Talk:Rhombohedron

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Polyhedra

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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)[reply]

Okay, I see a rhombic prism is an example of noncongruent rhombic faces. Tom Ruen (talk) 05:22, 28 June 2011 (UTC)[reply]

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)[reply]

Rhombisches Prisma.svg (#4) looks like a right rhombic prism, doesn't it? 8-)

- RavBol (talk) 20:50, 16 December 2019 (UTC)[reply]

Oops! No: its vertical faces are rectangles but not squares, so not rhombi... :-P

- RavBol (talk) 21:09, 16 December 2019 (UTC)[reply]

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)[reply]

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)[reply]

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»

\Leftrightarrow

«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)[reply]

Why guess when we can ask User:Steelpillow what he meant? —David Eppstein (talk) 18:18, 31 May 2024 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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. And in general I don't trust MathWorld on questions of nomenclature; they have too many neologisms.

—David Eppstein (talk) 21:25, 1 June 2024 (UTC)[reply]

@Steelpillow: Sorry, I don't understand why you have moved the «Lines» citation from one occurrence of «isohedral rhombohedron» to another (within the Rhombohedron article). Did you intend to move this citation from an occurrence of «isohedral rhombohedron» to one of «Trigonal trapezohedron», please? :-P —JavBol (talk) 22:45, 1 June 2024 (UTC)[reply]

@JavBol: I did not move any citation as such. There were two inline cites to Lines. One cited him for the term "isohedral trapezohedron", which he does not use, so I deleted that cite. The other cites him for the isohedral rhombohedron, which he does mention. Although he does not use the exact term "isohedral rhombohedron", he effectively includes isohedrality in his definition of a rhombohedron, by focusing on the angles, as "a parallepiped with three equal edges equally inclined to one another is called a rhombohedron (his bold). So I left that cite in but, in order to preserve the full reference, I had to move the template code from the deleted cite to the preserved one.

@David Eppstein: Lines does not discuss the rhombohedron beyond its definition (as above), although he does sketch out some properties of the lattices arising. With respect to the trapezohedron, he describes it as "bounded by congruent trapezoids, or quadrilaterals with two pairs of equal adjacent sides". He also discusses the Archimedean duals and lists some properties, without naming it as such. He lists its symmetry, common to the prisms, as F4₂n₁, and goes on to prove various theorems about the "duals of the facially regular solids". And I agree about Mathworld, it often perpetuates the small errors to be found in polyhedral folklore. — Cheers, Steelpillow (Talk) 10:54, 2 June 2024 (UTC)[reply]

Cromwell (Polyhedra) also discusses the rhombohedron. On pp.154-7 he notes Kepler's original study, strictly in the context of the rhombic triacontahedron's golden rhombs. On pp.319ff he covers Haûy's discovery of calcite crystals which Cromwell describes as "rhombohedral in shape". This latter is perhaps confusing, as this image suggests that the description "rhombohedral" does not in this case refer to equal edges, merely to a parallelepiped with equal-angled parallelograms. But adjectives are not definitions. — Cheers, Steelpillow (Talk) 12:47, 2 June 2024 (UTC)[reply]

@Steelpillow: Now I understand your edit on Rhombohedron better. But the cite that you have deleted was just after:

«Trigonal trapezohedron (also called isohedral rhombohedron)»,

so it referred to «isohedral rhombohedron»; didn't it? :-P

@David Eppstein: In the «Dictionnaire encyclopédique Larousse», among the «7 systèmes cristallins» (including the hexagonal one), if the angles

\alpha ,\beta ,\gamma

(between edges (or axes)) are not all equal, then the edge lengths

a,b,c

are not all equal either (also for the hexagonal system); so I'm pessimistic about finding a non-isohedral rhombohedron in crystallography (but I'm no expert at all).

@Both: PS: Suspect edits have been made very recently on Centroid.

