rings-chow-motives/
*Gabriele Vezzosi, [http://webusers.imj-prg.fr/~yue-tony.yu/rega/pdf/1314/09102013a.pdf An overview of derived algebraic geometry], October 2013

[[Category:Algebraic geometry]]
[[Category:Algebraic topology]]
[[Category:Ring theory]]
[[Category:Scheme theory]]

{{topology-stub}}

Cartan matrix

2017-12-31T03:54:14Z

Crasshopper:

In [[mathematics]], the term '''Cartan matrix''' has three meanings. All of these are named after the French [[mathematician]] [[Élie Cartan]]. In fact, Cartan matrices in the context of [[Lie algebra]]s were first investigated by [[Wilhelm Killing]], whereas the [[Killing form]] is due to Cartan.{{fact}}

== Lie algebras ==
{{Lie groups}}

A '''generalized Cartan matrix''' is a [[square matrix]] <math>A = (a_{ij})</math> with [[integer|integral]] entries such that

# For diagonal entries, <math>a_{ii} = 2 </math>.
# For non-diagonal entries, <math>a_{ij} \leq 0 </math>.
# <math>a_{ij} = 0</math> if and only if <math>a_{ji} = 0</math>
# <math>A</math> can be written as <math>DS</math>, where <math>D</math> is a [[diagonal matrix]], and <math>S</math> is a [[symmetric matrix]].

For example, the Cartan matrix for [[G2 (mathematics)#Dynkin diagram and Cartan matrix|''G''2]] can be decomposed as such:
:<math>
\left [
\begin{smallmatrix}
\;\,\, 2&-3\\
-1&\;\,\, 2
\end{smallmatrix}\right ]
= \left [
\begin{smallmatrix}
3&0\\
0&1
\end{smallmatrix}\right ]
\left [
\begin{smallmatrix}
2/3&-1\\
-1&\;2
\end{smallmatrix}\right ].
</math>

The third condition is not independent but is really a consequence of the first and fourth conditions.

We can always choose a ''D'' with positive diagonal entries. In that case, if ''S'' in the above decomposition is [[positive-definite matrix|positive definite]], then ''A'' is said to be a '''Cartan matrix'''.

The Cartan matrix of a simple [[Lie algebra]] is the matrix whose elements are the [[scalar product]]s

:<math>a_{ij}=2 {(r_i,r_j)\over (r_i,r_i)}</math>

(sometimes called the '''Cartan integers''') where ''ri'' are the [[root system|simple roots]] of the algebra. The entries are integral from one of the properties of [[root system|root]]s. The first condition follows from the definition, the second from the fact that for <math>i\neq j, r_j-{2(r_i,r_j)\over (r_i,r_i)}r_i</math> is a root which is a [[linear combination]] of the simple roots ''ri'' and ''rj'' with a positive coefficient for ''rj'' and so, the coefficient for ''ri'' has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let <math>D_{ij}={\delta_{ij}\over (r_i,r_i)}</math> and <math>S_{ij}=2(r_i,r_j)</math>. Because the simple roots span a [[Euclidean space]], S is positive definite.

Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See [[Kac–Moody algebra]] for more details).

=== Classification ===
An <math>n \times n</math> matrix ''A'' is '''decomposable''' if there exists a nonempty proper subset <math>I \subset \{1,\dots,n\}</math> such that <math>a_{ij} = 0</math> whenever <math>i \in I</math> and <math>j \notin I</math>. ''A'' is '''indecomposable''' if it is not decomposable.

Let ''A'' be an indecomposable generalized Cartan matrix. We say that ''A'' is of '''finite type''' if all of its [[principal minor]]s are positive, that ''A'' is of '''affine type''' if its proper principal minors are positive and ''A'' has [[determinant]] 0, and that ''A'' is of '''indefinite type''' otherwise.

Finite type indecomposable matrices classify the finite dimensional [[simple Lie algebra]]s (of types <math>A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 </math>), while affine type indecomposable matrices classify the [[affine Lie algebra]]s (say over some algebraically closed field of characteristic 0).

==== Determinants of the Cartan matrices of the simple Lie algebras ====
The determinants of the Cartan matrices of the simple Lie algebras given in the following table (Along with A1=B1=C1, B2=C2, D3=A3, D2=A1A1, E5=D5, E4=A4, and E3=A2A1)<ref>[https://deepblue.lib.umich.edu/bitstream/handle/2027.42/70011/JMAPAQ-23-11-2019-1.pdf Cartan-Gram determinants for the simple Lie Groups] Alfred C. T. Wu, J. Math. Phys. Vol. 23, No. 11, November 1982</ref>
{| class="wikitable" border="1"
|-
! A''n''
! B''n''
! C''n''
! D''n'', ''n''≥3
! E''n'', ''n''=3...8
! F4
! G2
|- align=center
| ''n''+1 || 2 || 2 || 4 || 9-''n'' || 1 || 1
|}
Another property of this determinant is that it is equal to the index of the associated root system, i.e. it is equal to <math>|P/Q| </math> where <math>P, Q </math> denote the weight lattice and root lattice, respectively.

== Representations of finite-dimensional algebras ==
In [[modular representation theory]], and more generally in the theory of representations of finite-dimensional [[associative algebra]]s ''A'' that are ''not'' [[Semisimple algebra|semisimple]], a '''Cartan matrix''' is defined by considering a (finite) set of [[principal indecomposable module]]s and writing [[composition series]] for them in terms of [[irreducible module]]s, yielding a matrix of integers counting the number of occurrences of an irreducible module.

== Cartan matrices in M-theory ==
In [[M-theory]], one may consider a geometry with [[Cycle graph|two-cycles]] which intersects with each other at a finite number of points, at the limit where the area of the two-cycles go to zero. At this limit, there appears a [[gauge group|local symmetry group]]. The matrix of [[intersection number]]s of a basis of the two-cycles is conjectured to be the Cartan matrix of the [[Lie algebra]] of this local symmetry group.<ref>{{cite journal|last=Sen|first=Ashoke|title=A Note on Enhanced Gauge Symmetries in M- and String Theory|journal=Journal of High Energy Physics|volume=1997|issue=9|pages=001|year=1997|publisher=IOP Publishing|doi=10.1088/1126-6708/1997/09/001|url=http://iopscience.iop.org/1126-6708/1997/09/001}}</ref>

This can be explained as follows. In M-theory one has [[soliton]]s which are two-dimensional surfaces called ''membranes'' or ''2-branes''. A 2-brane has a [[tension (physics)|tension]] and thus tends to shrink, but it may wrap around a two-cycles which prevents it from shrinking to zero.

