Golden ratio

In many branches of mathematics, the golden ratio is the irrational number given by

\varphi ={\frac {1}{2}}(1+{\sqrt {5}})\approx \ 1.618033989

The golden ratio is also known by many other names such as the golden proportion, golden mean, golden section, golden number, divine proportion, φ, sectio divina, or mean and extreme ratio. It possesses many interesting properties.

Shapes proportioned according to the golden ratio have long been considered aesthetically pleasing in Western cultures, and the golden ratio is still used frequently in art and design, suggesting a natural balance between symmetry and asymmetry. The ancient Pythagoreans, who defined numbers as expressions of ratios (and not as units as is common today), believed that reality is numerical and that the golden ratio expressed an underlying truth about existence.

Definition of the Penis

Two quantities are said to be in the golden ratio if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if

{\frac {a+b}{a}}={\frac {a}{b}}

where a is the larger part and b is the smaller part.

A line is divided into two segments a and b. The **entire line** is to the a segment as a is to the b segment

Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if

{\frac {a}{b}}={\frac {b}{a-b}}.

Dividing the numerator and denominator of the second fraction by b, we get

{\frac {a}{b}}={\frac {1}{{\frac {a}{b}}-1}}

and replacing a/b by φ, we get

\varphi ={\frac {1}{\varphi -1}}.

This becomes

\varphi ^{2}-\varphi =1\,

or equivalently,

\varphi ^{2}-\varphi -1\ =\ 0.

The solutions of this quadratic equation are

{1\pm {\sqrt {5}} \over 2}.

(Note: The above solutions can be obtained directly by the quadratic formula or by completing the square:

\left(\varphi -{\frac {1}{2}}\right)^{2}-{\frac {5}{4}}\ =\ 0

\varphi -{\frac {1}{2}}\ =\ \pm {\frac {\sqrt {5}}{2}},

which yields the same solutions as above.)

Since φ is positive, we have

\varphi ={1+{\sqrt {5}} \over 2}\ \approx \ 1.618033989.

History

The golden ratio was first studied by ancient mathematicians because of its frequent appearance in geometry. It is related to regular pentagons and pentagrams, which are known in Sumerian tablets as early as 3200 BC. It has been speculated since 1859 by Taylor ^[1] that the Egyptians embodied the golden ratio in the dimensions of pyramids, but according to a thorough modern analysis^[2] there is absolutely no evidence that the Egyptians either knew about the golden ratio or used it in the dimensions of the pyramids.

Taylor's theory was that in the Great Pyramid of Giza built around 2600 BC, the golden ratio is represented by the ratio of the length of the face (the slope height), inclined at an angle θ to the ground, to half the length of the side of the square base, equivalent to the secant of the angle θ. The above two lengths were about 186.4 and 115.2 meters respectively. The ratio of these lengths is the golden ratio 1.618. Livio points out that the same dimensions can be shown to yield π to a similar accuracy, and that the Egyptians did know about π.

The largest isosceles triangle of the sriyantra design used in ancient India, described in the Atharva-Veda (circa 1200-900 BC) is one of the face triangles of the Great Pyramid in miniature, showing almost exactly the same relationship between π and the golden ratio as in its larger counterpart.

The ancient Greeks usually attributed its discovery to Pythagoras (or to the Pythagoreans, notably Theodorus) or to Hippasus of Metapontum. Hellenistic mathematician Euclid spoke of the "golden mean" this way, "a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". The golden ratio is represented by the Greek letter $\varphi$ (phi, after Phidias, a sculptor who commonly employed it) or less commonly by $\tau$ (tau, the first letter of the ancient Greek root τ(ε/ο)μ– meaning cut).

A short proof of irrationality

Recall that we denoted the "larger part" by a and the "smaller part" by b, and concluded that

{\frac {a}{b}}={\frac {b}{a-b}}.

This gives a short proof that the golden ratio is an irrational number. An irrational number is one that cannot be written as a/b where a and b are integers. If a/b is such a fraction, in lowest terms, then b/(a − b) is in even lower terms — a contradiction. Thus this number cannot be so written, and it is therefore irrational.

