View source for Talk:List of regular polytopes

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

== 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. --[[User:Mike40033|mike40033]] 05:13, 26 Nov 2004 (UTC)

== 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!

[[User:Tomruen|Tom Ruen]] 07:45, 22 September 2005 (UTC)

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

== 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 S<sup>n</sup>, E<sup>n</sup> or H<sup>n</sup> (the positively-, zero- and negatively-curved spaces of n topological dimensions).  The relation between a polyhedron as normally understood and a tiling of S<sup>2</sup> could be clarified with a few pictures of tiled spheres.  &mdash;[[User:Tamfang|Tamfang]] 03:54, 23 January 2006 (UTC)

:: Sure, I agree "dimensions" is confusing as-is. As long {S<sup>n</sup>, E<sup>n</sup> or H<sup>n</sup>} 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.) [[User:Tomruen|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,<sup>5</sup>/<sub>2</sub>,3}, {4,3,<sup>5</sup>/<sub>2</sub>}, {<sup>5</sup>/<sub>2</sub>,3,4}, {<sup>5</sup>/<sub>2</sub>,3,<sup>5</sup>/<sub>2</sub>}
&mdash; [[User:Tamfang|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. &mdash;[[User:Tamfang|Tamfang]] 00:48, 8 February 2006 (UTC)

=== {p,q,r} table (forgive horrible pasted formating from Excel) ===
{|class="wikitable"
!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
|<span style="color:#ff00ff;">Grand 600-cell</span>  
|-
|2.5
|3
|3
|0.323639
|Nonconvex
|II
|<span style="color:#ff00ff;">Great grand stellated 120-cell</span>  
|-
|3
|3
|3
|0.25
|Convex
|Self-dual
|<span style="color:#ff0000;">5-cell</span>  
|-
|5
|2.5
|3
|0.20002
|Nonconvex
|I
|<span style="color:#ff00ff;">Great grand 120-cell</span>  
|-
|3
|2.5
|5
|0.20002
|Nonconvex
|II
|<span style="color:#ff00ff;">Great icosahedral 120-cell</span>  
|-
|4
|3
|2.5
|0.172499
|???
|I
|???  
|-
|2.5
|3
|4
|0.172499
|???
|II
|???  
|-
|4
|3
|3
|0.112372
|Convex
|I
|<span style="color:#ff0000;">8-cell</span>  
|-
|3
|3
|4
|0.112372
|Convex
|II
|<span style="color:#ff0000;">16-cell</span>  
|-
|2.5
|5
|2.5
|0.095492
|Nonconvex
|Self-dual
|<span style="color:#ff00ff;">Grand stellated 120-cell</span>  
|-
|5
|3
|2.5
|0.059017
|Nonconvex
|I
|<span style="color:#ff00ff;">Grand 120-cell</span>  
|-
|2.5
|3
|5
|0.059017
|Nonconvex
|II
|<span style="color:#ff00ff;">Great stellated 120-cell</span>  
|-
|3
|4
|3
|0.042893
|Convex
|Self-dual
|<span style="color:#ff0000;">24-cell</span>  
|-
|5
|2.5
|5
|0.036475
|Nonconvex
|Self-dual
|<span style="color:#ff00ff;">Great 120-cell</span>  
|-
|3
|5
|2.5
|0.014622
|Nonconvex
|I
|<span style="color:#ff00ff;">Icosahedral 120-cell</span>  
|-
|2.5
|5
|3
|0.014622
|Nonconvex
|II
|<span style="color:#ff00ff;">Small stellated 120-cell</span>  
|-
|5
|3
|3
|0.009037
|Convex
|I
|<span style="color:#ff0000;">120-cell</span>  
|-
|3
|3
|5
|0.009037
|Convex
|II
|<span style="color:#ff0000;">600-cell</span>  
|-
|4
|3
|4
|0
|FLAT
|Self-dual
|<span style="color:#008000;">Cubic honeycomb</span>  
|-
|3
|5
|3
| -0.05902
|HYPER
|Self-dual
|<span style="color:#0000ff;">Icosahedral hyperbolic honeycomb</span>  
|-
|5
|3
|4
| -0.08437
|HYPER
|I
|<span style="color:#0000ff;">Dodecahedron hyperbolic honeycomb</span>  
|-
|4
|3
|5
| -0.08437
|HYPER
|II
|<span style="color:#0000ff;">Cubic hyperbolic honeycomb</span>  
|-
|5
|3
|5
| -0.15451
|HYPER
|Self-dual
|<span style="color:#0000ff;">Dodecahedron hyperbolic honeycomb</span> 
|}
[[User:Tomruen|Tom Ruen]] 04:42, 18 January 2006 (UTC)

: Stripped out horrid html formatting above... [[User:Tomruen|Tom Ruen]] 21:30, 1 September 2006 (UTC)

:: whole hog! &mdash;[[User:Tamfang|Tamfang]] 06:51, 2 September 2006 (UTC)

== 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.