One may [[Compactification (physics)|compactify]] one dimension which is shared by all two-cycles and their intersecting points, and then take the limit where this dimension shrinks to zero, thus getting a [[dimensional reduction]] over this dimension. Then one gets type IIA [[string theory]] as a limit of M-theory, with 2-branes wrapping a two-cycles now described by an open string stretched between [[D-brane]]s. There is a [[U(1)]] local symmetry group for each D-brane, resembling the [[Degrees of freedom (physics and chemistry)|degree of freedom]] of moving it without changing its orientation. The limit where the two-cycles have zero area is the limit where these D-branes are on top of each other, so that one gets an enhanced local symmetry group.

Now, an open string stretched between two D-branes represents a Lie algebra generator, and the [[commutator]] of two such generator is a third one, represented by an open string which one gets by gluing together the edges of two open strings.
The latter relation between different open strings is dependent on the way 2-branes may intersect in the original M-theory, i.e. in the intersection numbers of two-cycles. Thus the Lie algebra depends entirely on these intersection numbers. The precise relation to the Cartan matrix is because the latter describes the commutators of the [[Simple root (root system)|simple root]]s, which are related to the two-cycles in the basis that is chosen.

Note that generators in the [[Cartan subalgebra]] are represented by open strings which are stretched between a D-brane and itself.

==See also==
* [[Dynkin diagram]]
* [[Exceptional Jordan algebra]]
* [[Fundamental representation]]
* [[Killing form]]
* [[Simple Lie group]]

==Notes==
{{reflist}}

==References==
* {{cite book | first1=William | last1=Fulton | authorlink=William Fulton (mathematician) | first2=Joe | last2=Harris |authorlink2=Joe Harris (mathematician) | title=Representation theory: A first course | series=[[Graduate Texts in Mathematics]] | volume=129 | publisher=Springer-Verlag | year=1991 | isbn=0-387-97495-4 | page=334 }}
* {{cite book | first=James E. | last=Humphreys | title=Introduction to Lie algebras and representation theory | series=[[Graduate Texts in Mathematics]] | volume=9 | publisher=Springer-Verlag | year=1972 | isbn=0-387-90052-7 | pages=55–56 }}
* {{Cite book|last=Kac|first= Victor G.|title=Infinite Dimensional Lie Algebras|edition=3rd|publisher=Cambridge University Press|year= 1990|isbn=978-0-521-46693-6}}.

== External links ==
* {{springer|title=Cartan matrix|id=p/c020530}}
* {{mathworld | urlname = CartanMatrix | title = Cartan matrix }}

{{DEFAULTSORT:Cartan Matrix}}
[[Category:Matrices]]
[[Category:Lie algebras]]
[[Category:Representation theory]]

Desargues's theorem

2017-06-09T06:35:03Z

Crasshopper:

[[Image:Desargues theorem alt.svg|thumb|350px|Perspective triangles. Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem states that the truth of the first condition is [[necessary and sufficient]] for the truth of the second.]]
In [[projective geometry]], '''Desargues's theorem''', named after [[Girard Desargues]], states:

:Two [[triangle]]s are in perspective ''axially'' [[if and only if]] they are in [[perspective (geometry)|perspective]] ''centrally''.

Denote the three [[vertex (geometry)|vertices]] of one triangle by {{math|''a'', ''b''}} and {{math|''c''}}, and those of the other by {{math|''A'', ''B''}} and {{math|''C''}}. Axial [[perspectivity]] means that lines {{math|{{overline|''ab''}}}} and {{math|{{overline|''AB''}}}} meet in a point, lines {{math|{{overline|''ac''}}}} and {{math|{{overline|''AC''}}}} meet in a second point, and lines {{math|{{overline|''bc''}}}} and {{math|{{overline|''BC''}}}} meet in a third point, and that these three points all lie on a common line called the ''axis of perspectivity''. Central perspectivity means that the three lines {{math|{{overline|''Aa''}}, {{overline|''Bb''}}}} and {{math|{{overline|''Cc''}}}} are concurrent, at a point called the ''center of perspectivity''.

This [[intersection theorem]] is true in the usual [[Euclidean plane]] but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly mathematicians "complete" the Euclidean plane by "adding" points at infinity following [[Jean-Victor Poncelet|Poncelet]]. This results in a [[projective plane]].

Desargues's theorem is true for the [[real projective plane]], for any projective space defined arithmetically from a [[field (mathematics)|field]] or [[division ring]], for any projective space of dimension unequal to two, and for any projective space in which [[Pappus's hexagon theorem|Pappus's theorem]] holds. However, there are some [[non-Desarguesian plane]]s in which Desargues's theorem is false.

==History==
Desargues never published this theorem, but it appeared in an appendix entitled ''Universal Method of M. Desargues for Using Perspective (Maniére universelle de M. Desargues pour practiquer la perspective)'' of a practical book on the use of perspective published in 1648<ref>{{harvtxt|Smith|1959|loc=pg.307}}</ref> by his friend and pupil Abraham Bosse (1602–1676).<ref>{{harvtxt|Katz|1998|loc=pg.461}}</ref>

==Projective versus affine spaces==
In an [[affine space]] such as the [[Euclidean plane]] a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the most basic of simple and intuitive geometric theorems whose natural home is in projective rather than affine space.

==Self-duality==
By definition, two triangles are [[Perspective (geometry)|perspective]] if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be [[similarity (geometry)|similar]].