—JavBol (talk) 15:45, 2 June 2024 (UTC)[reply]

If Larousse really says that then it is mistaken. Make yourself a stick cube with wobbly corners and see for yourself. — Cheers, Steelpillow (Talk) 16:32, 2 June 2024 (UTC)[reply]

It is plausible to me as a total non-expert in crystallography that the angles and edge lengths of a rhombic crystalline system are both related to the sizes of the atoms at the corners in such a way that unequal angles are only associated with unequal lengths. Obviously it is possible for mathematical polyhedra but maybe it is impossible or unlikely for crystals, at least to the extent that this is a special case the crystallographers don't care about. —David Eppstein (talk) 18:50, 2 June 2024 (UTC)[reply]

JavBol is right that there are no non-isohedral rhombohedra in crystallography though. The appropriate definition is that of the triclinic Bravais lattice or space group. This has three different axes (the side lengths) and three different (pairs of) angles. However it is something of a catch-all, in that while the lengths and angles are all independent, they need not all actually be different sizes. A lattice with different angles but equal sides (a = b = c, α ≠ β ≠ γ) would be classified as triclinic and not rhombohedral. It is possible that no examples are known in natural crystals, but even if that is true I would hesitate to assert that they never can occur. Certainly architects and civil engineers have to deal with such things, and they have a very human way of picking any old term that appeals to them. — Cheers, Steelpillow (Talk) 20:04, 2 June 2024 (UTC)[reply]

Of course, e.g. 900, 900×8, 900×27, 900×64, 900×125, or... parallelepipeds with not all equal angles

\alpha ,\beta ,\gamma

and with edge lengths 2, 3, 5, assembled face to face, can happen to form a parallelepiped still with not all equal angles

(\alpha ,\beta ,\gamma )

but with all equal edge lengths (30): a non-isohedral rhombohedron. But indeed, this is much rarer than e.g. (1), 8, 27, 64, 125, or... congruent cubes forming a cube, & the crystallographers probably don't name this very rare case (& otherwise they would have to name other very rare cases in other classes of other crystalline systems).

Remark in passing: my (physical) «Dictionnaire encyclopédique Larousse» is small; for the triclinic crystalline system, it mentions only

\alpha \neq \beta \neq \gamma \neq {\tfrac {\pi }{2}}

and

a\neq b\neq c.

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 ...)»?

If this question has already been answered, I don't understand where it has. :-P —JavBol (talk) 22:07, 2 June 2024 (UTC)[reply]

Your proposal has been answered more than once and the answer is "no". Through this discussion lt has become crystal clear (sic) that in both polyhedron theory and crystallography, the term "rhombohedron" refers to an isohedral figure (with all faces congruent). In the phrase "isohedral rhombohedron", the term "isohedral" is redundant, and the phrase should not be used.

This article should be refactored to confine it to the subject of its title, and any extraneous material worth keeping should be merged into Parallelepiped.

— Cheers, Steelpillow (Talk) 09:50, 3 June 2024 (UTC)[reply]

I AGREE that the term «rhombohedron» refers to an isohedral figure (with all faces congruent), that in the phrase «isohedral rhombohedron», the term «isohedral» is redundant, & that the phrase should not be used.

I've never proposed to add «isohedral rhombohedron» anywhere;

I've always proposed to add «RHOMBIC HEXAHEDRON» into the body of the article. :-P

— Cheers, JavBol (talk) 15:24, 3 June 2024 (UTC)[reply]

Glad to hear it. You used the other phrase so often I missed your intent. But we already introduce the term in the lead sentence, and it is better to use the page title throughout the article than drop in a synonym in arbitrary places. I think it best not to add "rhombic hexahedron" anywhere else, except in a direct quotation. — Cheers, Steelpillow (Talk) 18:49, 3 June 2024 (UTC)[reply]

In the following crystallography source section, rhombohedra are isohedral alright:

{{sfn|Spencer|1911|p=580, or p. 602 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, ''Rhombohedral Division'', HOLOSYMMETRIC CLASS (FIGS. 66 & 67)}}.^[1]

References

^ Spencer 1911, p. 580, or p. 602 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Rhombohedral Division, HOLOSYMMETRIC CLASS (FIGS. 66 & 67).

Spencer, Leonard James (1911). "Crystallography" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 07 (11th ed.). Cambridge University Press. pp. 569–591.

Could it enable to mention the (isohedral) rhombohedra in crystallography? (Where they seem to be very important: they form rhombohedra, but also other shapes.)

Moreover, the «Dictionnaire encyclopédique Larousse» states that in the rhombohedral system (which it considers as a standalone system),

\alpha =\beta =\gamma \ {\big (}\!\neq {\tfrac {\pi }{2}}{\big )}

and

a=b=c\!:

these rhombohedra really are isohedral alright.

What about merging Rhombohedron into Trigonal trapezohedron?