[[User:Tomruen|Tom Ruen]] 02:08, 13 January 2006 (UTC)

: Bowers nomenclature fills a need but I wish it weren't so, well, Orcish! &mdash; [[User:Tamfang|Tamfang]] 04:27, 18 January 2006 (UTC)

== 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 [http://sepang.nottingham.edu.my/~hmichael/atlas this Atlas] for an idea of the dangers of going down that route...  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Mike40033|Mike40033]] ([[User talk:Mike40033|talk]] • [[Special:Contributions/Mike40033|contribs]]) 15:46, 23 May 2006</span></small><!-- Template:Unsigned -->

: 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.
: [[User:Tomruen|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? [[User:Mike40033|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. [[User:Tomruen|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. [[User:Mike40033|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. [[User:Tomruen|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? [[User:Tomruen|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... --[[User:Mike40033|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}, ...
:: [[User:Tomruen|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... [[User:Tomruen|Tom Ruen]] 02:23, 20 September 2006 (UTC)

== 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 [http://discovermagazine.com/2007/apr/jarons-world-shapes-in-other-dimensions/?searchterm=polytope 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. — [[User:Loadmaster|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. — [[User:Loadmaster|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 [http://archone.tamu.edu/isama07]. Anyway, see discussion above, consideration for abstract forms. Overall I don't understand enough - what criteria define existence of abstract forms. [[User:Tomruen|Tom Ruen]] 03:57, 2 June 2007 (UTC)

:::Perhaps Séquin and Lanier did as well as possible, but their visualization of the 11-cell conveys virtually no information.  It was discovered by Branko Grünbaum (and later independently by H.S.M. Coxeter), though both of their publications are very difficult to locate, and as far as I know neither is available online, alas.

:::The most important thing to note, however, is that the 11-cell is '''not''' a regular polytope. Rather, it is an "abstract regular polytope" which is a '''much''' more general concept that includes regular polytopes, but is very far from limited to them.[[User:Daqu|Daqu]] ([[User talk:Daqu|talk]]) 06:08, 5 July 2012 (UTC)

== Nonconvex hyperbolic (infinite) regular polytopes in dimension 3 (regular tilings of star polygons in H2) ==

The table of regular polytope counts by dimension in the "Overview" section of the article lists the number of nonconvex hyperbolic tesselations in dimension 3 (the number of regular tilings of star polygons in H2) as infinite, but there is no subsection of the "Three-dimensional regular polytopes" section detailing those hyperbolic star tilings.  The "Euclidean star-tilings" subsection of that section, however, reads in part, "There are no regular plane tilings of [[star polygon]]s. There are many enumerations that fit in the plane (1/''p'' + 1/''q'' = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc, but none repeat periodically."  Are those enumerations of star polygons where 1/''p'' + 1/''q'' = 1/2 the infinite family of hyperbolic star tilings?  I wouldn't think so, if they fit in the Euclidean plane.  Are there an infinite number of regular hyperbolic tilings of regular star polygons?  Or is the table in the "Overview" section wrong in that respect, and if so, what is the actual number of regular tilings of star polygons in H2? [[User:Kevin Lamoreau|Kevin Lamoreau]] ([[User talk:Kevin Lamoreau|talk]]) 00:47, 4 April 2010 (UTC)

: The key to your question is in the equality 1/''p'' + 1/''q'' = 1/2 ; for H2 that '=' would be '&lt;'. —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 18:26, 5 April 2010 (UTC)

::Thank you, Tamfang.  But do any of that infinite number of enumerations repeat periodically?  If so, then those that do should be listed, or defined if they belong to an infinite family or families (the 1/''p'' + 1/''q'' &lt; 1/2 you mention if they all repeat periodically, otherwise some other definition of one or more families that covers all the "qualifying" enumerations).  If not, or if a non-infinite number of hyperbolic tilings of star polygons exist that do repeat perioically and can thus be considered regular tesselations of star polygons and thus regular infinite hyperbolic nonconvex polyhedra, then the table in the "Overview" section for the Dimension 3 row in the "Nonconvex hyperbolic tessellations" column should be corrected to show the correct number.  That number is currently listed as ∞ (infinity). [[User:Kevin Lamoreau|Kevin Lamoreau]] ([[User talk:Kevin Lamoreau|talk]]) 00:24, 11 April 2010 (UTC)