Under the standard [[duality (projective geometry)|duality of plane projective geometry]] (where points correspond to lines and collinearity of points corresponds to concurrency of lines), the statement of Desargues's theorem is self-dual:<ref>This is due to the modern way of writing the theorem. Historically, the theorem only read, "In a projective space, a pair of centrally perspective triangles is axially perspective" and the dual of this statement was called the [[theorem#Converse|converse]] of Desargues's theorem and was always referred to by that name. See {{harv|Coxeter|1964|loc= pg. 19}}</ref> axial perspectivity is translated into central perspectivity and vice versa. The Desargues configuration (below) is a self-dual configuration.<ref>{{harv|Coxeter|1964}} pp. 26–27.</ref>

==Proof of Desargues's theorem==
Desargues's theorem holds for projective space of any dimension over any field or division ring, and also holds for abstract projective spaces of dimension at least 3. In dimension 2 the planes for which it holds are called [[Desarguesian plane]]s and are the same as the planes that can be given coordinates over a division ring. There are also many [[non-Desarguesian plane]]s where Desargues's theorem does not hold.

===Three-dimensional proof===
Desargues's theorem is true for any projective space of dimension at least 3, and more generally for any projective space that can be embedded in a space of dimension at least 3.

Desargues's theorem can be stated as follows:

:If lines {{math|{{overline|''Aa''}}, {{overline|''Bb''}}}} and {{math|{{overline|''Cc''}}}} are concurrent (meet at a point), then
:the points {{math|{{overline|''AB''}} ∩ {{overline|''ab''}}, {{overline|''AC''}} ∩ {{overline|''ac''}}}} and {{math|{{overline|''BC''}} ∩ {{overline|''bc''}}}} are [[collinear]].

The points {{math|''A'', ''B'', ''a''}} and {{math|''b''}} are coplanar (lie in the same plane) because of the assumed concurrency of {{math|{{overline|''Aa''}}}} and {{math|{{overline|''Bb''}}}}. Therefore, the lines {{math|{{overline|''AB''}}}} and {{math|{{overline|''ab''}}}} belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point {{math|{{overline|''AB''}} ∩ {{overline|''ab''}}}} belongs to both planes. By a symmetric argument, the points {{math|{{overline|''AC''}} ∩ {{overline|''ac''}}}} and {{math|{{overline|''BC''}} ∩ {{overline|''bc''}}}} also exist and belong to the planes of both triangles. Since these two planes intersect in more than one point, their intersection is a line that contains all three points.

This proves Desargues's theorem if the two triangles are not contained in the same plane. If they are in the same plane, Desargues's theorem can be proved by choosing a point not in the plane, using this to lift the triangles out of the plane so that the argument above works, and then projecting back into the plane.
The last step of the proof fails if the projective space has dimension less than 3, as in this case it may not be possible to find a point outside the plane.

[[Monge's theorem]] also asserts that three points lie on a line, and has a proof using the same idea of considering it in three rather than two dimensions and writing the line as an intersection of two planes.

===Two-dimensional proof===
As there are [[non-Desarguesian projective plane]]s in which Desargues's theorem is not true,<ref>The smallest examples of these can be found in {{harvnb|Room|Kirkpatrick|1971}}.</ref> some extra conditions need to be met in
order to prove it. These conditions usually take the form of assuming the existence of sufficiently many [[collineation]]s of a certain type, which in turn leads to showing that the underlying algebraic coordinate system must be a [[division ring]] (skewfield).<ref>{{harv|Albert|Sandler|1968}}, {{harv|Hughes|Piper|1973}}, and {{harv|Stevenson|1972}}.</ref>

==Relation to Pappus's theorem==
[[Pappus's hexagon theorem]] states that, if a [[hexagon]] {{math|''AbCaBc''}} is drawn in such a way that vertices {{math|''a'', ''b''}} and {{math|''c''}} lie on a line and vertices {{math|''A'', ''B''}} and {{math|''C''}} lie on a second line, then each two opposite sides of the hexagon lie on two lines that meet in a point and the three points constructed in this way are collinear. A plane in which Pappus's theorem is universally true is called ''Pappian''.
{{harvtxt|Hessenberg|1905}}<ref>According to {{harv|Dembowski|1968|loc= pg. 159, footnote 1}}, Hessenberg's original proof is not complete; he disregarded the possibility that some additional incidences could occur in the Desargues configuration. A complete proof is provided by {{harvnb|Cronheim|1953}}.</ref> showed that Desargues's theorem can be deduced from three applications of Pappus's theorem.<ref>{{harvnb|Coxeter|1969|loc=p. 238, section 14.3}}</ref>

The [[Theorem#Converse|converse]] of this result is not true, that is, not all Desarguesian planes are Pappian. Satisfying Pappus's theorem universally is equivalent to having the underlying coordinate system be [[commutative]]. A plane defined over a non-commutative division ring (a division ring that is not a field) would therefore be Desarguesian but not Pappian. However, due to [[Wedderburn's little theorem]], which states that all ''finite'' division rings are fields, all ''finite'' Desarguesian planes are Pappian. There is no known completely geometric proof of this fact, although {{harvtxt|Bamberg|Penttila|2015}} give a proof that uses only "elementary" algebraic facts (rather than the full strength of Wedderburn's little theorem).

==The Desargues configuration==
{{main|Desargues configuration}}
[[Image:Mutually-inscribed-pentagons.svg|thumb|The Desargues configuration viewed as a pair of mutually inscribed pentagons: each pentagon vertex lies on the line through one of the sides of the other pentagon.]]
The ten lines involved in Desargues's theorem (six sides of triangles, the three lines {{math|{{overline|''Aa''}}, {{overline|''Bb''}}}} and {{math|{{overline|''Cc''}}}}, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines passes through three of the ten points, and each of the ten points lies on three of the ten lines. Those ten points and ten lines make up the [[Desargues configuration]], an example of a [[projective configuration]]. Although Desargues's theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more [[symmetry|symmetric]]: ''any'' of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.

