Talk:Uniform polytope

Untitled

I created this stub as I started some articles on higher dimensional uniform polytopes. It's crude, and I appreciate any help to improve it. Thanks!

Tom Ruen 04:41, 6 September 2006 (UTC)

I fleshed out some of the expressions of convex uniform polytopes and moved content under Schläfli symbol to here, since it doesn't well belong there. Still in progress, and I'll remove/reduce the text under Schläfli symbol when the restructuring is completed. Tom Ruen 07:32, 15 September 2006 (UTC)

Confusing

I added a {{confusing}} tag and merged some single-sentence paragraphs in the WP:LEAD. This article needs help from WP:MSM, specifically, it is seriously in need of {{prose}}. For example, there is a section, with eight sub-sections and dozens of diagrams, with only a single sentence of prose. That's unencyclopedic. Orange Knight of Passion (talk) 00:05, 14 September 2008 (UTC)

circumradii and v.f.

Uniform polytopes whose circumradius is equal to the edge length can be used as vertex figures for uniform tessellations. For example, the regular hexagon divides into 6 equilateral triangles and is the vertex figure for the regular triangular tiling. Also the cuboctahedron divides into 8 regular tetrahedra and 6 square pyramids (half octahedron), and it is the vertex figure for the alternated cubic honeycomb.

How useful is this paragraph? Any polytope that can be inscribed in a sphere is a candidate for a vertex figure, provided that its edge lengths are in appropriate ratios; but is it proven that all polytopes described above are v.f. of uniform tilings?

Five nonconvex polyhedra fit the description: octahemioctahedron, dodecadodecahedron, cubohemioctahedron, small dodecahemicosahedron, great dodecahemicosahedron. Are all of these known to be v.f. of (nonconvex) honeycombs? Also, 31 uniform polyhedra have circumradii smaller than their edge-length; are they all known to correspond to triangle-faced nonconvex polychora? —Tamfang (talk) 03:40, 3 June 2010 (UTC)

Quite obviously, the polytopes with H3 and H4 symmetry cannot be used as the vertex figures of a uniform tiling of Euclidean space. Even the regular pentagon cannot be the vertex figure of Euclidean space.

Additionally, the vertex figure of a tiling must be the dual polytope of the cell of the dual tiling. So, the snub dodecahedron can be the vertex figure of a uniform hyperbolic tiling if and only if the pentagonal hexecontahedron can tile hyperbolic space. Is that possible? It certainly seems implausible. Calcyman (talk) 15:50, 3 June 2010 (UTC)

Quite obviously, a hyperbolic tiling has no finite circumsphere, so it's not at issue here.

The regular pentagon's circumradius is less than its edge, so it is the v.f. of a tiling of S2, not E2. —Tamfang (talk) 18:25, 3 June 2010 (UTC)

I have been looking at uniform tessellations of Euclidean spaces, and found many had vertex figures as uniform polytopes of circumradius 1 (edge length). I scanned all the uniform polytopes of Klitzing's lists, took the subset with circumradius 1, and mapped them all onto single-ringed uniform tessellations. These subsets of uniform tessellations also correspond to sphere packings, with one sphere centered at each vertex, and the kissing number of each sphere packing is equal to the number of vertices in the vertex figure polytope. My test-list is at User:Tomruen/Uniform_lattices, and that was what I was getting at. Tom Ruen (talk) 19:38, 3 June 2010 (UTC)

The cubohemioctahedron and octahemioctahedra are indeed vertex figures of nonconvex Euclidean honeycombs. In the tetrahedral-octahedral honeycomb you can (as the name suggests) find tetrahedra (corresponding to the triangles of the cuboctahedron), octahedra (corresponding to the squares), but also triangular tilings as flat surfaces (not actually cells, corresponding to the equatorial hexagons). The honeycomb with octahemioctahedra as vertex figures has the tetrahedra and the triangular tilings; the one with cubohemioctahedra as vertex figures has the octrahedra and the triangular tilings. They're #4 and #5 on Eric Binnendyk's list of uniform honeycombs. The octahemioctahedron-verf one is on Richard Klitzing's site, but I can't find the cubohemioctahedron-verf one. Double sharp (talk) 08:10, 8 September 2022 (UTC)

Special case prismatic reductions

Special cases of products become hypercubes: ...

