Talk:Isohedral figure

Poll to move to face-transitive
Face-transitive is a standard term for polytopes. Face-uniform was an incorrect term derived from uniform polytopes which are are vertex-transitive, and their duals which are face-transitive.


 * Yes - Tom Ruen 22:27, 3 February 2007 (UTC)


 * Yes and no. I would prefer to move it to Isohedral because it's shorter and has been around longer, and redirect from Face-transitive. Steelpillow 13:05, 6 February 2007 (UTC)
 * It can be both with redirects, although I agree best to encourage one set of terms or other. How about starting by adding links/referenecs to all these stubs? Tom Ruen 19:33, 6 February 2007 (UTC)


 * I wasn't sure whether it was best to move the pages first, before doing all that. Yes I know it can be both with redirects, but we still need to decide the titles of the pages to which we redirect the others. Steelpillow 20:09, 6 February 2007 (UTC)

Move to Isohedral?
Looking behind the recent to-and-fro moves, it seems to me that Isohedral should be more than a redirect. It could be made a disambuguation page, or it could be basically this one, with links to isohedral numbers and the like. My vote is to move this page across. -- Cheers, Steelpillow (Talk) 19:35, 9 July 2008 (UTC)


 * Hi Guy. I don't have a strong preference what to do on the isohedral page. I'm not against moving this page there, or making it a disambuguation page. I do like using face-transitive better for being more descriptive to me, but that can as well exist as a redirect to isohedral.
 * Hmmmm.... a separate item, I added the isotopic term here, but unsure if it well belongs here, a bit different, meaning facet-transitive, so isotopic=isohedral for polyhedra only. Maybe it should be on its own page? I suppose it is too underused. The duals of uniform polytopes (isogonal/vertex-transitive) are isotopic, but they are pretty rarely given. Like the Disphenoid tetrahedral honeycomb is isotopic (cell-transitive), but might as well say that! Tom Ruen (talk) 20:24, 9 July 2008 (UTC)

Don't use an adjective
If this is to be moved, move it to face-transitive polytope or face-transitive polyhedron or isohedral polytope or isohedral polyhedron or some other appropriate noun phrase, not to an adjective. Michael Hardy (talk) 20:48, 9 July 2008 (UTC)


 * Or to face-transitivity. That is a noun phrase. Michael Hardy (talk) 20:49, 9 July 2008 (UTC)


 * In that case, I suggest Isohedrality. We should only use "polytope" in the title if we create a separate disambiguation page, which would be trivially small and nobody has voted for it. And hey, can you guys stop this ping-pong renaming until we reach consensus? It's making me dizzy ;-( -- Cheers, Steelpillow (Talk) 09:49, 10 July 2008 (UTC)


 * I'm mostly content with either. I see that edge-transitivity (isotoxality!?) and cell-transitivity (isochoricity?!) have already been moved now. And maybe facet-transitivity (isotopicity?!) Oh, I'm far over my head beyond -transitivity! Tom Ruen (talk) 15:34, 10 July 2008 (UTC)


 * "Isotoxality" only seems to appear in a single paper, unsurprisingly by Grunbaum and Shepherd (I don't know about "isotopic" vs. "facet-transitive", especially as "facet" means different things to different polyhedronists). In truth, only the adjectives are commonly used - in all cases the nouns are much rarer. I have asked the maths wikiproject about adjectives as article titles in this unusual circumstance. -- Cheers, Steelpillow (Talk) 19:46, 10 July 2008 (UTC)


 * From the wikiproject discussion, the best idea is to redirect "isohedral" etc. to the associated polyhedron article, e.g. "Isohedron", and at the top of the article to put disambiguation links to the other articles (tilings, etc) - which for the most part are there already. So if nobody objects, I shall move the "xxx-transitive/ity" pages over to "Iso-xxx polyhedron" when I get a moment. -- Cheers, Steelpillow (Talk) 21:34, 10 July 2008 (UTC)