==See also==
* [[Pascal's theorem]]

==Notes==
{{reflist}}

==References==
*{{Citation | last1 = Albert | first1 = A. Adrian | last2 = Sandler | first2 = Reuben | title = An Introduction to Finite Projective Planes | publisher = Holt, Rinehart and Winston | place = New York | year = 1968}}
* {{Citation | last1 = Bamberg | first1 = John | last2 = Penttila | first2 = Tim | title = Completing Segre's proof of Wedderburn's little theorem | journal = Bulletin of the London Mathematical Society | year = 2015 | volume = 47 | pages = 483–492 | doi = 10.1112/blms/bdv021}}
* {{Citation | last = Casse | first = Rey | title = Projective Geometry: An Introduction | publisher = Oxford University Press | place = Oxford | year = 2006 | isbn =0-19-929886-6 }}
* {{citation|last=Coxeter|first=H.S.M.|title=Projective Geometry|year=1964|publisher=Blaisdell|location=New York}}
*{{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Introduction to Geometry | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-50458-0 | mr=123930 | year=1969}}
* {{citation|last=Cronheim|first=A.|title=A proof of Hessenberg's theorem|journal=Proceedings of the American Mathematical Society|year=1953|volume=4|pages=219–221|doi=10.2307/2031794}}
*{{Citation | last = Dembowski | first = Peter | title = Finite Geometries | publisher = Springer Verlag | place = Berlin | year = 1968}}
*{{citation|title= Beweis des Desarguesschen Satzes aus dem Pascalschen
|journal=Mathematische Annalen
|publisher=Springer |place=Berlin / Heidelberg
|issn=1432-1807
|volume =61|issue= 2 |year= 1905
|doi=10.1007/BF01457558
|pages=161–172
|first=Gerhard|last= Hessenberg}}
* {{Citation
| author = [[David Hilbert|Hilbert, David]]; [[Stephan Cohn-Vossen|Cohn-Vossen, Stephan]]
| title = Geometry and the Imagination
| edition = 2nd
| year = 1952
| publisher = Chelsea
| isbn = 0-8284-1087-9
| pages = 119–128}}
*{{citation| last1=Hughes|first1=Dan|last2=Piper|first2=Fred| title=Projective Planes | publisher=Springer-Verlag | year=1973 | isbn=0-387-90044-6}}
*{{Citation | last = Kárteszi | first = F. | title = Introduction to Finite Geometries| publisher = North-Holland | place = Amsterdam | year = 1976 | isbn = 0-7204-2832-7}}
*{{Citation | last = Katz |first=Victor J.|title=A History of Mathematics:An Introduction|edition=2nd|publisher=Addison Wesley Longman|place=Reading, Mass.|year=1998|isbn=0-321-01618-1}}
*{{Citation | authorlink1=T. G. Room | last1 = Room | first1 = T. G. | last2 = Kirkpatrick | first2 = P. B. | title = Miniquaternion Geometry | publisher = Cambridge University Press | place = Cambridge | year = 1971 |isbn = 0-521-07926-8}}
*{{Citation |last= Smith|first=David Eugene|title=A Source Book in Mathematics|publisher=Dover|place=New York|year=1959|isbn=0-486-64690-4}}
*{{Citation | last = Stevenson | first = Frederick W. | title = Projective Planes | publisher = W.H. Freeman and Company | place = San Francisco |year = 1972 | isbn = 0-7167-0443-9}}
*{{eom|id=d/d031320|first=M.I.|last= Voitsekhovskii|title=Desargues assumption}}

== External links ==
*[http://mathworld.wolfram.com/DesarguesTheorem.html Desargues Theorem] at [[MathWorld]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/Desargues.shtml Desargues's Theorem] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Curriculum/Geometry/MongeTheorem.shtml Monge via Desargues] at [[cut-the-knot]]
* [http://planetmath.org/?op=getobj&from=objects&id=4514 Proof of Desargues's theorem] at [[PlanetMath]]
* [http://math.kennesaw.edu/~mdevilli/desargues.html Desargues's Theorem] at [http://math.kennesaw.edu/~mdevilli/JavaGSPLinks.htm Dynamic Geometry Sketches]

[[Category:Theorems in projective geometry]]
[[Category:Proof without words]]
[[Category:Theorems in geometry]]
[[Category:Theorems in plane geometry]]
[[Category:Euclidean plane geometry]]

Jetpack (Firefox project)

2017-06-01T14:45:18Z

Crasshopper:

{{Infobox software
| name = Jetpack
| logo =
| screenshot =
| caption =
| developer = [[Mozilla Corporation]]/[[Mozilla Foundation]]
| latest release version = 1.14<ref>{{cite web|url=https://wiki.mozilla.org/Jetpack/Release_Notes|title=Jetpack/Release Notes|accessdate=26 March 2013|publisher=mozilla.org}}</ref>
| latest release date = {{release date|2013|03|26}}
| frequently updated =
| programming language = [[HTML]], [[Cascading Style Sheets|CSS]] and [[JavaScript]]
| operating system = [[Cross-platform]]
| platform = [[Mozilla application framework|Mozilla]]
| language =
| genre = [[Software development kit]] (SDK)
| license = [[Mozilla Public License|MPL 2.0]]
| website = https://addons.mozilla.org/en-US/developers/
}}
{{Firefox TOC}}
'''Jetpack''' was a working group which wrote a [[software development kit]] for [[Firefox]] [[Add-on (Mozilla)|add-ons]]. They produced the Add-on SDK, a set of [[Application programming interface|APIs]], a [[Run-time system|runtime]], and a command-line tool for creating and running add-ons, and the Add-on Builder, a Web-based [[integrated development environment]] which used the SDK.<ref>{{cite web|url=https://addons.mozilla.org/en-US/developers/docs/sdk/latest/|title=Add-on SDK Documentation|accessdate=22 June 2011|publisher=mozilla.org}}</ref><ref>{{cite web|url=https://builder.addons.mozilla.org/|title=Add-on Builder|accessdate=16 June 2011|publisher=mozilla.org}}</ref>

Add-ons developed with the SDK were written in [[HTML]], [[Cascading Style Sheets|CSS]], and [[JavaScript]] using [[CommonJS]] conventions. They did not require the user to restart Firefox when they were installed or uninstalled. The SDK's APIs were high-level, task-oriented, and designed to insulate developers from changes across Firefox versions.<ref>{{cite web|url=https://wiki.mozilla.org/Jetpack/FAQ|title=Jetpack/FAQ|accessdate=22 June 2011|publisher=mozilla.org}}</ref>