:::There are two infinite series of nonconvex hyperbolic tilings that turn out to repeat periodically, which are {''p''/2, ''p''} and {''p'', ''p''/2}. Such tilings exist for all odd ''p'' ≥ 7, so the table is correct. (On a slightly irrelevant note, these two patterns extend into spherical space for odd ''p'' < 7; for ''p'' = 5, we obtain the [[small stellated dodecahedron]] and [[great dodecahedron]], while for ''p'' = 3, we obtain the [[tetrahedron]]. ''p'' cannot be even, as if [[greatest common divisor|gcd]](''p'', ''q'') > 1 for a star polygon {''p''/''q''}, the star polygon is degenerate.) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 07:04, 22 May 2012 (UTC)
:::For example, here are the two tilings generated when ''p'' = 7: the {7/2, 7} and {7, 7/2} tilings. The heptagons/heptagrams have been coloured by the colours of the [[visible spectrum]], going around a vertex marked by a black circle. Note that the {7/2, 7} is a [[stellation]] of the {7, 3} and that the {7, 7/2} is a [[faceting]] of the {3, 7} and a greatening of the {7, 3}, and their relationships can be compared to the relationship between the [[dodecahedron]], [[small stellated dodecahedron]] and the [[great dodecahedron]] (the tilings have ''p'' = 7, while the polyhedra have ''p'' = 5).
:::[[File:Hyperbolic tiling 7-2 7.png|200px]][[File:Hyperbolic tiling 7 7-2.png|200px]]
:::[[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 07:21, 22 May 2012 (UTC)

:::Here are the tilings for ''p'' = 9. This time, I have chosen a vertex-centered rather than face-centered view for the {''p''/2, ''p''} tiling, as a face-centered view would render the faces near the boundary of the disk to small to see clearly. Both the ''p'' = 7 and ''p'' = 9 images are now in the list, and the wording has been made clearer for the star-tilings of H<sup>2</sup>.
:::[[File:Hyperbolic tiling 9-2 9.png|200px]][[File:Hyperbolic tiling 9 9-2.png|200px]]
:::[[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:08, 23 May 2012 (UTC)

:::For comparison, the polyhedra that occur for ''p'' = 5 are shown below (these images are by [[User:Tomruen|Tomruen]]). Here, only one face is highlighted, but it is not hard to see the similarities between these and the above hyperbolic tilings.
:::[[File:Small stellated dodecahedron tiling.png|200px]][[File:Great dodecahedron tiling.png|200px]]
:::[[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:11, 23 May 2012 (UTC)

:::The similarities are very striking if we compare the vertex figures of the polyhedra and tilings:
:::[[File:Small stellated dodecahedron vertfig.png|200px]][[File:Hyperbolic tiling 7-2 7.png|200px]][[File:Hyperbolic tiling 9-2 9.png|200px]]
:::[[File:Great dodecahedron vertfig.png|200px]][[File:Hyperbolic tiling 7 7-2.png|200px]][[File:Hyperbolic tiling 9 9-2.png|200px]]
:::[[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:14, 23 May 2012 (UTC)

:::I've added more detail to the article. I've moved the rest of my comments, along with all the discussion in this section, to [[User:Double sharp/Hyperbolic star-tilings discussion]]. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:16, 23 May 2012 (UTC)

:Looks good DS. It's a bit confusing, pictures help some, but hard to improve with overlapping geometry. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 01:21, 25 May 2012 (UTC)
::Restored the discussion (because otherwise it is not clear what Tomruen's comment is referring to). [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 13:49, 3 March 2018 (UTC)

== shadows ==

Hm.  I thought we had pictures of the four principal orthographic 3-projections of the convex regular 4-polytopes, but can't find them. —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 08:38, 27 February 2011 (UTC)

: Some are grouped together, like [[tesseract]] at [[:File:Orthogonal_projection_envelopes_tesseract.png]] and [[16-cell]] at [[:File:Orthogonal_projection_envelopes_16-cell.png]]. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 01:18, 28 February 2011 (UTC)

{| class=wikitable
|+ 3 of 6
|-
|[[File:Orthogonal_projection_envelopes_5-cell.png|160px]]
|[[File:Orthogonal_projection_envelopes_tesseract.png|160px]]
|[[File:Orthogonal_projection_envelopes_16-cell.png|160px]]
|[[File:Orthogonal projection envelopes 24-cell.png|160px]]
|[[File:Orthogonal projection envelopes 120-cell.png|160px]]
|[[File:Orthogonal projection envelopes 600-cell.png|160px]]
|-
![[5-cell]]
![[Tesseract]]
![[16-cell]]
![[24-cell]]
![[120-cell]]
![[600-cell]]
|}

==Really?==
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. [[Special:Contributions/69.9.67.133|69.9.67.133]] ([[User talk:69.9.67.133|talk]]) 02:16, 5 May 2011 (UTC)

: Yes, it was vandalism, corrected now. The ''page history'' can show recent changes, and identify the vandalism, [http://en.wikipedia.org/w/index.php?title=List_of_regular_polytopes&action=historysubmit&diff=427501532&oldid=426467283] in this case. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 01:44, 6 May 2011 (UTC)
==Tesselations dimensions error?==
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'. [[User:Marsiancba|Marsiancba]] ([[User talk:Marsiancba|talk]]) 13:26, 24 June 2012 (UTC)
: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 diagram]]s have three nodes (the number of nodes denotes the number of dimensions). This should be explained in the article, though. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 09:10, 19 August 2012 (UTC)

