Talk:List of regular polytopes

message: Note: failed FLC can be found at Wikipedia:Featured list candidates/List of regular polytopes. Double sharp (talk) 04:37, 8 March 2015 (UTC)

message: 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)
thread_title:  Revert 

message: 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)
replies: text: Heh.  The topological degree is one lower!  —Tamfang (talk) 22:38, 5 January 2008 (UTC)
thread_title:  Infinite forms 

message: : 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)
replies: text:  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)
replies: text: * 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)

text: 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)

text: Stripped out horrid html formatting above... Tom Ruen 21:30, 1 September 2006 (UTC)
replies: text:  whole hog! —Tamfang 06:51, 2 September 2006 (UTC)
thread_title:  Move away from "dimensions" for group headers? 

message: 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)
replies: text: Stripped out horrid html formatting above... Tom Ruen 21:30, 1 September 2006 (UTC)
replies: text:  whole hog! —Tamfang 06:51, 2 September 2006 (UTC)
thread_title:  {p,q,r} table (forgive horrible pasted formating from Excel) 

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

message: Hm.  I thought we had pictures of the four principal orthographic 3-projections of the convex regular 4-polytopes, but can't find them. —Tamfang (talk) 08:38, 27 February 2011 (UTC)
replies: text: Some are grouped together, like tesseract at :File:Orthogonal_projection_envelopes_tesseract.png and 16-cell at :File:Orthogonal_projection_envelopes_16-cell.png. Tom Ruen (talk) 01:18, 28 February 2011 (UTC)
thread_title:  shadows 

message: Hi I know nothing about fixing wikipedia but I was reading this article and was wondering how legitimate the 'fuck symbol' is... seems like some vandalism to me. Mind you I also know nothing about fifth dimensions so I could be dead wrong it just looks hoaxy to me. Sorry if I'm doing the talk page thing wrong, too like I said I'm very ignorant here. 69.9.67.133 (talk) 02:16, 5 May 2011 (UTC)
replies: text: Yes, it was vandalism, corrected now. The page history can show recent changes, and identify the vandalism,  in this case. Tom Ruen (talk) 01:44, 6 May 2011 (UTC)
thread_title: Really?

message: Hi. I'm not expert in this, but it seems that there is a problem in column 'Convex Euclidean tessellations' in the first table. Seems to be shifted by 1 row. E.g. '3 tillings' should be listed under 'dimension=2' and not 'dimension=3'. Marsiancba (talk) 13:26, 24 June 2012 (UTC)
replies: text: No, it's correct (if unintuitive). A polyhedron in 3D, and an infinite polyhedron would be an infinite tiling of the plane. Besides, like the 3D polyhedra, their Coxeter-Dynkin diagrams have three nodes (the number of nodes denotes the number of dimensions). This should be explained in the article, though. Double sharp (talk) 09:10, 19 August 2012 (UTC)
thread_title: Tesselations dimensions error?

message: "A 5-polytope has been called a polyteron, . . ."
Egomaniacs should not inflict their coinages on the whole world, using Wikipedia as their vehicle.  Please: Get your own blog. Spare us your coinages.Daqu (talk) 05:56, 5 July 2012 (UTC)
replies: text: I don't care about this as much as I do for polychoron, but it does seem to be becoming a standard among the main current polytope geometers. I wouldn't feel much for it if it were deleted though: "polychoron" is becoming a standard outside those polytope geometers, but not "polyteron", "polypeton", "polyexon" (it's been changed to "polyecton" anyway for some reason), "polyzetton" and "polyyotton". Double sharp (talk) 09:24, 20 July 2012 (UTC)
thread_title:  Polyhogwash 

message: There would be an infinite number of regular compound polygons and five regular compound polyhedra (these are easily referenced). In 4D and above I don't think they're known, or at least referenceable. Double sharp (talk) 07:26, 24 January 2015 (UTC)
replies: text: I"m open either way, but I've seen nothing above 4D. But I've never checked what Stella4D can construct from this list, regular 4-polytope compound. Tom Ruen (talk) 07:59, 24 January 2015 (UTC)
replies: text: You'd have to find the vertex figure for the compound and then press "u". For example, starting with the stella octangula and pressing "u" gives you the 3-tesseract compound (select "4" as the edge length). The dual is the compound of three 16-cells. Double sharp (talk) 07:46, 25 January 2015 (UTC)
replies: text: Oh right, you don't need that one: it's already in the library, along with 75 tesseracts/16-cells, 25 24-cells, and 10 grand 600-cells. Now playing around to find more... Double sharp (talk) 07:50, 25 January 2015 (UTC)
replies: text: Taking the 10-tetrahedra compound and pressing "u" (select 4) gives you 75 tesseracts. The dual is of course 75 16-cells. Double sharp (talk) 13:48, 25 January 2015 (UTC)

