Talk:Rhombohedron
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Trigonal trapezohedron?
As best I can tell, the rhombohedron and trigonal trapezohedron should be identical. All edges are equal length, but hard to see how the acute angles can be different. The MathWorld link is the source of this claim. Tom Ruen (talk) 05:16, 28 June 2011 (UTC)
- Okay, I see a rhombic prism is an example of noncongruent rhombic faces. Tom Ruen (talk) 05:22, 28 June 2011 (UTC)
Rhombohedral symmetry?
This removed gallery doesn't seem to reflect this polyhedron, even if they have rhombohedral symmetry! Tom Ruen (talk) 05:39, 28 June 2011 (UTC)
- Rhombisches Prisma.svg (#4) looks like a right rhombic prism, doesn't it? 8-)
- Oops! No: its vertical faces are rectangles but not squares, so not rhombi... :-P
Can «rhombic hexahedron» mean «non-isohedral rhombohedron»?
- Can «Rhombic hexahedron» really mean «non-isohedral rhombohedron»? See Officer781's «03:50, 28 April 2022» edit summary:
- if rhombic hexahedron refers to the general case, then it should not refer to the isohedral case of trigonal trapezohedron.
- —JavBol (talk) 01:48, 31 May 2024 (UTC)
- The sources I found give examples which happen to be isohedral but they do not clarify whether the term refers only to the isohedral case or to the general case. —David Eppstein (talk) 05:00, 31 May 2024 (UTC)
- The sources that David Eppstein found are meant for scholar use, & the images they provide have much regularity, so they very probably refer to the isohedral case.
- Moreover, «rhombic dodecahedron» & «rhombic triacontahedron» refer to isohedral polyhedra, & «rhombic icosahedron» refers to an equifacial polyhedron (& «equifacial rhombohedron» «isohedral rhombohedron»).
- So, moving «(also called rhombic hexahedron)»
- into «● Trigonal trapezohedron (also called isohedral rhombohedron or ...)»
- would be more prudent; wouldn't it?
- —JavBol (talk) 15:55, 31 May 2024 (UTC)
- Why guess when we can ask User:Steelpillow what he meant? —David Eppstein (talk) 18:18, 31 May 2024 (UTC)
- These terms generally arise as descriptive labels, within the context of the author's current area of interest. In my case the context was a particular isohedral rhombohedron or trigonal trapezohedron; I did not consider the non-isohedral case. The edit summary quoted above is quite misplaced - most if not all occurrences in the literature refer to isohedral examples, so we cannot claim that to be "wrong". I would note that Coxeter's cited term, as a trigonal trapezohedron, also makes a tacit assumption of isohedrality - one can readily construct non-isohedral trigonal trapezohedra which are longer at one end than the other. This illustrates the point that the literature on polyhedra is not rigorous in its naming conventions, so seeking to cite rigour is a lost cause. The best we can do is to follow the literature and fail to be rigorous as to whether "rhombic hexahedron" applies also to the non-isohedral case. To treat it any other way cannot be supported from RS. — Cheers, Steelpillow (Talk) 19:47, 31 May 2024 (UTC)
- About non-isohedral trapezohedra: one can even construct inverted trapezohedra, i.e. with both ends on the same «side» of (e.g. above) the skew polygon base (& with one end longer than the other); one can even construct twisted inverted trapezohedra!
- About following the sources: what about:
- leaving «(also called rhombic hexahedron)» as it is in Top,
- AND copying it into «● Trigonal trapezohedron (also called isohedral rhombohedron or ...)»?
- —JavBol (talk) 16:27, 1 June 2024 (UTC)
- First, my small mistake: Coxeter defines a rhombohedron as "a parallepiped bounded by six equal rhombs", not as a trigonal trapezohedron as I stated. And the relevant citation is to Lines anyway. Lines also defines a trapezohedron as having congruent sides (i.e. faces), so I have deleted the cite for a term he did not use.
- More significantly to this article, both authors define a rhombohedron as having equal faces. So the article's assertion that a rhombohedron may have differing pairs of rhombs requires better citation - or, if that cannot be found, some heavy rewriting to conform to RS. — Cheers, Steelpillow (Talk) 20:10, 1 June 2024 (UTC) — Cheers, Steelpillow (Talk) 20:10, 1 June 2024 (UTC)
- Certainly there exist polyhedra in which opposite sides are congruent rhombi, with two or three distinct shapes of rhombi. The question is whether there are sufficient publications on those specific shapes to support a standalone article (in which case we need to follow them in naming) or not (in which case maybe parallelepiped is a possible merge target). Maybe the crystallographic literature might have something on this?
- Among our current sources:
- Miller and Inchbald don't discuss the possibility of having more than one face shape
- Court has a line about "a parallelepiped whose faces are rhombuses", calling it a rhomboid, without enough depth of coverage to support notability.
- Lines is an offline book and I don't know what is in it.
- MathWorld Vector Addition is almost entirely off-topic.
- MathWorld Rhombohedron does discuss this class of shapes, using the name "rhombohedron", but again without enough depth of coverage to support notability.
- —David Eppstein (talk) 21:25, 1 June 2024 (UTC)
- These terms generally arise as descriptive labels, within the context of the author's current area of interest. In my case the context was a particular isohedral rhombohedron or trigonal trapezohedron; I did not consider the non-isohedral case. The edit summary quoted above is quite misplaced - most if not all occurrences in the literature refer to isohedral examples, so we cannot claim that to be "wrong". I would note that Coxeter's cited term, as a trigonal trapezohedron, also makes a tacit assumption of isohedrality - one can readily construct non-isohedral trigonal trapezohedra which are longer at one end than the other. This illustrates the point that the literature on polyhedra is not rigorous in its naming conventions, so seeking to cite rigour is a lost cause. The best we can do is to follow the literature and fail to be rigorous as to whether "rhombic hexahedron" applies also to the non-isohedral case. To treat it any other way cannot be supported from RS. — Cheers, Steelpillow (Talk) 19:47, 31 May 2024 (UTC)
- Why guess when we can ask User:Steelpillow what he meant? —David Eppstein (talk) 18:18, 31 May 2024 (UTC)
- The sources I found give examples which happen to be isohedral but they do not clarify whether the term refers only to the isohedral case or to the general case. —David Eppstein (talk) 05:00, 31 May 2024 (UTC)
