Talk:Quasiregular polyhedron

message: By the definition here, "A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular", all regular polyhedra are quasiregular.  By the definition at Polyhedron, "vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive", no regular polyhedron is quasiregular.  Thus these two definitions are inconsistent.  Furthermore, the enumeration of convex quasiregular polyhedra given here, which includes the octahedron but excludes the other regular polyhedra, is inconsistent with both definitions.
In fact the two external sites cited disagree as to whether the octahedron should be regarded as quasiregular.  But both agree that to be quasiregular, faces with n sides must alternate with faces with m sides at each vertex -- implying an even number of faces at each vertex.  Then the octahedron is quasiregular or not, depending on whether you allow n=m or not, but the other regular polyhedra (having 3 or 5 faces at each vertex) are not.
(posted by 24.58.33.52 on 06:06, 3 January 2008).
:Hi, you are quite right about the inconsistencies. The literature on quasiregular polyhedra is pretty sparse, but it appears to be one of those many areas where geometers define one thing and then describe something else. One of the difficulties is that there are several interesting quasiregular properties, but nobody has ever methodically figured out which properties are fundamental and which are consequences of these. Feel free to make clarifications as to the nature of the muddle.
:As for the octahedron, it can be seen as quasiregular by the definition at Polyhedron, if we colour alternate faces black and white so there are two kinds - a figure sometimes also called the tetratetrahedron.
:And as for other web sites, they are for the most part a terrible load of rubbish. Mathworld perpetuates many shameful myths about polyhedra. George Hart is a lot more reliable, though his classification of quite so many star polyhedra as quasiregular is unusual (I happen to agree with him, but that is a long and continuing story).
:HTH -- Steelpillow (talk) 12:55, 3 January 2008 (UTC)
thread_title: Inconsistencies

message: If non-convex polyhedra with non-convex vertex figures count, there's a number missing, firstly the edge-sharing forms (is there an "official" term?) of the pictured five:
*tetrahemihexahedron
*cubohemioctahedron & octahemioctahedron
*small dodecahemidodecahedron & small icosihemidodecahedron
*great dodecahemidodecahedron & great icosihemidodecahedron
*small dodecahemicosahedron & great dodecahemicosahedron
Also, these have only singly symmetrical but nevertheless alternating vertex figures. Dunno if that's an additional problem.
*small rhombihexahedron, great rhombihexahedron
*small rhombidodecahedron, great rhombidodecahedron
*rhombicosahedron
*small dodecicosahedron, great dodecicosahedron
Didn't want to add these directly for two reasons - the section for non-convex quasiregulahedrons is titled with "examples", and I notice the discussion on what the authorities consider quasiregular anyway...
--Tropylium (talk) 19:28, 20 February 2008 (UTC)
replies: text: You have the right idea. The first list are indeed quasiregular (though the second list are not - they do not have edges all wthin a single symmetry orbit). We can also find quasiregular examples among the apeirohedra - plane tessellations and infinite skew polyhedra. Trouble is, none of this seems to have been published in an adequate reference (It's on George Hart's website but AIUI personal websites are deemed inadequate). A few months ago I submitted a paper on exactly this to the Mathematical Intelligencer, so we shall have to wait and see. -- Steelpillow (talk) 20:58, 22 February 2008 (UTC)
thread_title: Missing entries?

message: under the list of convex quasi-regular polyhedra, it says the the cuboctohedron and the icosahedron are the only two. However in the category "Quasiregular Polyhedra", it lists the rhombic dodecahedron and rhombic tricontahedron as well. Shouldn't they be included then?  —Preceding unsigned comment added by Timeroot (talk • contribs) 17:39, 4 July 2008 (UTC)
replies: text: They're given here under Quasiregular_polyhedron#Quasiregular_duals. Tom Ruen (talk) 05:18, 5 July 2008 (UTC)

text: See also the remark in that section, about differences of opinion as to whether these two polyhedra are quasiregular or not. Cromwell regards them as quasiregular, while Coxeter et. al. do not. -- Cheers, Steelpillow (Talk) 11:50, 5 July 2008 (UTC)
thread_title:  More missing entries 

message: The lead for this uses too much technical jargon. Move the mathematical definition into the article and make the lead into an introduction which is accessible to an intelligent, well educated non-mathematician.
I'm trained as an engineer and I couldn't make head nor tail of this intro. filceolaire (talk) 09:51, 23 November 2009 (UTC)
replies: text: A worthy complaint! Hopefully better now? Tom Ruen (talk) 10:22, 23 November 2009 (UTC)
replies: text:  Much better. Thanks. filceolaire (talk) 12:54, 23 November 2009 (UTC)
thread_title:  Lead needs to be more accessible 

