Talk:List of regular polytopes

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Contents 1 Revert 2 Infinite forms 3 Move away from "dimensions" for group headers? 3.1 {p,q,r} table (forgive horrible pasted formating from Excel) 4 Table format 5 Broadening the Definition? 6 Regular 11-cell 4-D polytope

[edit] Revert

I reverted the phrase "There are no other non-convex polytopes in dimensions greater than four" back to the original "There are no non-convex polytopes in dimensions greater than four". It strikes me that including the word "other" might give the (false) impression that the cube, simplex and cross are non-convex. --mike40033 05:13, 26 Nov 2004 (UTC)

[edit] Infinite forms

I extended this article to include "infinite forms" - polyhedra/polytopes with a zero angle defect. Technically they exist in a one-lower dimension, but I grouped them under the topological degree. I also added subcategories for each dimension: Convex, Stellated, and infinite.

I also linked all the pictures I could find - of course they should be presented better with labels, but a start!

Tom Ruen 07:45, 22 September 2005 (UTC)

Heh. The topological degree is one lower! —Tamfang (talk) 22:38, 5 January 2008 (UTC)

[edit] Move away from "dimensions" for group headers?

I'm not happy with classifying the tilings of n-space as (n+1)-dimensional; it would be less bad, imho, to describe all the polytopes as tilings of Sⁿ, Eⁿ or Hⁿ (the positively-, zero- and negatively-curved spaces of n topological dimensions). The relation between a polyhedron as normally understood and a tiling of S² could be clarified with a few pictures of tiled spheres. —Tamfang 03:54, 23 January 2006 (UTC)

Sure, I agree "dimensions" is confusing as-is. As long {Sⁿ, Eⁿ or Hⁿ} are grouped together I'm happy. I'm not prepared to tackle renaming, but glad if you want a go. Only issue - a number of links including header anchors need to all be carefully changed. (Visible in "What links here" list.) Tom Ruen 04:41, 23 January 2006 (UTC)

• P.S. I've got a list of enumerated star-polytopes permutations {p,q,r,s} in a spreadsheet, 23 total nonconvex candidates, but unsure about existence. I could email you a copy if you want to play with it (or repeat your own count) or look for references for confirmation.

---

Four missing forms: {3,⁵/₂,3}, {4,3,⁵/₂}, {⁵/₂,3,4}, {⁵/₂,3,⁵/₂} — Tamfang 04:27, 18 January 2006 (UTC)

What are these? They APPEAR to be NONCONVEX star-polychora, but I can't generate them by vertex figures, unsure why.

I'm told by George Olshevsky that such forms have "infinite density": if I understand right, each cell covers an irrational fraction of the hypersphere and so they never close. —Tamfang 00:48, 8 February 2006 (UTC)

[edit] {p,q,r} table (forgive horrible pasted formating from Excel)

p	q	r	Test	Result	Dual	Name
3	2.5	3	0.440983	???	Self-dual	???
2.5	3	2.5	0.404508	???	Self-dual	???
3	3	2.5	0.323639	Nonconvex	I	Grand 600-cell
2.5	3	3	0.323639	Nonconvex	II	Great grand stellated 120-cell
3	3	3	0.25	Convex	Self-dual	5-cell
5	2.5	3	0.20002	Nonconvex	I	Great grand 120-cell
3	2.5	5	0.20002	Nonconvex	II	Great icosahedral 120-cell
4	3	2.5	0.172499	???	I	???
2.5	3	4	0.172499	???	II	???
4	3	3	0.112372	Convex	I	8-cell
3	3	4	0.112372	Convex	II	16-cell
2.5	5	2.5	0.095492	Nonconvex	Self-dual	Grand stellated 120-cell
5	3	2.5	0.059017	Nonconvex	I	Grand 120-cell
2.5	3	5	0.059017	Nonconvex	II	Great stellated 120-cell
3	4	3	0.042893	Convex	Self-dual	24-cell
5	2.5	5	0.036475	Nonconvex	Self-dual	Great 120-cell
3	5	2.5	0.014622	Nonconvex	I	Icosahedral 120-cell
2.5	5	3	0.014622	Nonconvex	II	Small stellated 120-cell
5	3	3	0.009037	Convex	I	120-cell
3	3	5	0.009037	Convex	II	600-cell
4	3	4	0	FLAT	Self-dual	Cubic honeycomb
3	5	3	-0.05902	HYPER	Self-dual	Icosahedral hyperbolic honeycomb
5	3	4	-0.08437	HYPER	I	Dodecahedron hyperbolic honeycomb
4	3	5	-0.08437	HYPER	II	Cubic hyperbolic honeycomb
5	3	5	-0.15451	HYPER	Self-dual	Dodecahedron hyperbolic honeycomb

Tom Ruen 04:42, 18 January 2006 (UTC)

Stripped out horrid html formatting above... Tom Ruen 21:30, 1 September 2006 (UTC)

whole hog! —Tamfang 06:51, 2 September 2006 (UTC)

[edit] Table format

I extended this article section, making consistent tables for each subsection, and a short "existence" condition for dimensions 3, 4, 5 from Coxeter's "Regular Polytopes" book.