== Polyhogwash ==

"''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.[[User:Daqu|Daqu]] ([[User talk:Daqu|talk]]) 05:56, 5 July 2012 (UTC)
: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". [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 09:24, 20 July 2012 (UTC)

Just to nail the coffin, here's a calculator with said ego, rendering further non-standardised coinage unimpressive: [http://thenameofyoursitehere.com/calc/ Polytope Name Calculator] [[Special:Contributions/86.152.102.2|86.152.102.2]] ([[User talk:86.152.102.2|talk]]) 11:42, 4 January 2013 (UTC)

== line segment ==

What is the basis for the idea that every bounded line segment must always be a polytope? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 23:18, 12 January 2015 (UTC)

: By definition? But we may need to be more careful, since the [[line segment]] article seems to imply a Euclidean definition, while its more of a [[geodesic]] in general, like an arc on an n-sphere, or a flat torus, and outside of Euclidean space there are ambiguities since there can be more than one shortest interval between them.
:: Norman's book says "A 1-polytope (or ditel) <AB> = {0;A,B;AB} comprises the empty set, two points A and B, and the segment or arc AB joining them." as well as "The nullitope )( is trivially regular, as are all monons () and any ditel {} with two ordinary or two absolute vertices (but not one of each)."
: [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 06:58, 13 January 2015 (UTC)

::By definition? Show me where a general mathematical text defines a line segment as a polytope. The article on the [[line segment]] covers the standard ground without mention of polytopes. Why don't you add there that all line segments are polytopes and see how that goes down? Until then, why should we accept that same claim here? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 11:40, 13 January 2015 (UTC)

::: The obvious answer is to turn around the sentence: [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 21:31, 13 January 2015 (UTC)
:::* A one-dimensional polytope or ''1''-polytope is a closed [[line segment]], bounded by its two endpoints.

== Do compounds belong here? ==

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. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 07:26, 24 January 2015 (UTC)

: 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]]. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 07:59, 24 January 2015 (UTC)

::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. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 07:46, 25 January 2015 (UTC)

:::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... [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 07:50, 25 January 2015 (UTC)
::::Taking the 10-tetrahedra compound and pressing "u" (select 4) gives you 75 tesseracts. The dual is of course 75 16-cells. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 13:48, 25 January 2015 (UTC)

::Cobbled together a compounds section from [[star polygon]] and [[polytope compound]]. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 08:45, 25 January 2015 (UTC)

:::So the regular star compounds are just 5 or 10 of each Schläfli-Hess polychoron? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 13:44, 25 January 2015 (UTC)

What about the compound of two 16-cells (the 4D version of a [[stella octangula]]), that can be inscribed in a tesseract? Does that not qualify as regular? Could it be vertex-transitive but not cell-transitive? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 13:53, 25 January 2015 (UTC)

More compounds: 225 24-cells, 675 tesseracts, 675 16-cells, 720 5-cells (convex hull is 120-cell in all cases). Not sure if they're regular. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 13:58, 25 January 2015 (UTC)
:OK, I think I see the problem with 225 24-cells: it's not cell-transitive. (As pointed out by [http://software3d.com/Forums/viewtopic.php?t=26&view=previous&sid=06a6556b3383ed96593eaf1a516f82c1 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.) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:08, 25 January 2015 (UTC)
::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?) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:21, 25 January 2015 (UTC)
:::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. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:23, 25 January 2015 (UTC)
Finally, is it known if this list is complete? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:21, 25 January 2015 (UTC)

To answer the original question, no, compounds do not belong in a list of polytopes. Either the compounds need moving to their own [[List of regular polytope compounds]], or the article needs moving to a [[List of regular polytopes and compounds]]. Which do you think is better? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 15:16, 25 January 2015 (UTC)
:I'd prefer the latter: the compounds are still closely related, after all. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 12:13, 26 January 2015 (UTC)

== apeirogonal hosohedron? ==

{2,∞} is not mentioned; is there a reason why it doesn't work? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 08:42, 25 January 2015 (UTC)

: Its dual was listed, so I added it. [[List_of_regular_polytopes#Euclidean_tilings]]. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 11:35, 25 January 2015 (UTC)