 * Ummmm... what about the issue that these terms apply to polytopes in general, AND include infinite forms as the honeycomb/tilings? I'd prefer Isohedrality (geometry) if needed for clarity, or something like that, just like Truncation (geometry), etc. See Category:Polytopes for other example. Tom Ruen (talk) 21:57, 10 July 2008 (UTC)


 * "Isohedrality (geometry) would be little more than a very short disambig page: policy guidelines do not approve of spawning mini-pages unnecessarily, and I do not think that it is necessary here. Also, "Isotoxality (geometry)" has only one documented usage - that's hardly encyclopedic. Isohedral polychora are of comparably minor interest to isotoxal polyhedra - the interesting ones being the isochora. So unless and until "Isohedral polychoron" spawns its own article, it should at best be no more than a redirect to Isohedron which would have the disambig bit at the top, or maybe to polychoron. It's all about reconciling common sense with Wikipedia policy. I actually believe that the insistence on nouns is in this instance misplaced, but I'm not about to start a policy war with the self-appointed Wikipedia thought police. -- Cheers, Steelpillow (Talk) 08:57, 12 July 2008 (UTC)


 * I can't really say I follow much of this logic, BUT I'd be okay with something like this: (And adding a (geometry) qualifier anywhere it is needed.)
 * Isogonality, redirected there by Vertex-transitivity/Vertex-transitive (0 faces)
 * Isotoxality, redirect there by Edge-transitivity/Edge-transitive (1 faces)
 * Isohedrality, redirect there by Face-transitivity/Face-transitive (2 faces)
 * isochoricity(?), redirected there by Cell-transitivity/Cell-transitive (3 faces)
 * isotopical(?), redirected there by Facet-transitivity/Facet-transitive (n-1 faces)

Anyway I DEFINITE dislike adding the word "polyhedron" to terms that apply to all polytopes/tilings/honeycombs. Tom Ruen (talk) 18:12, 12 July 2008 (UTC)


 * Every scheme so far seems to have its problems:
 * "Xxxxxality" fails because "Isotoxality" is not used enough to warrant inclusion.
 * "Xxxxx-transitive" fails because "Cell-transitive" suffers the same problem (Google returns just 1 significant hit).
 * "Xxxxxal polytope" fails because tilings, numbers and other things can have these properties.
 * "Xxxxxal" is what everybody uses but fails politically because it is an adjective.
 * "Xxxxxal figure" is perhaps a workable alternative? It includes the commonly used form and manages to be a noun.

Would that last one be OK by you? -- Cheers, Steelpillow (Talk) 19:23, 12 July 2008 (UTC)


 * Good suggestion. Gosset used the term figure for both polytopes and tilings. Tom Ruen (talk) 21:18, 12 July 2008 (UTC)
 * Isogonal figure (0-face transitive).
 * Isotoxal figure (1-face transitive)
 * Isohedral figure (2-face transitive)
 * Isochoric figure (3-face transitive)
 * Isotopic figure ((n-1)-face transitive)


 * OK I have moved the first three, and started a poll here to delete the fourth, as it is so little-used. I know nothing about "isotopic", so have done nothing about it. If it is as unused as cell transitivity, i suggest that it should go too. -- Cheers, Steelpillow (Talk) 19:08, 13 July 2008 (UTC)

k-isohedral
I am confused by the section. The definitions for polytopes and tilings appear to be very different: The former makes no mention of symmetry orbits and explicitly forbids different shapes. Strangely, it does not refer to any k. The latter defines k as the number of symmetry orbits and explicitly allows different shapes. But why m<k? Apparently confirming my suspicion that the definition for polytopes is simply wrong, the example is a polytope that has 2 or 3 symmetry orbits. (An ambiguity, that, btw, doesn't make it a good choice for an example.) But if the definition for tilings is the correct one for any figures, we run into another problem: The example says it is "k-isohedral but not isohedral", which seems to contradict my understanding that isohedral is the same as 1-isohedral. Is my understanding wrong or is there a special condition that k > 1? Why would that be? &mdash; Sebastian 16:24, 19 August 2015 (UTC)