Mozillians running the project made a tool called the Jetpack Prototype. APIs provided by the Jetpack Prototype were not compatible with the Add-on SDK.<ref>{{cite web|url=https://jetpack.mozillalabs.com/prototype.html |title=Jetpack Prototype |accessdate=22 June 2011 |publisher=mozillalabs.org |deadurl=yes |archiveurl=https://web.archive.org/web/20110708192300/https://jetpack.mozillalabs.com/prototype.html |archivedate= 8 July 2011 |df= }}</ref>

==References==
{{reflist}}

==External links==
*[https://wiki.mozilla.org/Jetpack Jetpack Project Page]
*[https://addons.mozilla.org/en-US/developers/builder Add-on Builder and SDK]
*[https://addons.mozilla.org/en-US/developers/docs/sdk/latest/ Add-on SDK Documentation]

{{Mozilla}}

[[Category:Mozilla add-ons]]

{{Compu-library-stub}}

Jetpack (Firefox project)

2017-06-01T14:44:36Z

Crasshopper:

{{Infobox software
| name = Jetpack
| logo =
| screenshot =
| caption =
| developer = [[Mozilla Corporation]]/[[Mozilla Foundation]]
| latest release version = 1.14<ref>{{cite web|url=https://wiki.mozilla.org/Jetpack/Release_Notes|title=Jetpack/Release Notes|accessdate=26 March 2013|publisher=mozilla.org}}</ref>
| latest release date = {{release date|2013|03|26}}
| frequently updated =
| programming language = [[HTML]], [[Cascading Style Sheets|CSS]] and [[JavaScript]]
| operating system = [[Cross-platform]]
| platform = [[Mozilla application framework|Mozilla]]
| language =
| genre = [[Software development kit]] (SDK)
| license = [[Mozilla Public License|MPL 2.0]]
| website = https://addons.mozilla.org/en-US/developers/
}}
{{Firefox TOC}}
'''Jetpack''' was a working group which wrote a [[software development kit]] for [[Firefox]] [[Add-on (Mozilla)|add-ons]]. They produced the Add-on SDK, a set of [[Application programming interface|APIs]], a [[Run-time system|runtime]], and a command-line tool for creating and running add-ons, and the Add-on Builder, a Web-based [[integrated development environment]] which used the SDK.<ref>{{cite web|url=https://addons.mozilla.org/en-US/developers/docs/sdk/latest/|title=Add-on SDK Documentation|accessdate=22 June 2011|publisher=mozilla.org}}</ref><ref>{{cite web|url=https://builder.addons.mozilla.org/|title=Add-on Builder|accessdate=16 June 2011|publisher=mozilla.org}}</ref>

Add-ons developed with the SDK were written in [[HTML]], [[Cascading Style Sheets|CSS]], and [[JavaScript]] using [[CommonJS]] conventions. They do not require the user to restart Firefox when they are installed and uninstalled. The SDK's APIs are high-level, task-oriented, and designed to insulate developers from changes across Firefox versions.<ref>{{cite web|url=https://wiki.mozilla.org/Jetpack/FAQ|title=Jetpack/FAQ|accessdate=22 June 2011|publisher=mozilla.org}}</ref>

Mozillians running the project made a tool called the Jetpack Prototype. APIs provided by the Jetpack Prototype were not compatible with the Add-on SDK.<ref>{{cite web|url=https://jetpack.mozillalabs.com/prototype.html |title=Jetpack Prototype |accessdate=22 June 2011 |publisher=mozillalabs.org |deadurl=yes |archiveurl=https://web.archive.org/web/20110708192300/https://jetpack.mozillalabs.com/prototype.html |archivedate= 8 July 2011 |df= }}</ref>

==References==
{{reflist}}

==External links==
*[https://wiki.mozilla.org/Jetpack Jetpack Project Page]
*[https://addons.mozilla.org/en-US/developers/builder Add-on Builder and SDK]
*[https://addons.mozilla.org/en-US/developers/docs/sdk/latest/ Add-on SDK Documentation]

{{Mozilla}}

[[Category:Mozilla add-ons]]

{{Compu-library-stub}}

Jetpack (Firefox project)

2017-06-01T14:42:35Z

Crasshopper: jetpack is not defunct. web extensions is preferred

{{Infobox software
| name = Jetpack
| logo =
| screenshot =
| caption =
| developer = [[Mozilla Corporation]]/[[Mozilla Foundation]]
| latest release version = 1.14<ref>{{cite web|url=https://wiki.mozilla.org/Jetpack/Release_Notes|title=Jetpack/Release Notes|accessdate=26 March 2013|publisher=mozilla.org}}</ref>
| latest release date = {{release date|2013|03|26}}
| frequently updated =
| programming language = [[HTML]], [[Cascading Style Sheets|CSS]] and [[JavaScript]]
| operating system = [[Cross-platform]]
| platform = [[Mozilla application framework|Mozilla]]
| language =
| genre = [[Software development kit]] (SDK)
| license = [[Mozilla Public License|MPL 2.0]]
| website = https://addons.mozilla.org/en-US/developers/
}}
{{Firefox TOC}}
'''Jetpack''' was a project that develops tools and frameworks to ease development of [[Firefox]] [[Add-on (Mozilla)|add-ons]]. The project produced the Add-on SDK, a set of [[Application programming interface|APIs]], a [[Run-time system|runtime]], and a command-line tool for creating and running add-ons, and the Add-on Builder, a Web-based [[integrated development environment]] which used the SDK.<ref>{{cite web|url=https://addons.mozilla.org/en-US/developers/docs/sdk/latest/|title=Add-on SDK Documentation|accessdate=22 June 2011|publisher=mozilla.org}}</ref><ref>{{cite web|url=https://builder.addons.mozilla.org/|title=Add-on Builder|accessdate=16 June 2011|publisher=mozilla.org}}</ref>

Add-ons developed with the SDK were written in [[HTML]], [[Cascading Style Sheets|CSS]], and [[JavaScript]] using [[CommonJS]] conventions. They do not require the user to restart Firefox when they are installed and uninstalled. The SDK's APIs are high-level, task-oriented, and designed to insulate developers from changes across Firefox versions.<ref>{{cite web|url=https://wiki.mozilla.org/Jetpack/FAQ|title=Jetpack/FAQ|accessdate=22 June 2011|publisher=mozilla.org}}</ref>