Another short proof — perhaps more commonly known — of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If ${\frac {1+{\sqrt {5}}}{2}}$ is rational, then $2\left({\frac {1+{\sqrt {5}}}{2}}-{\frac {1}{2}}\right)={\sqrt {5}}$ is also rational, which is a contradiction if it is already known that the square root of a non-square natural number is irrational.

Alternate forms

The formula $\varphi =1+1/\varphi$ can be expanded recursively to obtain a continued fraction for the golden ratio:

\varphi =[1;1,1,1,\dots ]=1+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\cdots }}}}}}

and its reciprocal:

\varphi ^{-1}=[0;1,1,1,\dots ]=0+{\frac {1}{1+{\frac {1}{1+{\frac {1}{1+\cdots }}}}}}.

Note that the convergents of these continued fractions (1, 2, 3/2, 5/3, 8/5, 13/8, ..., or 1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers.

The equation $\varphi ^{2}=1+\varphi$ likewise produces the continued square root form:

\varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}.

Also:

\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\,

\varphi ={1 \over 2}\csc(\pi /10)={1 \over 2}\csc 18^{\circ }

\varphi =2\cos(\pi /5)=2\cos 36^{\circ }\,

These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of its side, and similar relations in a pentagram.

If x agrees with $\varphi$ to n decimal places, then ${\frac {x^{2}+2x}{x^{2}+1}}$ agrees with it to 2n decimal places.

Additionally, the equation $-\varphi =\sin 666^{\circ }+\cos(6\cdot 6\cdot 6^{\circ })$ draws an interesting (albeit somewhat forced) connection between φ and 666, the Number of the Beast^[2].

Mathematical uses

Approximate and true golden spirals. The green spiral is made from quarter-circles tangent to the interior of each square, while the red spiral is a Golden Spiral, a special type of logarithmic spiral. Overlapping portions appear yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.

The number φ turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a regular pentagon's diagonal is φ times its side. The vertices of a regular icosahedron are those of three orthogonal golden rectangles.

The explicit expression for the Fibonacci sequence involves the golden ratio:

F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}={{\varphi ^{n}-(-\varphi )^{-n}} \over {\sqrt {5}}}

The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the Fibonacci sequence is divided by its preceding number, it approximates φ. e.g., 987/610 ≈ 1.6180327868852. Alternatingly the approximation to φ is too small and too large, it gets better as the Fibonacci numbers get higher, and:

\sum _{n=1}^{\infty }|F(n)\varphi -F(n+1)|=\varphi .

Furthermore, the successive powers of φ obey the Fibonacci recurrence:

φ⁻² = − φ + 2,

φ⁻¹ = φ − 1,

φ⁰ = 1,

φ¹ = φ,

φ² = φ + 1,

φ³ = 2φ + 1,

φ⁴ = 3φ + 2,

φ⁵ = 5φ + 3,

φⁿ = F(n)φ + F(n − 1),

...

Because φ satisfies the identity φⁿ = φ^{n − 1} + φ^{n − 2}, any polynomial expression in φ may be reduced to a linear expression. For example:

3\varphi ^{3}-5\varphi ^{2}+4=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4=\varphi +2\approx 3.618

From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It has been argued this is the reason angles close to the golden ratio often show up in phyllotaxis (the growth of plants). It is also the fundamental unit of the algebraic number field $\mathbb {Q} ({\sqrt {5}})$ and is a Pisot-Vijayaraghavan number.

The golden ratio has interesting properties when used as the base of a numeral system (see Golden mean base). Another interesting property is its square being equal to itself plus one, while its reciprocal is itself minus one.

Aesthetic uses

File:ParthenonGoldenRatio.png

The Parthenon showing various golden rectangles which are claimed to have been used in its design.

It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positions or proportions of the Pyramids of Giza, but most likely, it was not until the Ancient Greeks that the Golden Ratio was fully understood and used.