 * Yes, wording could use improvement. They should be the same. Isohedral means 1-isohedral. While k-isohedral doesn't require all k types of faces to be identical, hence the m, which is called monohedral if m=1. For instance the k-uniform tilings have k-types of vertices, and are also t-isohedral (with t types of regular polygon faces). So examples here really should show different types of faces to not give the wrong impression. Tom Ruen (talk) 17:26, 19 August 2015 (UTC)


 * It does seem fundamentally correct, although the wording is awful and Tom has already changed what you were asking about so I cannot be sure. Tiles or faces of a given shape may lie in more than one symmetry orbit, i.e. in distinct groups which permute with their own members but not with each other's despite being congruent. Thus, the number of distinct shapes m may be less than the number of orbits k. This applies both to polyhedra and to tilings. Also, the section title should not be an adjective, it is an embarrassment to whoever put it there. &mdash; Cheers, Steelpillow (Talk) 17:48, 19 August 2015 (UTC)


 * Thanks a lot for your clarification and edits to the article; that's a huge improvement! Now my only question remaining is why there is the condition m<k. I wouldn't be surprised if this strict inequality could be proven to hold in reality, but I don't see why it is part of the definition. &mdash; Sebastian 17:57, 19 August 2015 (UTC)
 * Your intuition is correct, it is not part of the definition - for polyhedra or for tilings. It needs to be explained separately, in a paragraph relevant to both polyhedra and tilings. &mdash; Cheers, Steelpillow (Talk) 18:02, 19 August 2015 (UTC)


 * I added some varied examples. The text still could use improvement and unification between polyhedra and tilings. The m is the number of unique shaped-faces, and the k is the number of symmetry positions, with different colors used to distingish geometrically identical faces. I don't care of the m commentary is removed. I agree it is confusing. Tom Ruen (talk) 18:06, 19 August 2015 (UTC)


 * Thank you! And, yes, I meant to thank you for the edits! &mdash; Sebastian 18:19, 19 August 2015 (UTC)


 * p.s. The pseudo-deltoidal icositetrahedron article says it is 2-isogonal, but 3-isogonal if only rotational symmetry is considered. I've not seen any definitions that say "k-isohedral in rotational symmetry". I actually uploaded a new image with 2 colors, but the original had 3. Tom Ruen (talk) 18:15, 19 August 2015 (UTC)


 * Yes, that's what I referred to. I realize the word "ambiguous" isn't exactly correct, since for each definition it is unambiguous. I just meant that it adds another aspect to consider, which doesn't help when explaining a simple concept. Your other examples already do that well, so we could just leave out the pseudo-deltoidal icositetrahedron. On the other hand, the confusion also gets diluted thanks to the other examples, so it might as well stay. &mdash; Sebastian 18:23, 19 August 2015 (UTC)