text: Cobbled together a compounds section from star polygon and polytope compound. Double sharp (talk) 08:45, 25 January 2015 (UTC)
replies: text: So the regular star compounds are just 5 or 10 of each Schläfli-Hess polychoron? Double sharp (talk) 13:44, 25 January 2015 (UTC)

text: OK, I think I see the problem with 225 24-cells: it's not cell-transitive. (As pointed out by George Olshevsky). But crucially, this is the fault of a compound of 25 24-cells within it! The rest of the cells work fine, so if you remove those 25 24-cells, you get the compound of 200 24-cells, which Coxeter does list. (It's still not regular though, unfortunately.) Double sharp (talk) 14:08, 25 January 2015 (UTC)
replies: text: 720 5-cells is not cell-transitive either. (But the compounds of 675 I mentioned are both vertex- and cell-transitive. I guess they fail with faces or edges?) Double sharp (talk) 14:21, 25 January 2015 (UTC)
replies: text: 675 = 225 × 3, so perhaps the 300- and 600- tesseract/16-cell compounds are related instead to the more regular 200 24-cell compound. In that case I don't see anything missing from the list. Double sharp (talk) 14:23, 25 January 2015 (UTC)

text: I'd prefer the latter: the compounds are still closely related, after all. Double sharp (talk) 12:13, 26 January 2015 (UTC)
thread_title:  Do compounds belong here? 

message: {2,∞} is not mentioned; is there a reason why it doesn't work? Double sharp (talk) 08:42, 25 January 2015 (UTC)
replies: text: Its dual was listed, so I added it. List_of_regular_polytopes#Euclidean_tilings. Tom Ruen (talk) 11:35, 25 January 2015 (UTC)
thread_title:  apeirogonal hosohedron? 

message: Are things like {4,3}/2 and friends suitable for this list? They have "improper examples" too, being of the form {1,2q} and {2p,1}. Double sharp (talk) 12:02, 1 February 2015 (UTC)
replies: text: It seems like this article might be a better place to work on: Regular_map_(graph_theory). But I don't feel comfortable doing anything without reading more. Tom Ruen (talk) 14:14, 1 February 2015 (UTC)
replies: text: We already have the examples {1,2} and {2,1}. But {1} seems to behave rather badly. In projective geometry (which you can get by just identifying antipodal points on the sphere) you would be able to realize {1,2q} and {2p,1}, but they don't seem to want to behave well as spherical polytopes except for {1,1}, {1,2}, {2,1}! Double sharp (talk) 15:20, 1 February 2015 (UTC)
thread_title:  projective regulars 

message: What does {p,2,r} end up like? Double sharp (talk) 14:48, 1 February 2015 (UTC)
replies: text: Dihedral cells: {p,2}, and hosohedral vertex figure {2,q}? Tom Ruen (talk) 15:46, 1 February 2015 (UTC)
replies: text: OK, added a mention of these to the 4D improper forms. Double sharp (talk) 14:26, 2 February 2015 (UTC)
thread_title:  just a thought re improper 4-polytopes 

message: Thinking of doing one for the S4/E4/H4 case: it suffices to keep p, q, r, s as 3, 4, or 5 as all the other cases instantly end up as noncompact. Then you'd have a 3×3 array of individual 3×3 little tables.
::  Double sharp (talk) 08:00, 5 February 2015 (UTC)
thread_title:  summary tables 

message: Feel free to add or edit pro/con reasons below. Tom Ruen (talk) 18:26, 7 March 2015 (UTC)
replies: text: Sources for this approach, that immediately come to mind, include Grünbaum's "Are Your Polyhedra the Same as My Polyhedra?", the paper by McMullen and Schulte cited in the apeirotopes section (and probably many of their other papers), and I would expect the general principle to be endorsed in the Johnson ms you think so highly of. Also, I am aware of the importance of the abstract approach in contemporary work and that kind of editorial knowledge is expected to help assess the reliability of sources and shape the presentation of material. The key point about abstract theory is that it helps in this shaping: indeed, there is a strong case to make that this kind of classification issue is exactly the kind of thing that motivated abstract theory in the first place. Grünbaum set the ball rolling half a century ago, Danzer, McMullen and Schulte had pretty much cracked the mathematics by the turn of the millennium and their approach has been going like a train since then. Seriously, if you are so ignorant of the subject matter to this extent, you should not be throwing your editorial weight around so confidently. — Cheers, Steelpillow (Talk) 21:04, 7 March 2015 (UTC)
replies: text: See above, both this topic and previous ones. I am weary of saying the same thing over and over. It never seems to sink in, why should I repeat what will not sink in this time either. — Cheers, Steelpillow (Talk) 21:04, 7 March 2015 (UTC)
replies: text:  Perhaps I should be insulted, seriously. All we have is your opinion that this is an important topic, which I am not denying. I am merely challenging you that this article requires that section. I have no evidence a single other person would agree with your editorial opinion. Tom Ruen (talk) 21:20, 7 March 2015 (UTC)
thread_title:  Discussion to rename this article to [[List of flat regular polytopes, honeycombs and compounds]] 