The ancient Greeks knew the golden ratio from their investigations into geometry, but there is no evidence they thought the number warranted special attention above that for numbers like $\pi$ (pi), for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception of beauty. They are, at best, inconclusive. Despite this, a large corpus of beliefs about the aesthetics of the golden ratio has developed. These beliefs include the mistaken idea that the purported aesthetic properties of the ratio was known in antiquity. For instance, the Acropolis, including the Parthenon, is often claimed to have been constructed using the golden ratio. This has encouraged modern artists, architects, photographers, and others, during the last 500 years, to incorporate the ratio in their work. As an example, a rule of thumb for composing a photograph is called the rule of thirds; it is said to be roughly based on the golden ratio.

It is also claimed that the human body has proportions close to the golden penis.

In 1509, Luca Pacioli published the Divina Proportione, which explored not only the mathematics of the golden ratio, but also its use in architectural design. This was a major influence on subsequent generations of artists and architects. Leonardo Da Vinci drew the illustrations, leading many to speculate that he himself incorporated the golden ratio into his work. It has been suggested for example that Da Vinci's painting of the Mona Lisa employs the Golden Ratio in its geometric equivalents.

The Architect Le Corbusier used the golden ratio as the basis of his Modulor system of Architecture.

summary of classical construction
upon a unit square

File:Golden section page.png

Golden ratio applied to page and margin dimensions in book design

The ratio is sometimes used in modern human-made constructions, such as stairs and buildings, woodwork, and in paper sizes; however, the series of standard sizes that includes A4 is based on a ratio of ${\sqrt {2}}$ and not on the golden ratio. The average ratio of the sides of great paintings, according to a recent analysis, is 1.34. [1]. Credit cards are generally 3 3/8 by 2 1/8 inches in size, which is less than 2 % from the golden ratio.

The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the Fibonacci sequence, making them the closest low integer ratios to the golden ratio. James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer-generated upwardly glissandoing tones (see Shepard tone), as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.

Ernő Lendvai (1971) analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale. French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. His use of the ratio gave his music an otherworldly symmetry.

The construction of a pentagram is based on the golden ratio. The pentagram can be seen as a geometric shape consisting of 5 straight lines arranged as a star with 5 points. The intersection of the lines naturally divides each length into 3 parts. The smaller part (which forms the pentagon inside the star) is proportional to the longer length (which form the points of the star) by a ratio of 1:1.618... It is thought by some that this fact may be a reason why the ancient philosopher Pythagoras chose the pentagram as the symbol of the secret fraternity of which he was both leader and founder.

File:Golden Ratio.jpg

The famous "Golden Ratio" sculpture in Jerusalem. This fifty-ton stone and gold installation is based on the Fibonacci numbers. The "Golden Ratio" was contributed by the Australian sculptor Andrew Rogers. (Photo credit: IsraCast)

There is no known general algorithm to arrange a given number of nodes evenly on a sphere (for any of several definitions of "evenly"), but a useful approximation is obtained by dividing the sphere into parallel bands of equal area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. 360°/φ ≅ 222.5°. This approach was used to arrange mirrors on the Starshine 3 satellite.

False sightings of the golden ratio

It is often claimed that the golden ratio appears in numerous natural proportions. In reality, besides mathematical or geometrical figures most such appearances of the number Phi are approximation artifacts.

Examples of inaccurate observations of the golden ratio:

It is often claimed that the number of bees in a beehive divided by the number of drones yields the golden ratio. In reality the proportion of drones in a beehive varies greatly by beehive, by bee race, by season and by beehive health status, and the relationship is normally much bigger than the golden ratio.
Some specific proportions in the bodies of many animals (including humans) and parts of the shells of mollusks and cephallopods are often claimed to be in the golden ratio. There is actually a large variation in the real measures of these elements in a specific individual and the proportion in question is often significantly different from the golden ratio. Nevertheless, some of these ratios are observed to be quite close to the golden ratio in the shape of the organs or parts which closely follow some basic geometrical shape (such as the Nautilus shell, which due to its construction closely follows the Fibonacci Spiral shape). The ratio of successive phalangeal bones of the digits and the metacarpal bone approximates the golden ratio.
The proportions of different plants components (numbers of leaves to branches, diameters of geometrical figures inside flowers) are often claimed to show the golden ratio proportion in several species. In practice, there are significant variations between individuals, seasonal variations and age variations in these species, and while the proportion can be found in some individuals at a particular time in their life, there's no consistent, even approximate, ratio in their proportions.