== "Improper" hyperbolic honeycombs ==

So what happens to things like {7,3,2}, {2,3,7}, {5,4,2}, {2,4,5}, etc.? (The lower-dimensional equivalent ought to be a noncompact order-2 pseudogonal tiling on the hyperbolic plane, [iπ/λ,2], if I'm imagining this correctly.) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 11:56, 1 February 2015 (UTC)
: They are valid Coxeter group, [7,3,2], [2,3,7], [5,4,2], [2,4,5], etc, but I content to leave them unrecognized unless we had some sort of visualization. I think I remember discussing this with Roice Nelson when he made the uniform hyperbolic honeycomb images. [[User:Tamfang]] might be able to make some H2 tilings from [iπ/λ,2], at least he made some progress on one example, tr{&infin;,iπ/λ}, [[:File:H2 tiling 2iu-7.png]], but something got him stuck. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 14:12, 1 February 2015 (UTC)
::[[zh:二階超無限邊形鑲嵌|Chinese Wikipedia]] has a set of articles on the [iπ/λ,2] family, complete with pictures. How would {7,3,2} and {2,3,7} look? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:31, 1 February 2015 (UTC)
:::Is this related to the [[(2,3,7) triangle group]]? I am not sure I am interpreting the notation correctly. [[User:Tkuvho|Tkuvho]] ([[User talk:Tkuvho|talk]]) 14:53, 1 February 2015 (UTC)
:::: (2,3,7) is Coxeter group [7,3] in '''H'''<sup>2</sup>, or {{CDD|node|7|node|3|node}}, while [7,3,2] is in '''H'''<sup>3</sup>, or {{CDD|node|7|node|3|node|2|node}}. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 02:25, 2 February 2015 (UTC)
::::I'm using [[Schläfli symbol]]s, so {7,3,2} means a polytope with [[heptagonal tiling|{7,3}]] facets, meeting 2 per edge. {7,3} is hyperbolic, so I imagine this would be a noncompact honeycomb. Related to it and perhaps slightly easier to understand would be {7,3}×{} = t<sub>0,3</sub>{7,3,2}, a heptagonal tiling prism, which is again a noncompact honeycomb according to [http://bendwavy.org/klitzing/dimensions/hyperbolic.htm Klitzing] (though not regular, so not within the scope of this list.)
::::(It's not regular, but how does {7,3}×{&infin;} work out?) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 15:15, 1 February 2015 (UTC)

:: I got stuck on a desire to rewrite my code to remove complex arithmetic (for speed) and calls to NumPy (for portability); that would be a lot easier if I knew how to get the inradius of a triangle from its angles, which I don't. —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 21:21, 1 February 2015 (UTC)
::: Try again sometime! :) Can you iterate a center? [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 02:49, 2 February 2015 (UTC)
:::: What does that mean? —[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 17:07, 3 March 2018 (UTC)

BTW, what exactly is the [[circumradius]] of para- and noncompact hyperbolic honeycombs? It seems logical that spherical polytopes have positive circumradius, Euclidean polytopes have zero circumradius, and compact hyperbolic honeycombs have purely imaginary circumradius. But then paracompact honeycombs like {3,&infin;} would seem to need to have circumradius 0''i'', looking at the circumradius of {3,7}, {3,8}, etc.: this seems strange to me. And it would get even weirder for noncompact honeycombs... [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 15:25, 1 February 2015 (UTC)

== projective regulars ==

Are things like [[hemi-cube|{4,3}/2]] and friends suitable for this list? They have "improper examples" too, being of the form {1,2q} and {2p,1}. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 12:02, 1 February 2015 (UTC)
: 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. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 14:14, 1 February 2015 (UTC)
::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}! [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 15:20, 1 February 2015 (UTC)

== just a thought re improper 4-polytopes ==

What does {p,2,r} end up like? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:48, 1 February 2015 (UTC)
: Dihedral cells: {p,2}, and hosohedral vertex figure {2,q}? [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 15:46, 1 February 2015 (UTC)
::OK, added a mention of these to the 4D improper forms. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:26, 2 February 2015 (UTC)
What about {&infin;,2,2} and {2,2,&infin;}? [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 15:30, 2 February 2015 (UTC)

== summary tables ==

Thinking of doing one for the S<sup>4</sup>/E<sup>4</sup>/H<sup>4</sup> 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.
:{{done}}: {{tlx|Regular tetracomb table}} [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 08:00, 5 February 2015 (UTC)
The trouble comes for S<sup>5</sup>/E<sup>5</sup>/H<sup>5</sup>: there are cases in each, so a table makes sense, but I can't imagine how it would have to look! At least p, q, r, s, and t can be restricted as either 3 or 4. There's no need for higher dimensions, as the only regular polytopes there are α<sub>n</sub> (simplices), β<sub>n</sub> (cross-polytopes), γ<sub>n</sub> (hypercubes), and δ<sub>n</sub> (hypercubic honeycombs). (They cannot form hyperbolic honeycombs for n&nbsp;≥&nbsp;6: attempting to use them as facets or vertex figures only results in α<sub>n+1</sub>, β<sub>n+1</sub>, γ<sub>n+1</sub>, and δ<sub>n+1</sub>.) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 15:23, 2 February 2015 (UTC)

== Regular skew aperitopes? ==

It seems like the [[List_of_regular_polytopes_and_compounds#Apeirotopes|apeirotope]] sections should be focused on a SKEW section, along with the FINITE regular skew polytopes as well. That is the "flat" aperitopes are already covered as regular honeycombs, so if we want to be comprehensive, we'd need to add section such as these...