message: In the diagram "Regular and quasiregular figures" there are two errors.
In the third column, second row, the Schläfli symbol is missing. It should be {3 3} arranged vertically.
In the fourth column, fourth row, the caption should not be "pentagonal tiling". This thing is quasiregular not regular. Maybe "tetrapentagonal tiling"?
I haven't time to fix these now. Maproom (talk) 13:59, 18 February 2010 (UTC)
replies: text: Thanks. I have fixed the tetrapentagonal tiling. The Schläfli symbol poses a problem. These symbols are supposed to be unique to a given figure. Clearly, \begin{Bmatrix} 3 \\ 3 \end{Bmatrix} applies to two of the examples given, thus providing a counter-example to long-held mathematical folk lore. I think that all we can do here is to faithfully reflect published sources, imperfect as they are. -- Cheers, Steelpillow (Talk) 16:38, 18 February 2010 (UTC)
replies: text:  The 5.4.5.4 tiling does not belong in that table!  It ought to show {5,4} as \begin{Bmatrix} 5 \\ 5 \end{Bmatrix}, the rectification of {5,5}. —Tamfang (talk) 22:06, 18 February 2010 (UTC)
replies: text:  Yes, it belongs in the table above - that will mess up somebody's nice layout. The appropriate "single-height" symbol is {5.4}2, and none other. -- Cheers, Steelpillow (Talk) 20:34, 19 February 2010 (UTC)
replies: text: with (), not {} —Tamfang (talk) 02:43, 3 July 2023 (UTC)
thread_title:  Two errors in second diagram 

message: I have moved this from the article because the references given do not include the majority of these figures. Indeed, they do not consistently recognise all the original "parents" as quasiregular - the most reliable sources such as Coxeter do not.
As an example of the problem, the tetrahemihexahedron does not have a Schläfli symbol distinct from that of the cuboctahedron. Coxeter did not regard it as quasiregular, even though he sometimes gave definitions of quasiregularity which are consistent with it. We are once again into the inconsistencies of modern theory, and we must be careful to avoid wp:or. -- Cheers, Steelpillow (Talk) 18:44, 28 February 2010 (UTC)
thread_title: Gallery of duals

message: I'll bite.  Why did  add Chiral polytope to the See also? —Tamfang (talk) 22:46, 30 August 2012 (UTC)
replies: text: Because his discussion of abstract chiral polytopes discusses their relation to quasiregularity? — Cheers, Steelpillow (Talk) 13:02, 1 September 2012 (UTC)
thread_title:  chiral link 

message: In Regular Polytopes (book) Coxeter says {4,3,4},  is the only regular honeycomb in Euclidean 3-space, and h{4,3,4},  or  is the only quasiregular honeycomb. So I'm not sure of the full definition, but in this case, it has a quasiregular vertex figure. So from that case alone here's a list of qualifying forms, each has alternating of two types of cells, although they are also regular if the two cells are the same type (like a checkerboard). Tom Ruen (talk) 01:23, 13 October 2013 (UTC)
replies: text: Well, Tom, that's all very fascinating but it appears to be unpublished. As such it is WP:OR and has no place here. — Cheers, Steelpillow (Talk) 09:14, 14 October 2013 (UTC)

text: As I said I only have explicit source statement for the Euclidean honeycomb h{4,3,4}, so that is defendable on the article. But for the full list above, that's why I put it here on the talk page. I'll continue looking for what else Coxeter said about Quasiregular honeycombs. Tom Ruen (talk) 09:30, 14 October 2013 (UTC)
replies: text: Eh? Coxeter does give a full definition in Regular Polytopes, p.69 (3rd ed.)! "A honeycomb is said to be quasiregular if its cells are regular while its vertex figures are quasiregular. This definition (cf. § 2·3) implies that the vertex figures are all alike, and that the cells are of two kinds, arranged alternately." Double sharp (talk) 11:35, 4 March 2015 (UTC)
replies: text:  That's helpful, at least works for me for 3D honeycombs, Euclidean and hyperbolic, and 4D polychora as spherical honeycombs. Tom Ruen (talk) 12:02, 4 March 2015 (UTC)
thread_title:  Quasiregular polychora and honeycombs 

message: (p.q)³ where p≠q, starting with (3.4)³. —Tamfang (talk) 20:30, 13 June 2016 (UTC)
replies: text: I see. I added row cases for (p p 3) and (p p 4) triangles. Tom Ruen (talk) 03:01, 14 June 2016 (UTC)
thread_title:  missing