I used Schläfli symbols for a compact notation for cell types and vertex figures. I consider this easier to read, and not too annoying, since you can scroll up and reference the lower forms.

I refrained from adding Jonathan Bowers's nickname notation, although I like it. I also kept with N-cell naming rather than <prefix>-choron as short and clear.

It needs a bit more clean up and details for completeness, but I consider this a good quick reference source now for all the regular polytopes.

Tom Ruen 02:08, 13 January 2006 (UTC)

Bowers nomenclature fills a need but I wish it weren't so, well, Orcish! — Tamfang 04:27, 18 January 2006 (UTC)

[edit] Broadening the Definition?

On re-reading this article after so long, I am reminded of Lakatos' 'Proofs and Refutations'. What exaclty is a regular polytope? I think last I looked, the tesselations in euclidean and hyperbolic space were not included. Since they are now, there are some other infinite forms in euclidean space that are not tesselations that should also be included.

And are we going to move all the way to a 'list of abstract regular polytopes' ?

See this Atlas for an idea of the dangers of going down that route... —Preceding unsigned comment added by Mike40033 (talk • contribs) 15:46, 23 May 2006

Okay, agreed - no abstract regular polytopes here. That was easy!

Seriously I accept polytope traditionally doesn't include infinite forms, but I believe all these belong together.

One worthy issue unrepresented is like the difference between a polyhedron {p,q} in 3-space versus {p,q} as a regular tiling of a sphere.

If you define polytopes as tilings on a sphere (or n-sphere more generally), you can add other regular polyhedra which have p,q<3, like {2,n} and {n,2} for instance. I'd be interested in expanding there, as well as showing all the regular polyhedra as spherical projections.

I suppose an article could be made list of regular tessellations which would contain this article content as-is (with spherical tiling polyhedra, etc), and then back down this article as polytopes as traditionally defined.

Tom Ruen 06:28, 26 May 2006 (UTC)

Perhaps, then, the article should be sectioned not according to rank (dimension), but according to the chronology of when the concept of polytope was broadened to include the objects? So the classical convex polytopes would come first, then stellated polyhedra, then tesselations of euclidean and hyperbolic space, then apeirogons (such as the zig-zag and higher-dimensional equivalents), then abstrat polytopes at the bottom? mike40033 01:22, 6 September 2006 (UTC)

I don't have time to consider a major reorganization. If you'd like try, one way is to make a TEST parallel version as a user subpage, and reorganize a copy there in a way you think is better, and then link it here for comments. Tom Ruen 19:12, 6 September 2006 (UTC)

For example: User:Mike40033/List of regular polytopes

Done. Please take a look, and give comments here. mike40033 02:13, 13 September 2006 (UTC)

It has potential. My only quick suggestion is the "existence constraint" equation should be repeated under each dimensional grouping. Tom Ruen 04:36, 13 September 2006 (UTC)

Maybe a "header section" can contain all the existence equations with anchor links to each section than enumerates the solution? Tom Ruen 04:38, 13 September 2006 (UTC)

Well, I've done it. I thought I'd better do it soon, before too many edits put the two versions out of synch. It still need pictures of the apeirohedra though... --mike40033 01:51, 20 September 2006 (UTC)

Hi Mike. Looks very good. Should be tessellation for spelling. I'm content if you want to drop it back here.

One small extension, I've been looking at the degenerate regular forms that exist as spherical tessellations: digon {2}, hosohedron {2,n}, dihedron {n,2}. Obviously extendable, although I've not seen any enumerations printed: {2}, {2,n}, {n,2}, {2,p,q}, {p,q,2}, {n,2,m}, ...

Tom Ruen 02:04, 20 September 2006 (UTC)

I changed the tesselation spelling, although maybe both are standard. I see the degenerate 2/3D forms are listed, but not in the tables. Enough for now... Tom Ruen 02:23, 20 September 2006 (UTC)

[edit] Regular 11-cell 4-D polytope

Anyone care to add information about the 11-cell regular 4-D polytope (hendecatope) described in the Apr 2007 issue (Apr or May 2007) of Discover magazine, (re)discovered by Donald Coxeter, to this article? An image or two would be especially appreciated. Coxeter also discovered a 57-cell regular 4-D polytope, also mentioned in the article. — Loadmaster 03:08, 2 June 2007 (UTC)

Well, I see that 11-cell is an article, which calls it a hendecachoron, and it has a link to the very same Discover article. But there does not seem to be a link from this article to it. — Loadmaster 03:14, 2 June 2007 (UTC)

I did some updates from 11-cell after seeing a copy of the Discover article, and didn't have a URL link to the Discover article then (yes, worth adding!). I also have a secondary article as PDF from Séquin and Lanier called Hyperseeing the Regular Hendecachoron, published at ISAMA'07 [1]. Anyway, see discussion above, consideration for abstract forms. Overall I don't understand enough - what criteria define existence of abstract forms. Tom Ruen 03:57, 2 June 2007 (UTC)