Talk:Wythoff construction

Generalizing

The discussion and examples here can be much generalized:

• Nonconvex Schwarz's triangles on sphere. (For nonconvex uniform polyhedra)
• Convex triangles on plane. (For regular tilings)
• Convex triangles on hyperbolic plane. (For uniform hyperbolic tilings)
• Convex tetrahedra on 3-space honeycombs. (For convex uniform honeycombs)
• Convex tetrahedra on hypersphere. (For uniform polychora)
• etc!!!

I gotta do some more reading before I add anything! Tom Ruen 05:19, 3 August 2006 (UTC)

Wythoff symbol

I finally got the nerve to add a section for Wythoff symbol since most links to this page are for this.

My new section isn't integrated well with the old text. Partly I wonder if a new article Wythoff symbol ought to be made, but I'll leave this as is for now.

Tom Ruen 08:31, 6 January 2007 (UTC)

A new article at Wythoff symbol, moved all the content I worked on there. I can see Wythoff construction is categorically more general than the Wythoff symbol, which is limited to polyhedra and tilings. Coexeter sells this term for all the Uniform polytope contructions, although uses the Coxeter-Dynkin diagram to represent them. Tom Ruen 01:13, 8 January 2007 (UTC)

Wythoff's symbol works only in three dimensions, although i did experiment with four dimensions (which uses six numbers!). Wythoff's construction works in any mirror-group, but can only be reliably used to produce uniform polytopes when the group is a simplex, or a prism-product of the same in Euclidean space.

The general class of alternating symmetries to produce a snub is typically counted as wythoffian. It works in all dimensions, but there are more edges to set than the degrees of freedom (eg six equations in four variables as in 4d usually has no solution). It's only in 3d, and a limited number of cases in higher dimensions, where a solution comes. The only non-wythoff figure on the main page is the banded tiling of triangles and squares. --Wendy.krieger (talk) 10:19, 29 March 2010 (UTC)

composites

I keep hoping for an explanation of p q (r s) beyond "it somehow magically combines p q r and p q s triangles". —Tamfang (talk) 20:52, 7 September 2009 (UTC)

Me too! :) Tom Ruen (talk) 22:57, 7 September 2009 (UTC)

I don't know exactly what the brackets are doing, eg p q (r s), but it has a clear meaning without the brackets, ie p q r s, such as miller's monster. It's simply a quadralateral, for example, 2 2 2 2 is a generalised rectangle as reflective region. In the snub case, squares, not triangles appear between the various polygons. --Wendy.krieger (talk) 10:12, 29 March 2010 (UTC)

Looking through Wenninger's book I noticed something: the Wythoff symbols with four numbers, two of them stacked, belong to the models whose vertex figures are bowties (crossed isosceles trapezoids) with no parallel sides: 74, 86, 90, 96, 101, 103, 109. —Tamfang (talk) 18:04, 11 June 2010 (UTC)

Merge proposal

I propose that the contents of Uniform star polyhedron/Uniform polyhedra by Wythoff construction should be merged into Wythoff construction, as both articles appear to have the same subject. Gandalf61 (talk) 08:55, 11 June 2010 (UTC)

I think maybe I started that subpage that was never completed. Part of the attempted compilation was to shows all the various constructions of visually identical polyhedra. But if it was completed, it would better to be moved into uniform star polyhedron. In contrast Wythoff construction (Coxeter's terminology) really is a very large topic of higher dimensions, and includes euclidean and hyperbolic tilings and honeycombs, so this article ought to be written to express that. PLUS, the uniform star polyhedra represent another level of wythoff construction where face(t)s and vertex figures can be overlapping star forms, which isn't explained at all in this article, and isn't very intuitive. WORSE still, not all uniform star polyhedra are produced simply by simple wythoff constructions, but rather require composite constructions of two Wythoff symbols. So, in summary, the subpage is UNWORTHY as exists, and this article ought to be expandd at some point for more general theory, considering the categories of uses and repeated extended uses by Coxeter. Tom Ruen (talk) 00:20, 12 June 2010 (UTC)

I have a related attempt at User:4/Uniform polyhedron Wythoff. 4 T C 08:22, 20 August 2011 (UTC)

Non-Wythoffian constructions

Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian. They generally can be derived from Wythoffian forms either by alternation (deletion of alternate vertices) or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry. Sometimes snub forms are considered Wythoffian, even though they can only be constructed by the alternation of omnitruncated forms.

While "non-Wythoffian" does seem to be used in the literature, if rarely, are there any references for the rest of this paragraph? Deltahedron (talk) 18:55, 3 August 2014 (UTC)

I'm suspicious of the second sentence. Does 'generally' have its vernacular meaning or its mathematical meaning? Can such constructions give Wendy Krieger's exceptional H3 tilings? —Tamfang (talk) 09:09, 25 August 2014 (UTC)

Needing much work!

The intro and the section describing the Wythoff construction are utterly incomprehensible to anyone who does not already know the concept. A much greater effort is needed in order to make this even minimally adequate, by someone with a much better knowledge and abiity to express themselves.2602:306:CF5D:1270:E4D6:73D5:CCA7:6ABD (talk) 23:44, 22 August 2017 (UTC)

Glossary for Hyperspace

Some editors ([1], [2]) consider this link might misinform readers since it was written by an expert who took the time to write it up, so I include it here with this warning to the world that experts can't always be trusted. Maybe someone can help edit this article since it is "utterly incomprehensible" according to the last comment above. Tom Ruen (talk) 19:19, 27 December 2017 (UTC)

• Olshevsky, George. "Wythoff construction". Glossary for Hyperspace. Archived from the original on 4 February 2007.