2D
* regular skew apeirogons in 2D (zig-zag) [[Skew_polygon#Regular_skew_apeirogons_in_2-dimensions]] 
3D
* regular skew polygons in 3D (zig-zag) [[Skew_polygon#Zig-zag_skew_polygon_in_3-dimensions]]
* regular skew apeirogons in 3D (helical) [[Skew_polygon#Helical_apeirogons_in_3-dimensions]]
* regular skew apeirohedra in 3D [[Regular_skew_polyhedron#Infinite_regular_skew_polyhedra]]
4D
* regular skew polygons in 4D (flat torus) [[Skew_polygon#Regular_skew_polygons_in_4-dimensions]]
* regular skew polyhedra in 4D [[Regular_skew_polyhedron#Finite_regular_skew_polyhedra_of_4-space]]
* regular skew apeirogons in 4D (duocylindrical helix)
* regular skew apeirohedra in 4D ???
5D
* regular skew polygons in 5D (Petrie polygons at least)
* regular skew polyhedra in 5D ???
* regular skew polychora in 5D ???
* regular skew apeirogons in 5D ???
* regular skew apeirohedra in 5D ???
* regular skew polychora in 5D ???
* etc.

Alternately, I'm open to saying ALL skew forms should be excluded, being VAST, mostly unexplored and perhaps as large or larger than the "flat" possibilities given degrees of skew-geometric freedom, and so don't belong in this list. So then the intro can just say that skew forms are excluded. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 15:05, 5 March 2015 (UTC)
:The solution is so simple. Is is reliably sourced? If yes, include it. If not, exclude it. If you are unsure about the source, ask here. Let [[WP:POLICY]] be your guide. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 19:07, 5 March 2015 (UTC)

: I disagree. We have a decision whether to include skew forms here or not. That's not a sourcing issue, but an organizational one. Are regular skew polytopes categorically different from nonskew ones? We don't even know how many FINITE "regular skew polyhedra" there are since they can exist in any number of dimensions and unlimited degrees of freedom, as Coxeter's 4D search shows, and his "hole" forms {p,q|r} only works well on a fraction of them. 
: My vote might be to ask "Does it have a linear unmodifed [[Schläfli symbol]] that uniquely defines it in a given space?" If the answer is no, and I think the answer is no on all skew forms, then they don't belong. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 19:27, 5 March 2015 (UTC)
: The intro to this article says this. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 19:40, 5 March 2015 (UTC)
:: "The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each."
::You ask, "Are regular skew polytopes categorically different from nonskew ones?" Simple - what do the sources tell us? You wonder how many finite skew polyhedra there are, but you miss the point that Wikipedia's policy will not permit you to list any that cannot be reliably sourced. I'm sorry, but if you disagree with Wikipedia's policy so strongly, what do you think you are doing here? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 20:04, 5 March 2015 (UTC)
: This article is a list, and it exists as a list because its contents are finite and enumerable (or countably). Adding skew that have not been enumerated doesn't make clear sense. We have to make a decision, and your idea that someone else is there as arbitrator what should go into our article list is senseless to me. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 22:02, 5 March 2015 (UTC)
::Then please explain what part of this, from [[WP:VERIFIABILITY]], is senseless: "In Wikipedia, verifiability means that people reading and editing the encyclopedia can check that the information comes from a reliable source. Wikipedia does not publish original research. Its content is determined by previously published information rather than the beliefs or experiences of its editors. Even if you're sure something is true, it must be verifiable before you can add it". Oh, and you'll need to argue the senselessness at [[Wikipedia talk:Verifiability]], there's no point doing so here. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 08:53, 6 March 2015 (UTC)
: We seem to be talking about completely different things. I'm trying to decide whether "regular skew polytopes/aperiotopes" belong here, and you're trying to explain that some book is going to tell us the answer to an opinion. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 10:17, 6 March 2015 (UTC)
::No, I'm  trying to explain that some book is going to tell us the answer to your question. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 18:09, 6 March 2015 (UTC)
:::Whoa. I think Tom is trying to say that he is deciding whether to put any regular skew polytopes ''that have been enumerated in reliable sources'' in this list, or put them in a separate article ([[regular skew apeirohedron]]). Given that Tom says "Adding skew that have not been enumerated doesn't make clear sense", I doubt he has any intentions to add any not found in reliable sources, even more so as I see the list of references in [[regular skew apeirohedron]]. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 14:06, 7 March 2015 (UTC)