message: I am trying to translate article to russian.
Some sentence I cannot understand.... What means not covered above (p,q,r,s,... natural numbers above 2, or infinity)? Jumpow (talk) 21:27, 9 January 2016 (UTC)
replies: text: It could be worded better. It's saying all higher integer values not listed give noncompact hyperbolic honeycombs, like {7,3,3} for example. Tom Ruen (talk) 21:36, 9 January 2016 (UTC)
thread_title:  Difficult language 

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thread_title:  External links modified 

message: Coxeter proved that, but assumed compact or paracompact honeycombs. It does not appear that we are making this assumption, judging from all the pictures of honeycombs like {5,3,7}.
So, could one not stellate the dodecahedra of {5,3,4}, say, to produce a noncompact honeycomb {5/2,5,4}? (Presumably the stellation "pokes out past infinity".) Double sharp (talk) 09:48, 14 August 2021 (UTC)
replies: text: (Answering myself late: yes you can, but I've not found an RS for it.) Double sharp (talk) 23:22, 31 January 2023 (UTC)
thread_title:  No regular star-honeycombs in H<sup>3</sup>? 

message: One sentence reads as follows:
"There are ten flat regular honeycombs of hyperbolic 3-space: (previously listed above as tessellations) ..."
1. I do not know what "flat" means in this context, and there is no explanation or link to helpful text. Can someone please include a clarification somewhere in the article?
2. A number of tessellations in the article are called "paracompact". There is no explanation of what this means or any link to helpful text. The term "paracompact" has a standard meaning in mathematics having nothing directly to do with tessellations: Paracompact.
I strongly suggest not using standard mathematical terminology in a nonstandard way. And in any case explaining what the meaning is in this article. I hope someone knowledgeable will do this.
3. There is nothing wrong with calling a tessellation of a simply connected space like Euclidean n-space or hyperbolic n-space a "honeycomb". But it most definitely is a tessellation, so there is something seriously wrong with not calling it a tessellation.
:# I think flat is meant informally in this context. Flat has a technical definition in Abstract Regular Polytopes, but its not this. It should probably be removed or replaced.
:# Paracompact has a standard definition in the context of Coxeter groups which is being used here. It's used this way in the literature. A Paracompact Coxeter group is one such that its fundamental domain has finite area, alternatively it is a Coxeter group such that all the groups generated by any strict subset of their generators are at most Euclidean.
:# Honeycomb has a technical definition in McMullen. A honeycomb is a apeirotope of full rank. So e.g. the apeir tetrahedron is a honeycomb but the mucube isn't.
: This article definitely has issues with using technical terminology inconsistently and without clarification. AquitaneHungerForce (talk) 14:36, 24 January 2024 (UTC)
thread_title:  Terminology problems 

message: whut Ayen2022-3 (talk) 12:04, 31 March 2022 (UTC)
thread_title:  [a] 

message: McMullen paper from 2007. Double sharp (talk) 23:22, 31 January 2023 (UTC)
thread_title:  More regular polyhedra in 4-space 

message: The definition this page gives for a regular compound is the same as the one given by the page Polyhedron compound. I am disputing the accuracy of this definition. Please use the thread I started on that page to read my issue and discuss. AquitaneHungerForce (talk) 04:58, 20 January 2024 (UTC)
thread_title:  Inaccurate compound definition 

message: This article currently covers Regular polytopes and Regular polytope compounds (or simply Regular compounds). These concepts are related, however the name of these concepts implies a stronger relationship between them then there really is between these concepts. A regular polytope has a flag-transitive symmetry group, however many regular compounds are not flag transitive (e.g. the compound of 5 tetrahedra) and some flag-transitive compounds are not regular compounds (e.g. the dual compound of the 5-cell).
Since these concepts are not analogous, I am proposing that this article be split into List of regular polytopes and List of regular polytope compounds as to cover them separately. This would be pretty easy since the content on the two is already pretty well separated in the text. AquitaneHungerForce (talk) 15:51, 22 January 2024 (UTC)
replies: text: Since no discussion happened I just went ahead and did it. WP:BEBOLD. AquitaneHungerForce (talk) 18:44, 15 February 2024 (UTC)
thread_title:  Proposal to split of "Regular compounds" from this article