Mozillians running the project made a tool called the Jetpack Prototype. APIs provided by the Jetpack Prototype were not compatible with the Add-on SDK.<ref>{{cite web|url=https://jetpack.mozillalabs.com/prototype.html |title=Jetpack Prototype |accessdate=22 June 2011 |publisher=mozillalabs.org |deadurl=yes |archiveurl=https://web.archive.org/web/20110708192300/https://jetpack.mozillalabs.com/prototype.html |archivedate= 8 July 2011 |df= }}</ref>

==References==
{{reflist}}

==External links==
*[https://wiki.mozilla.org/Jetpack Jetpack Project Page]
*[https://addons.mozilla.org/en-US/developers/builder Add-on Builder and SDK]
*[https://addons.mozilla.org/en-US/developers/docs/sdk/latest/ Add-on SDK Documentation]

{{Mozilla}}

[[Category:Mozilla add-ons]]

{{Compu-library-stub}}

The Power of Myth

2017-04-03T18:23:01Z

Crasshopper: /* Companion book */ who is surprised here?

{{Infobox television |
| show_name = Joseph Campbell and the Power of Myth
| image = The Power of Myth.jpg
| caption =
| camera =
| picture_format =
| runtime = 360 minutes (60 minutes per episode)
| creator =
| starring =[[Joseph Campbell]] [[Bill Moyers]]
| country = United States
| network = [[Public Broadcasting Service|PBS]]
| first_aired = June 21, 1988
| last_aired = June 26, 1988
| num_episodes = 6
|}}
'''''The Power of Myth''''' is a book based on the 1988 [[Public Broadcasting Service|PBS]] documentary '''''Joseph Campbell and the Power of Myth'''''. The documentary was originally broadcast as six one-hour conversations between mythologist [[Joseph Campbell]] (1904–1987) and journalist [[Bill Moyers]]. It remains one of the most popular series in the history of American public television.<ref>{{Cite news|url=http://billmoyers.com/series/joseph-campbell-and-the-power-of-myth-1988/|title=Joseph Campbell and The Power of Myth {{!}} Shows {{!}} BillMoyers.com|newspaper=BillMoyers.com|language=en-US|access-date=2017-01-18}}</ref>

==Overview==
The interviews in the first five episodes were filmed at [[George Lucas]]'s [[Skywalker Ranch]] in California, with the sixth interview conducted at the [[American Museum of Natural History]] in New York, during the final two summers of Campbell's life. (The series was broadcast on television the year following his death.) In these discussions Campbell presents his ideas about [[comparative mythology]] and the ongoing role of [[myth]] in human society. These talks include excerpts from Campbell's seminal work ''[[The Hero with a Thousand Faces]]''<ref name=":0" />

==Documentary series==
The documentary series ''Joseph Campbell and the Power of Myth'' was broadcast in six parts:

*''Episode 1: The Hero's Adventure'' (first broadcast June 21, 1988 on PBS)
About Campbell, [[hero]] types, hero deeds, [[Jesus Christ]], the [[Gautama Buddha|Buddha]], [[Krishna]], movie heroes, [[Star Wars]] as a metaphor, an [[Iroquois]] story: the refusal of suitors, dragons, dreams and [[Jungian psychology]], “follow your bliss,” consciousness in plants, Gaia, [[Chartres cathedral]], spirituality vs. economics, emerging myths, “Earthrise” as a symbol.

*''Episode 2: The Message of the Myth'' (first broadcast June 22, 1988 on PBS)
[[Creation myth]]s, transcending duality, pairs of opposites, God vs. Nature, sin, morality, participation in sorrow, the Gospel of Thomas, Old Time Religion, computers, religion as “software,” the story of [[Indra]]: “What a great boy am I!,” participation in society.

*''Episode 3: The First Storytellers'' (first broadcast June 23, 1988 on PBS)
Animal memories, harmonization with body and life-cycle, consciousness vs. its vehicle, killing for food, story: “The Buffalo's Wife,” buffalo massacre, initiation ritual, rituals diminishing, crime increasing, artists, the Shaman, the center of the world.

*''Episode 4: Sacrifice and Bliss'' (first broadcast June 24, 1988 on PBS)
[[Chief Seattle]], the sacred Earth, agricultural renewal, human sacrifice, sacrifice of the Mass, transcendence of death, story: “The Green Knight,” societal dictates vs. following bliss, “hidden hands” guiding life's work.

*''Episode 5: Love and the Goddess'' (first broadcast June 25, 1988 on PBS)
The [[Troubadour]]s, [[Eros]], romantic love, [[Tristan]], libido vs. credo, separation from love, [[Satan]], loving your enemy, the [[Crucifixion]] as atonement, virgin birth, the story of [[Isis]], [[Osiris]] and [[Horus]], [[the Madonna]], the [[Big Bang Night in Canada|Big Bang]], the correlation between the earth or mother Goddess and images of fertility (the sacred feminine).

*''Episode 6: Masks of Eternity'' (first broadcast June 26, 1988 on PBS)
Identifying with the infinite, the [[circle]] as a symbol, [[clown]]s and [[mask]]s, epiphanies and [[James Joyce]], artistic arrest, the monstrous as sublime, the dance of [[Shiva]], that which is beyond words.

==Companion book==
{{Original research|section|date=June 2010}}
{{Infobox book
| name = The Power of Myth
| image =
| caption =
| author = [[Joseph Campbell]] [[Bill Moyers]] [[Betty Sue Flowers]] (ed.)
| cover_artist = Cathy Saksa
| country = United States
| language = English
| subject = [[Myth]]
| genre = Non-Fiction
| publisher = Doubleday
| release_date = 1988
| media_type = Print ([[hardcover]], [[paperback]])
| pages = 231 (hardcover)
| isbn = 0-385-24773-7 |isbn_note= (hardcover) ISBN 0-385-24774-5 (paperback)
| dewey= 291.1/3 19
| congress= BL304 .C36 1988
| oclc= 17650532
}}