—JavBol (talk) 17:08, 4 June 2024 (UTC)[reply]

Aargh! No no no. The rhombohedron may well be a special case of the trigonal trapezohedron, but it is far more notable and should definitively be kept separate. — Cheers, Steelpillow (Talk) 17:23, 4 June 2024 (UTC)[reply]

Huh? It doesn't look to me that this is a special case of that. It looks like "trigonal trapezohedron" is what we name our existing article on polyhedra with six congruent rhombus faces (the important case) and maybe that article should be named "rhombohedron" instead, regardless of what we end up naming the other article (the one we're on the talk page for) about polyhedra with six not-necessarily-congruent rhombus faces. —David Eppstein (talk) 22:39, 4 June 2024 (UTC)[reply]

I think you have it the wrong way round. Where do you get your definition of a "trigonal trapezohedron" from? That article cites no source for its definition. It mistakenly cited Lines for "isohedral rhombohedron", as this one did, so I just corrected that too. For the lulz I checked Mathworld's Rhombohedron, which allows unequal rhombs but then cites only Coxeter and his collaboration with Ball - both of which are explicit that the figure is only a rhombohedron if all faces are congruent - it's as bad as these articles here. All I see is poorly-cited Internet myths. Finding sources that consider trapezia other than the Archimedean duals is hard; Cundy & Rollett (p.117) are unusual in explicitly referring to them as "Archimedian trapezia", by implication allowing non-Archimedean examples. Cromwell (Polyhedra pp.302,304) describes a "scalene trapezohedron" whose faces are convex quadrilaterals, not even kite trapezia, though later (p.367) he does demand face-transitivity (congruence). So we can presumably have a non-Archimedean and scalene trigonal trapezohedron. On the other hand, this article (rhombohedron) is no longer about non-congruent forms, and most of it never was. So it is the article on the trigonal trapezohedron which should be covering asymmetric cases - subject to sufficient sourcing to make a coherent topic. — Cheers, Steelpillow (Talk) 19:59, 5 June 2024 (UTC)[reply]

I am by no means stuck to the current terminology of our current Wikipedia articles, I was merely confused by your previous message suggesting that this article (on polyhedra with rhombic faces) was a special case of a different article (on polyhedra with congruent rhombic faces). Given that we have two articles on these two topics, the questions should be whether each should exist separately, if not where their content should go, and which names we should assign or reassign to them. —David Eppstein (talk) 20:43, 5 June 2024 (UTC)[reply]

As I said, that is not correct. You continue to assume falsehoods. Both articles are concerned with congruent faces, but according to Cromwell the trapezohedra also include non-rhombic "scalene" examples. And we don't really have two articles, we have one viable article and one inconsistent mess which does not even cite a definition for its title; until that mess is cleared up, questions of what its subject is, or ought to be, remain moot. This conversation should be held on that article's talk page, not here. — Cheers, Steelpillow (Talk) 07:53, 6 June 2024 (UTC)[reply]

Your combative attitude is unhelpful. I care not at all about the names. There are multiple classes of shapes here. Some may be notable, others not. We should determine which shapes are notable, first, focus an article on each, and avoid the current confusion where we focus an article on a name and then argue about all the different shapes that might be called that name. Only after we have a focus can we determine what the right name for each article might be. WP:NOTDICT applies. Articles are on topics not on their names. —David Eppstein (talk) 17:57, 6 June 2024 (UTC)[reply]

My sincere apologies if I appear combative, that is not my intention. Here, I merely seek to keep this discussion focused on the facts and policies which will help improve this article. For the other article, I have now opened a couple of discussion topics there. Anyway, I am done here - the field is all yours. — Cheers, Steelpillow (Talk) 19:21, 6 June 2024 (UTC)[reply]

I would rather you continued since you obviously have actual expertise in this area, sorely needed. —David Eppstein (talk) 04:53, 7 June 2024 (UTC)[reply]

What is the length of the line joining the two extremities?

I don't seem to have found a mathematical expression in the text for this distance. Where can I find it? Macrocompassion (talk) 13:21, 16 August 2024 (UTC)[reply]

[FOOTNOTESpencer1911580,_or_p._602_on_Wikisource,_CRYSTALLOGRAPHY,_6._HEXAGONAL_SYSTEM,_''Rhombohedral_Division'',_HOLOSYMMETRIC_CLASS_(FIGS._66_&_67)-1] Spencer 1911, p. 580, or p. 602 on Wikisource, CRYSTALLOGRAPHY, 6. HEXAGONAL SYSTEM, Rhombohedral Division, HOLOSYMMETRIC CLASS (FIGS. 66 & 67).

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