::::I'm sorry, but I missed the bit where Tom said he wanted to preserve the material elsewhere. What I did see was his [https://en.wikipedia.org/w/index.php?title=List_of_regular_polytopes_and_compounds&diff=650155921&oldid=650128919 edits] including summary deletion and bald statement in the lead that it was not covered.
::::Also, so that you can understand what I have to put up with, here is a post of Tom's back, that I deleted just above:
{{quotation|
Ah, found it: [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 09:15, 7 March 2015 (UTC)
* "When writing articles on Wikipedia which list regular polytopes, the article should exclude skew regular forms which are not geometrically fixed, and fail to be expressed by Schläfli symbols. Please tell SteelPillow, this decision has my blessing." - [[H. S. M. Coxeter]], Regular polytopes, p.223}}
::::I find such gratuitous insult to be grossly offensive. "Whoa!" indeed. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 16:12, 7 March 2015 (UTC)
::::: I was demonstrating how I understood your request, with a small exaggeration for comic appeal. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 16:27, 7 March 2015 (UTC)
::::::Well, don't. Your comment was ill-judged and came across as crass and offensive. I would ask you to apologise for the offence (see [[WP:IUC]]) caused. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 18:03, 7 March 2015 (UTC)

I decided to be bold and removed the skew apeirotopes from this article with a statement in the intro of the exclusion. I moved the apeirotope paragraph to [[regular skew apeirohedron]] where serious work is yet needed to rectify Coxeter's older list(s) to McMullen's newer ones, given divergent notations and terminology. If at some point clarity is established on the skew forms, I'm open to moving summary enumerations back here. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 11:14, 6 March 2015 (UTC)

:I removed the apeirogon section again, re-added by Steelpillow. They don't clearly belong. They have no more significant than the finite forms which are also excluded. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 15:09, 7 March 2015 (UTC)

: Steelpillow restored the section again. There's NOTHING listed there. This article list a LIST! Why can't that material go elsewhere? [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 15:58, 7 March 2015 (UTC)

::Your revert was unwarranted. I did indeed begin refactoring it as a list. You cannot just go deleting properly cited material because you don't want it. What you have done amounts to warring. Let's just calm down, take a deep breath and talk this through - politely, per [[WP:CIVIL]]. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 16:01, 7 March 2015 (UTC)

::: I moved the only section with any unique content to [[Regular_skew_apeirohedron#Expanded_regular_apeirohedra_in_Euclidean_3-space]] although as described there is absolutely no content expressing what these forms are. How about putting this vague content at [[Regular_polytope#Apeirotopes_.E2.80.94_infinite_polytopes]]? [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 16:11, 7 March 2015 (UTC)

We have "flat" tessellations in the list. Tessellations are one form of apeirotope. If we are to have any apeirotopes, we need to be consistent and have them all. Do we want to remove tessellations from this list? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 16:01, 7 March 2015 (UTC)

: You are wrong. We have flat polytopes. We have flat honeycombs. We don't have skew polytopes. We don't need to have skew apeirotopes. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 16:07, 7 March 2015 (UTC)
::I am sorry, what is your basis ''in policy'' for avoiding any polytopes, whether skew or otherwise, that are attested in the literature? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 16:19, 7 March 2015 (UTC)
:: All you have to do is describe in the intro of the article what content is included and what is not included. That's an editorial decision, NOT a literature decision. I'm not deciding what material deserves to be on Wikipedia. I'm deciding what material should go into this article. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 16:20, 7 March 2015 (UTC)
::No, you have to ''agree through consensus'' what content is to be included. "I'm deciding what material should go into this article." you claim. Indeed you are, and you most certainly should not be. You do not [[WP:OWN]] this article. If you want a '''List of non-skew polytopes''' that is a different list, but Heavens to Betsy, why would anyone want that? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 18:03, 7 March 2015 (UTC)
::: Yes, I spent 2 days trying to extract an opinion from you, and failed. All I got from you was lecturing. And Yes, I'm completely open to renaming this article to [[List of flat regular polytopes and compounds]] if that would make you happy. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 18:07, 7 March 2015 (UTC)
:::: Steelpillow wrote, "Heavens to Betsy, why would anyone want that?" TomRuen replied, "Yes ... if that would make you happy." I find that a truly bizarre response, but perhaps there is reason behind it? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 19:26, 7 March 2015 (UTC)

== Discussion to rename this article to [[List of flat regular polytopes, honeycombs and compounds]] ==
Feel free to add or edit pro/con reasons below. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 18:26, 7 March 2015 (UTC)

'''Reasons to focus this list article to flat polytopes and honeycombs:'''
# They can all expressed by the [[Schläfli symbol]]s.
# They have a rigid geometry, only varying in scale.
# They are all regular tessellations (regular polytopes are tessellations of n-spheres.)
# They are completely documented.