The companion book for the series, ''The Power of Myth'' (Joseph Campbell, Bill Moyers, and editor [[Betty Sue Flowers]]), was released in 1988 at the same time the series aired on PBS. In the editor's note to ''The Power of Myth'', Flowers credits [[Jacqueline Kennedy Onassis]], as "the [[Doubleday (publisher)|Doubleday]] editor, whose interest in the ideas of Joseph Campbell was the prime mover in the publication of this book."<ref name=":1">{{Cite book|title=The Power of Myth|last=Campbell|first=Joseph|publisher=Anchor|year=1991|isbn=9780385418867|location=New York|pages=64|quote=|via=}}</ref> The book follows the format of the documentary and provides additional discussions not included in the original six-hour release. Chapters:

# Myth and the Modern World
# The Journey Inward
# The First Storytellers
# Sacrifice and Bliss
# The Hero's Adventure
# The Gift of the Goddess
# Tales of Love and Marriage
# Masks of Eternity
# The Tale of Buddha

''The Power of Myth'' is based on the interviews between Joseph Campbell and Bill Moyers that were the basis for the acclaimed television series. It deals with the universality and evolution of myths in the history of the human race and the place of myths in modern society. Campbell blends accounts of his own upbringing and experience with stories from many cultures and civilizations to present the reader with his most compelling thesis that modern society is going through a transition from the old mythologies and traditions to a new way of thinking where a global mythology will emerge.

Some of the material in the first chapter comes from Campbell's previously published books, ''The Hero with a Thousand Faces'' and ''The Masks of God''. The main theme of the book is the universality of myths—what Campbell calls "mankind's one great story"— that occur throughout the history of mankind, no matter which epoch or whichever culture or society is considered.<ref name=":1" /> Myths are the body of stories and legends that a people perceive as being an integral part of their culture. Before the invention of writing, these stories and legends were handed down from generation to generation in the form of rituals and oral traditions. The reappearance of certain themes, time and again, in different mythologies, leads to the realization that these themes portray universal and eternal truths about mankind.

Campbell defines the function of a mythology as the provision of a cultural framework for a society or people to educate their young, and to provide them with a means of coping with their passage through the different stages of life from birth to death. In a general sense myths include religion as well and the development of religion is an intrinsic part of a society's culture. A mythology is inevitably bound to the society and time in which it occurs and cannot be divorced from this culture and environment. This is true even though Western society previously learnt from, and was informed by, the mythology of other cultures by including the study of Greek and Roman writings as part of its heritage.

The record of the history of the development of a culture and society is embodied in its mythology. For example, the Bible describes the evolution of the Judeo-Christian concept of God from the time when the Jews were in Babylon and the god they worshiped corresponded to a local tribal god, to when the concept became that of a world savior as a result of the Hebrews becoming a major force in the East Mediterranean region. The geographic context of a specific mythology also plays a role in its evolution. The physical scope of Biblical mythology was limited to the general area of the Middle East but in other parts of the world, Chinese and Aztec religions and cultures emerged as separate and distinct belief systems. When different cultures expand their spheres of influence they eventually come into contact with each other, and the outcome of the collision, be it conquest, subjugation, or amalgamation, will be evident in the resultant mythology.

The form and function of mythology in the modern world is the main topic of this chapter and to illustrate his ideas, Campbell recounts aspects of his own earlier life. Without specifically stating it, the assumption is made that the modern world under consideration is that of Campbell's world—the Christian-based, urbanized culture of North America, the so-called Modern Western Society.

Campbell describes his own upbringing as a Roman Catholic and his early fascination with the myths and stories of the American Indians. He recalls the excitement he felt when he realized that the motifs of creation, death, resurrection, and ascension into heaven, which the nuns were teaching him at his school, also occurred in American Indian myths. This was the beginning of his lifelong interest in comparative mythology. Later on in life he found the same universal themes in Hinduism and in the medieval Arthurian legends.

The discussion considers the role of myth and ritual in contemporary society. Contemporary rituals are carried out to mark special events in private lives, such as an individual's marriage or enlistment in a branch of the armed forces and, on public occasions such as the inauguration of civil and national leaders. In the Introduction to the book, Moyers recalls Campbell's description of the solemn state funeral after the assassination of John F. Kennedy, as an "illustration of the high service of ritual to a society," and where Campbell identifies the ritualized occasion as fulfilling a great social necessity.

In general, however, Campbell and Moyers, reach the conclusion that there is a lack of effective mythology and ritual in modern American society. They find nothing that compares with the powerful puberty rituals of primitive societies. They claim that the exclusion of classical studies from the modern educational syllabus has led to a lack of awareness of the mythological foundations of western society's heritage. This, combined with an increased materialism and emphasis on technology, has led to modern youth in New York, becoming alienated from the mainstream of society and inventing their own morality, initiations and gangs.

Marriage, as an example of a paramount modern social institution, becomes the next subject of discussion. Campbell differentiates between marriage and love affairs and imparts some very lofty ideals to marriage, in contrast to love affairs, that he categorically states inevitably end in disappointment. True marriage, in Campbell's opinion, embodies a spiritual identity and invokes the image of an incarnate God. Campbell and Moyers agree that the main objective of marriage is not the birth of children and the raising of families. They discard the concept of perpetuation of the human species as being the primary function of marriage and relegate this to a first stage. This first stage is followed by a second one where the offspring have departed into the world and only the couple is left. Campbell invokes the image of marriage as being an ordeal in which the ego is sacrificed to a relationship in which two have become one. This, he states, is a mythological image that embodies the sacrifice of the visible for a transcendent good. Campbell labels this stage of marriage as the alchemical stage. On the subject of the ritual of marriage, Campbell and Moyers complain that it has lost its force and has become a mere remnant of the original; they contend that the ritual that once conveyed an inner reality is now merely form.

The interviews between Campbell and Moyers are recorded at George Lucas' Skywalker Ranch. Campbell and Lucas became friends when Lucas publicly acknowledged the influence Campbell's writings had on the development of his hugely successful film "Star Wars." Campbell expresses great enthusiasm for this film; a film that he says conforms to classical mythological legends. So it is not surprising that there are many references to the characters from "Star Wars" throughout the book. In a similar fashion, John Wayne is identified as a modern myth and Campbell recalls Douglas Fairbanks as having been a boyhood hero.