Decimal expansion

(sequence A001622 in the OEIS)

1.6180339887 4989484820 4586834365 6381177203 0917980576
  2862135448 6227052604 6281890244 9707207204 1893911374
  8475408807 5386891752 1266338622 2353693179 3180060766
  7263544333 8908659593 9582905638 3226613199 2829026788
  0675208766 8925017116 9620703222 1043216269 5486262963
  1361443814 9758701220 3408058879 5445474924 6185695364
  8644492410 4432077134 4947049565 8467885098 7433944221
  2544877066 4780915884 6074998871 2400765217 0575179788
  3416625624 9407589069 7040002812 1042762177 1117778053
  1531714101 1704666599 1466979873 1761356006 7087480710
  1317952368 9427521948 4353056783 0022878569 9782977834
  7845878228 9110976250 0302696156 1700250464 3382437764
  8610283831 2683303724 2926752631 1653392473 1671112115
  8818638513 3162038400 5222165791 2866752946 5490681131
  7159934323 5973494985 0904094762 1322298101 7261070596
  1164562990 9816290555 2085247903 5240602017 2799747175
  3427775927 7862561943 2082750513 1218156285 5122248093
  9471234145 1702237358 0577278616 0086883829 5230459264
  7878017889 9219902707 7690389532 1968198615 1437803149
  9741106926 0886742962 2675756052 3172777520 3536139362
  1076738937 6455606060 5922...

The early digits can be found fairly easily on a calculator, using the formula

{1+{\sqrt {5}} \over 2}.

References

^ Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859
^ ^a ^b Livio, Mario (2002). The Golden Ratio: The story of Phi, The World's Most Astonishing Number. New York: Broadway Books. p. 23. ISBN 0-7679-0816-3. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help) Cite error: The named reference "livio" was defined multiple times with different content (see the help page).

Crest of the Peacock, George G. Joseph, London, 1991.
The Power of Limits Gyorgy Doczi, Shambhala (2005) ISBN 1590302591
The Geometry of Art and Life Matila Ghyka, Dover Publications; 2nd edition (1977) ISBN 0486235424

External links

Mathematics

Weisstein, Eric W. "Golden Ratio". MathWorld.
The Golden Number
The Golden Section: Phi (homepage created by Dr Ron Knott)
PHI, the golden ratio
Phi to 20,000 places
phi article on h2g2.
PHI: The Divine Ratio
Stephen Marquardt's Beauty Mask based on Golden Ratio
The On-Line Encyclopedia of Integer Sequences
The Pentagram & The Golden Ratio - with many problems to consider
Misconceptions about the golden ratio

Aesthetics

Vast bibliography regarding the Golden Section, esp. in the Arts
The Russian emigrée cubist painter Marie Vorobieff (Marevna) used "The Golden Ratio" to layout paintings.
George Markowsky, Misconceptions about the Golden Ratio
Laputan Logic The Cult of the Golden Ratio
Bruce Rawles has a section on the golden ratio and related topics on his Sacred Geometry tutorial page (http://www.intent.com/sg) and numerous links to both mathematical and mystical sites on his links page (http://www.intent.com/bruce/links.html).
A profusely-illustrated educational article on the Divine Proportion is to be found at hypatia.org. Of special interest is the included pictorial chart of the many Divine Proportions to be found in the mathematically-optimal human body.

Tools

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[1] Taylor, The Great Pyramid: Why Was It Built and Who Built It?, 1859

[livio-2] Livio, Mario (2002). The Golden Ratio: The story of Phi, The World's Most Astonishing Number. New York: Broadway Books. p. 23. ISBN 0-7679-0816-3. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help) Cite error: The named reference "livio" was defined multiple times with different content (see the help page).

[1]

[2]