True, but the notation shown is for the symmetry groups, not for the polytopes themselves; this is wrong — the symmetry group [4,3] is bigger than the product []×[]×[]. —Tamfang (talk) 19:09, 12 June 2010 (UTC)

Rewrite if you like. By symmetry they are all listable, but by geometry (of uniform solutions) and topological result, they're identical, so the question is how far you want to expand the lists. Tom Ruen (talk) 03:08, 13 June 2010 (UTC)

single-ring v.f.

Vertex figures for single-ringed Coxeter-Dynkin diagrams can be constructed from the diagram by removing the ringed node, and ringing neighboring nodes. Such vertex figures are also uniform polytopes, being prismatic if the ringed node was in contact with more than one node.

I'm going to remove the latter sentence. Such figures are vertex-transitive, but not uniform (faces are not regular) if the arcs from the ringed node are not alike; and they're not prismatic if the parent CD is multiply connected. —Tamfang (talk) 16:27, 6 May 2011 (UTC)

Summary table

I'm still working on organization for the higher dimensional uniform polytopes, but here's a summary of finite and convex uniform polytope articles. Tom Ruen (talk) 22:43, 3 June 2011 (UTC)

n	Coxeter group	Regular t0	Quasiregular t1	Truncated t0,1	Cantellated t0,2	Runcinated t0,3	Stericated t0,4	Pentellated t0,5	Hexicated t0,6	Heptellated t0,7
3	A3=[3,3] BC3=[3,4] H3=[3,5]	tetrahedron octahedron cube dodecahedron icosahedron	cuboctahedron icosidodecahedron	Truncated_tetrahedron Truncated_octahedron Truncated_cube Truncated_dodecahedron Truncated_icosahedron	Rhombicuboctahedron Rhombicosidodecahedron Truncated cuboctahedron Truncated icosidodecahedron Snub cube Snub dodecahedron
4	A4=[3,3,3] F4=[3,4,3] BC4=[3,3,4] H4=[3,3,5]	5-cell 16-cell tesseract 24-cell 120-cell 600-cell	Rectified_5-cell Rectified_24-cell Rectified_tesseract Rectified_120-cell Rectified_600-cell	Truncated_5-cell Truncated_24-cell Truncated_tesseract Truncated_120-cell	Cantellated_5-cell Cantellated_24-cell Cantellated_tesseract Cantellated_120-cell	Runcinated_5-cell Runcinated_24-cell Snub_24-cell Runcinated_tesseract Runcinated_120-cell
5	A5=[3,3,3,3] BC5=[3,3,3,4] D5=[32,1,1]	5-simplex 5-orthoplex 5-cube	Rectified_5-simplex Rectified_5-orthoplex Rectified_5-cube 5-demicube	Truncated_5-simplex Truncated_5-orthoplex Truncated_5-cube Truncated_5-demicube	Cantellated_5-simplex Cantellated_5-orthoplex Cantellated_5-cube Cantellated_5-demicube	Runcinated_5-simplex Runcinated_5-orthoplex Runcinated_5-cube Runcinated_5-demicube	Stericated_5-simplex Stericated_5-orthoplex Stericated_5-cube
6	A6=[35] BC6=[34,4] D6=[33,31,1] E6=[32,2,1]	6-simplex 6-orthoplex 6-cube	Rectified_6-simplex Rectified_6-orthoplex Rectified_6-cube 6-demicube 221, 122	Truncated_6-simplex Truncated_6-orthoplex Truncated_6-cube Truncated_6-demicube Truncated 221 Truncated 122	Cantellated_6-simplex Cantellated_6-orthoplex Cantellated_6-cube Cantellated_6-demicube Cantellated 221	Runcinated_6-simplex Runcinated_6-orthoplex Runcinated_6-cube Runcinated_6-demicube	Stericated_6-simplex Stericated_6-orthoplex Stericated_6-cube Stericated_6-demicube	Pentellated_6-simplex Pentellated_6-orthoplex Pentellated_6-cube
7	A7=[36] BC7=[35,4] D7=[34,31,1] E7=[33,2,1]	7-simplex 7-orthoplex 7-cube	Rectified_7-simplex Rectified_7-orthoplex Rectified_7-cube 7-demicube 321, 231, 132	Truncated_7-simplex Truncated_7-orthoplex Truncated_7-cube Truncated_7-demicube Truncated 321 Truncated 231 Truncated 132	Cantellated_7-simplex Cantellated_7-orthoplex Cantellated_7-cube Cantellated_7-demicube Cantellated 321 Cantellated 231	Runcinated_7-simplex Runcinated_7-orthoplex Runcinated_7-cube Runcinated_7-demicube	Stericated_7-simplex Stericated_7-orthoplex Stericated_7-cube Stericated_7-demicube	Pentellated_7-simplex Pentellated_7-orthoplex Pentellated_7-cube Pentellated_7-demicube	Hexicated_7-simplex Hexicated_7-orthoplex Hexicated_7-cube
8	A8=[37] BC8=[36,4] D8=[35,31,1] E8=[34,2,1]	8-simplex 8-orthoplex 8-cube	Rectified_8-simplex Rectified_8-orthoplex Rectified_8-cube 8-demicube 421, 241, 142	Truncated_8-simplex Truncated_8-orthoplex Truncated_8-cube Truncated_8-demicube Truncated 421 Truncated 241 Truncated 142	Cantellated_8-simplex Cantellated_8-orthoplex Cantellated_8-cube Cantellated_8-demicube Cantellated 421 Cantellated 241	Runcinated_8-simplex Runcinated_8-orthoplex Runcinated_8-cube Runcinated_8-demicube Runcinated 421	Stericated_8-simplex Stericated_8-orthoplex Stericated_8-cube Stericated_8-demicube	Pentellated_8-simplex Pentellated_8-orthoplex Pentellated_8-cube Pentellated_8-demicube	Hexicated_8-simplex Hexicated_8-orthoplex Hexicated_8-cube Hexicated_8-demicube	Heptellated_8-simplex Heptellated_8-cube
9	A9=[38] BC9=[37,4] D9=[36,31,1]	9-simplex 9-orthoplex 9-cube	Rectified_9-simplex Rectified_9-orthoplex Rectified_9-cube 9-demicube
10	A10=[39] BC10=[38,4] D10=[37,31,1]	10-simplex 10-orthoplex 10-cube	Rectified_10-simplex Rectified_10-orthoplex Rectified_10-cube 10-demicube