At the beginning of this chapter, and in other parts of the book, Campbell states that modern society lacks the stability it previously derived from being educated in the mythology and legends of the Greek and Roman classics. Campbell and Moyers agree that there is no effective mythology in modern society by which individuals can relate to their role in the world. An analysis of the national symbols of the United States is used by Campbell to illustrate the ability for myths to incorporate the beliefs of a whole society and to provide the mythology to unify a nation. More recently, when the image of the earth, taken from the lunar landings, was published, it led to the universal realization that human beings must identify with the entire planet. This concept of the emergence of a new mythology based on global aspects of life is reiterated several times by Campbell.

==''Star Wars''==
In the first episode of the series, ''The Hero's Adventure'',<ref name=":0">[http://www.tv.com/joseph-campbell-and-the-power-of-myth/the-heros-adventure/episode/432756/summary.html Joseph Campbell and the Power of Myth: The Hero's Adventure - TV.com]</ref> and the fifth chapter of the book, "The Hero's Adventure," Moyers and Campbell discuss [[George Lucas]]' report that Campbell's work directly influenced the creation of the ''[[Star Wars]]'' films.
Moyers and Lucas filmed an interview 12 years later in 1999, modeled after ''The Power of Myth''. It was called the ''Mythology of Star Wars with George Lucas & Bill Moyers'' and further discussed the impact of Campbell's work on Lucas' films.<ref>[http://www.films.com/id/11017/The_Mythology_of_Star_Wars_with_George_Lucas_and_Bill_Moyers.htm Films for the Humanities and Sciences - The Mythology of Star Wars with George Lucas and Bill Moyers]</ref>

==Errata==
The words attributed to Chief Seattle, read by Campbell in the fourth episode of the series, were actually written by Ted Perry for a 1972 [[ecology]] film called ''Home''. Perry adapted the text from newspaper accounts that were, in turn, published years after Chief Seattle delivered the actual speech.<ref>[http://www.snopes.com/quotes/seattle.asp snopes.com: Chief Seattle Speech]</ref><ref>Newsweek May 4, 1992 p 68 [subject] the Arts "Just Too Good to Be True: another reason to beware of false eco-prophets" by Malcolm Jones Jr. with Ray Sawhill</ref>

== Campbell's views on religious fundamentalism in the "Power of Myth" ==
Campbell did respond to the yearning by religious fundamentalists for a return to the 'old-time religion'. The following part of the interview reveals Campbell's views on religious fundamentalism which, in the modern world, is conceptualised by the desire by some adherents of world's disparate religions, to return to the old practices of a given religion.

Bill Moyers: You've seen what happens when primitive societies are unsettled by white man's
civilization. They go to pieces, they disintegrate, they become
diseased. Hasn't the same thing been happening to us since myths began to disappear?

Joseph Campbell: Absolutely, it has.

Bill Moyers: Isn't that why conservative religions today are calling for the old-time religion?

Joseph Campbell: Yes, and they're making a terrible mistake. They are going back to something that is vestigial {something that has passed}, that doesn't serve life.

Bill Moyers: But didn't it serve us?

Joseph Campbell: Sure it did.

Bill Moyers: I understand the yearning. In my youth I had fixed stars. They comforted me with their permanence. They gave me a known horizon. And they told me there
was loving, kind, and just father out there looking down on me, ready to receive me, thinking of my concerns all the time. Now, Saul Bellow says that science has made a housecleaning of beliefs. But there was value in these things for me. I am today what I am because of those beliefs. I wonder what happens to children who don't have those fixed stars, that known horizon - those myths?

Joseph Campbell: Well, as I said, all you have to do is read the newspapers. It's a mess. On this immediate level of life and structure, myths offer life models. But the models have to be appropriate to the time in which you are living, and our time has changed so fast that what was proper fifty years ago is not proper today. ''The virtues of the past are the vices of today. And many of what were thought to be the vices of the past are the necessities of today. The moral order has to catch up with the moral necessities of actual life in time, here and now''. ''And that is what we are not doing. The old-time religion belongs to another age, another people, another set of human values, another universe. By going back you throw yourself out of sync with history. Our kids lose their faith in the religions that were taught to them, and they go inside."<ref>{{Cite web|title = Myths-Dreams-Symbols|url = http://mythsdreamssymbols.com/functionsofmyth.html|website = mythsdreamssymbols.com|accessdate = 2015-11-10}}</ref>

==Related media==
'''Print'''
: Moyers, Bill and Joseph Campbell. ''The Power of Myth'' (1988). [[Betty Sue Flowers]] (ed.). New York: Doubleday, hardcover: ISBN 0-385-24773-7

'''Audio'''
: ''Joseph Campbell and the Power of Myth with Bill Moyers'' (2001). Penguin/Highbridge, ISBN 1-56511-510-4.

'''Video'''
: ''Joseph Campbell and the Power of Myth with Bill Moyers'' (2010). Bonus Interview with George Lucas on Mythology from ''The Mythology of Star Wars'', conversation with Campbell from 1981 ''Bill Moyers Journal''. Acorn Media/Athena.

==See also==
* ''[[The Hero's Journey|The Hero's Journey: The World of Joseph Campbell]]'' (1987)
* [[Comparative Mythology]]

==References==
{{Reflist}}

==External links==
*[http://www.jcf.org/new/index.php?categoryid=83&p9999_action=details&p9999_wid=765 Joseph Campbell Foundation ''Power of Myth'' page]
*[http://homevideo.about.com/od/dvdrevi3/fr/Review-Joseph-Campbell-and-The-Power-of-Myth-DVD-a.htm About.com review of 2011 DVD]
*[http://www.sfgate.com/cgi-bin/article.cgi?file=/g/a/2002/11/15/notes111502.DTL ''San Francisco Chronicle'' review]
*[http://www.tv.com/joseph-campbell-and-the-power-of-myth/show/37853/summary.html tv.com]
*[http://www.powells.com/cgi-bin/biblio?inkey=62-0385247745-0 Powell's review (book)]
* {{IMDb title|0296362|Joseph Campbell and the Power of Myth}}
{{Bill Moyers}}
{{Joseph Campbell}}

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[[Category:Books by Joseph Campbell]]
[[Category:Comparative mythology]]
[[Category:American documentary television films]]
[[Category:PBS network shows]]
[[Category:American documentary television series]]