 * p.s. All the examples are edge-to-edge, but we could add a non-edge-to-edge example tiling like the Herringbone pattern that has 2 or more short edges along a longer one. I guess polyhedra also don't have to be edge-to-edge, but not very common. ... I replaced the isohedral tiling example with herringbone. k-isohedral tilings could also have curved edges, like this File:Wallpaper_group-p6m-1.jpg could be 3-isohedral by the black boundaries.Tom Ruen (talk) 18:27, 19 August 2015 (UTC)
 * Good choice, the herringbone is a great example because it's well known. It's also a good idea to just replace it, since I like the nice symmetry between your two tables. Question, though: Are regular-faced and monohedral mutually exclusive? &mdash; Sebastian 18:38, 19 August 2015 (UTC)
 * Not mutually exclusive. There are 3 regular tilings (and 5 platonic solids that are 1-isohedral and monohedral.
 * The Voderberg tiling is monohedral, but perhaps &infin;-isohedral since it doesn't have translational symmetry! Tom Ruen (talk) 18:33, 19 August 2015 (UTC)
 * Wow, that's an amazing pattern! If I'm not mistaken, it doesn't even have rotational symmetry except the obvious C2 in the center and then perhaps locally for some tiles. &mdash; Sebastian 18:41, 19 August 2015 (UTC)
 * Yes, just C2 inversion symmetry. I actually don't know what the coloring implies. There are frieze groups also with infinite tiles, like the isohedral File:E2_tiling_22i-8_dual.png, but fortunately not many examples there. And we could also show hyperbolic k-symmetry, also not many examples. No, I'm not really suggesting more examples, unless there's a source that says Voderberg is &infin;-isohedral. AND penrose tilings would also be like this has 4 tile shapesFile:Penrose Tiling (P1).svg, but &infin;-isohedral based on radial symmetry. Tom Ruen (talk) 18:56, 19 August 2015 (UTC)
 * I think it's good enough as it is; it explains the concept as far as it's useful. That's all I feel needs to be done here for now. Now you got me hooked with the Voderberg tiling. I'll continue the conversation on that there. &mdash; Sebastian 19:18, 19 August 2015 (UTC)

Monohedral figures
I see r-hedral apparently means r shapes of tiles, monohedral (r=1), dihedral (r=2), trihedral (r=3)... So maybe this could be defined somewhere else, and then monohedral could be moved there. Monohedral tiling directs to Tessellation. Tom Ruen (talk) 19:09, 19 August 2015 (UTC)
 * We don't usually use redirect pages for discussions, but how about moving this to talk:Monohedral tiling? I could undelete that page, which contained just the maths rating template. &mdash; Sebastian 19:18, 19 August 2015 (UTC)
 * It wouldn't be good to have talk page on a redirect. Monohedral figure directs here at the moment, so I'll just add a new section for now. Tom Ruen (talk) 19:24, 19 August 2015 (UTC)

Factual correction and proposal for a merge with Isohedron
The lede says: "Isohedron redirects here", but it doesn't. This article actually links to Isohedron! (If that statement were correct, this link would create a circular – and useless – self-reference.)

Also, Pentagonal tiling links to an old name for this page, Isohedral, which redirects to this article's page, i.e. to Isohedral figure. (According to earlier talk on this discussion page, the expression "Isohedral figure" was a compromise between alternatives such as (IIRC) "Isohedral polytope" and "Isohedral polygon", whilst trying to avoid continuing to use the adjective "Isohedral" for a page title.)

Another difficulty for readers at present is that the discussion of Isohedral figures and Isohedral is disjointed, being split across both of the named pages. Readers would get a clearer picture from a single, well-organised page that combines information from both.

Here's a plan:
 * 1) Remove the erroneous redirection statement I quoted above.
 * 2) Merge this page, Isohedral figure, with Isohedron.
 * 3) Choose a sensible name for the resulting page, one which meets users' expectations that page names are the names of things, i.e. nouns, and also one which reflects the broader context, rather than:
 * one restricted context (e.g. "Isohedral polygon"), or
 * one particular detail (e.g. "Isohedron").

Based on those constraints, I suggest that the best title we could use would be Isohedral polytope.

I look forward to a fruitful discussion with interested readers and editors. yoyo (talk) 14:50, 5 November 2015 (UTC)


 * There could be some needed corrections, but I think "isohedral figure" is more compehensive than isohedron (or isohedral tilings), since it applies to higher dimensional polytopes and honeycombs. Also I don't think "Isohedral polygon" makes sense as a polygon has no faces. Tom Ruen (talk) 16:29, 5 November 2015 (UTC)
 * We have series of articles on isotoxal figure, isogonal figure, etc. Unless someone want to overturn that, I think that merging the isohedron in here is the best approach. &mdash; Cheers, Steelpillow (Talk) 17:37, 5 November 2015 (UTC)
 * Isohedron should be merged with this article because all it has are examples of isohedra. The examples on isohedron should be moved to the examples on this page. Eli355 (talk) 14:46, 7 June 2018 (UTC)
 * The merge has been performed. Eli355 (talk) 00:40, 27 June 2018 (UTC)

Isohedron became a redirect to Isohedral figure (here) on 2018-06-26.