Definition/Lede

"A uniform polytope is a vertex-transitive polytope made from uniform facets of a lower dimension. The uniform polytopes in two dimensions are the regular polygons."

If the definition is the first sentence only, a requirement that all the edges must congruent should be added for consistency within the article. I'm not aware if there's a competing definition of uniformity that permits faces with an even number of edges to have two different alternating lengths, but if there is it's not the most commonly accepted one. If the second sentence is part of the definition, this should be rephrased to make that more clear. 173.227.48.5 (talk) 23:40, 11 April 2013 (UTC)

Yes, wording could be improved. Its a recursive definition with defining uniform polygons as regular polygons as the smallest nontrivial dimension, so congruent edges are implied from 2 and up. Tom Ruen (talk) 02:57, 12 April 2013 (UTC)

This might be excessively pedantic, but the article appears to consider the line segment to be a uniform polytope. A single vertex and the null polytope might be considered (uniform) polytopes as well. The revised definition only refers to dimensions 2 and up. 173.227.48.5 (talk) 17:07, 12 April 2013 (UTC)

Actually a apeirogon can be considered uniform (rather than regular) if it has alternating 2 colors (types) of edges, which is also a simple truncation t{∞}=t_0,1{∞}=. Tom Ruen (talk) 03:55, 26 September 2013 (UTC)

"Scaliform"

The "scaliform" term is really only used by Bowers – unless it is adapted by someone more notable, I don't believe it belongs on this article. I deleted the corresponding section with this justification. Hope it isn't an issue. – OfficialURL (talk) 21:53, 6 April 2020 (UTC)