This page started as [Face-uniform] on (2006-08-05); moved to [Face-transitive] on 2007-02-10; moved to [Face-transitive polyhedron] and back to [Face-transitive] on 2008-07-08; moved to [Isohedral figure] on 2008-07-13, where it is today. - A876 (talk) 23:57, 3 April 2019 (UTC)

Counterexamples
Counterexamples might be worth mentioning, if only on this talk page.

The following are not isohedral figures. They have all identical (congruent) faces, but they are not "face-transitive":
 * Rhombic icosahedron (dual of a modified Johnson solid) It has 20 faces that are [golden] rhombi, but is not face-transitive.
 * Bilinski dodecahedron (dual of cuboctahedron) It has 12 faces that are [golden] rhombi, like the rhombic dodecahedron (a Catalan solid (all of which are face-transitive)), but is not face-transitive.

These lead to some questions:
 * Is the superset: polyhedra that have identical faces (with no other requirement) significant, and does it have a name?
 * Is there a larger superset if reflection of asymmetrical faces is also allowed?
 * Is there a name for the difference set (polyhedra that have identical faces but are not isohedral)?

Addition?
 * The excavated dodecahedron (which is not currently listed as isohedral) has 60 equilateral triangle faces. It is like the pentakis dodecahedron (which is listed), except that its 12 pyramids extend inward rather than outward.

Subtraction??
 * The rhombic triacontahedron (which is listed as isohedral), to me, looks different from all the rest. It has two kinds of faces, as does the rhombic icosahedron (which is not isohedral). Mathworld includes it too. I don't understand how it is isohedral. (It's probably me.) - A876 (talk) 23:57, 3 April 2019 (UTC)

A876, those are good questions. I also am looking for a name for the superset you asked for -- polyhedron composed entirely of identical faces (mirror-image faces also allowed). In addition to the polyhedra you already listed, more congruent-face but not face-transitive polyhedron include:
 * pseudo-deltoidal icositetrahedron
 * snub disphenoid
 * "some spherical tilings" by Robert Dawson has a collection of ways to tile the sphere with lots of identical spherical triangles. Some (all?) can be "flattened" to a congruent-face polyhedron.
 * the infinite family of gyroelongated bipyramids, which are arguably the "best" approximation to a sphere when we require where every face to have the same shape and we are forced to use more than 120 faces (as suggested at "Can the sphere be partitioned into small congruent cells?" and "tile the sphere as many (identical) face as possible" and "Some Polyhedra with Identical Triangular Faces" ).
 * If you include non-convex shapes, some of the shapes mentioned in toroidal polyhedron
 * If you include non-convex shapes, any quantity greater than 1 of rhombic dodecahedra attached face-to-face, or similarly any other space-filling polyhedra that either has only one kind of face, or can somehow be attached so only one kind of face is visible from the outside.
 * skew apeirohedron ?

Perhaps monohedrons (monohedral figure) are a good name for that superset, "polyhedron composed entirely of identical faces, whether or not they are face-transitive"? (Alas, plesiohedron uses "monohedral" to mean something entirely different -- apparently a kind of polyhedron that may have a variety of irregular faces, but large numbers of identical copies of the entire polyhedron tile space with no gaps).

It's apparently still an open question in 2020 as to how many kinds of monohedral but non-isohedral polyhedrons exist (see "What are the known convex polyhedra with congruent faces?", unanswered as of 2020). --DavidCary (talk) 20:47, 4 December 2020 (UTC)