'''Reasons to include regular skew polytopes and apeirogons:'''
# They categorically overlap with the flat honeycombs.
# They are also regular by certain definitions.

'''Reasons to exclude regular skew polytopes and apeirogons:'''
# They have no unique symbolic names, like [[Schläfli symbol]]s.
# They have a varied geometry.
# They are not regular tessellations.
# They are not completely known or documented.
# They categorically overlap with the flat honeycombs. (Should we also list all the polytopes twice, first as polytopes and second as spherical honeycombs?)
# They can be described equally well in other articles of wikipedia, and linked in the introduction.

*'''Comment''' Sources are clear that the regular skew apeirotopes belong here alongside the honeycombs. In particular, they satisfy the modern definition in terms of faithful realisations of regular abstract polytopes. If anything, the compounds should be removed to a separate list since they fail to meet the modern definition. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 19:42, 7 March 2015 (UTC)
*: What sources? What makes abstract polytopes important specifically for this article? And why can't this list exist with only flat forms included? What Wikipedia policy is being harmed by putting skew forms elsewhere? What is the harm to readers? All I see is your opinion against my opinion. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 20:17, 7 March 2015 (UTC)
:::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. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 21:04, 7 March 2015 (UTC)
*: As an experiment I copied your section to [[apeirotope]]. Previously we had no article for it. It was just a redirect to [[apeirohedron]]. Tell me why you can expand this content there? Tell me why this article can't reference that article in the introduction when they are curious about going beyond flat regular polytopes? Why is this a bad solution? [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 20:55, 7 March 2015 (UTC)
:::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. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 21:04, 7 March 2015 (UTC)
:::: 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. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 21:20, 7 March 2015 (UTC)

== Apeirotopes ==

{{Ping|Tomruen}} I am astonished, dismayed and dare I say disgusted that you deleted the apeirotope material yet again without any consensus here. It is cited, yet you deleted those citations. This article is a list - its job is to list stuff that is more fully described elsewhere. Taking a hard line is one thing, but breaking editing policies and guidelines is quite another. If you muck about like that again I shall take you back to ANI. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 10:53, 10 March 2015 (UTC)
: There's no consensus with you. I moved the material to [[apeirotope]] for you to work out the details. I left a fair and good summary. There was no list removed, nothing at all to list, except what you imagine was there. I'm sorry you disagree, but your work here is senseless to me. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 11:59, 10 March 2015 (UTC)

::So, let's take two dimensions as an example. What is it about this content (that you deleted) that differs from a list?

{{quotation|
A regular '''[[apeirogon]]''' is a regular division of an infinitely long line into equal segments, joined by vertices. 

A regular [[skew apeirogon]] in two dimensions forms a zig-zag. 

In three dimensions, a regular [[skew apeirogon]] traces out a helical spiral and may be either left- or right-handed.}}

::Now, let's compare that to the section on abstract polytopes, which you saw fit to leave undisturbed:

{{quotation|The [[abstract polytope]]s arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, Euclidean and hyperbolic space, tessellations of other [[manifold]]s, and many other objects that do not have a well-defined topology, but instead may be characterised by their "local" topology. There are infinitely many in every dimension. See [http://www.abstract-polytopes.com/atlas/ this atlas] for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the [[11-cell]] and the [[57-cell]].}}

::Perhaps you could explain why that ''is'' (presumably) supposed to be a list?
::As a wider issue on definitions, I would suggest that not every definition of a polytope admits apeirotopes such as tilings. In particular there is an ambiguity over the "dimension" of an apeirotope as an analogue of a polytope, for example a two-dimensional tiling is an analogue of a three-dimensional polyhedron. This list article can be more rationally structured by moving all such apeirotopes to their own section. If a separate [[List of apeirotopes]] article then appears necessary, then we can have that discussion, but of course it would mean taking all the tilings from this list with it. But the first step is to gather all the apeirotope material together, and this is what you are refusing to allow.
::In reply to your communications problem. My work here is based on building an encyclopedia which enshrines the '''[[WP:FIVEPILLARS|Five Pillars]]''' and I take [[WP:POLICY]] very seriously. What are you here for? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]])
::&mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 13:16, 10 March 2015 (UTC)
::P.S. You say that you unilaterally created the [[Apeirotope]] article for me to work on. I did not ask you to, nor was I yet ready to, yet you now use it against me. That comes across to me as an indication that you have a sense of ownership of this list article, that I "should" not be shaping it as you are. Do you indeed feel that you have a greater right to shape it than I do? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 15:16, 10 March 2015 (UTC)

: I don't have any idea why you want to hide your apeirotope work in a section of a huge article of flat polytopes/honeycombs. But at least the abstract section links the 11-cell and 57-cells. Probably we should also list the hemi polyhedra {p,q}/2, but given the size of this article I'd prefer it stay focused on "real" forms on the sphere/Euclidean/Hyperbolic planes.
: If keep we skew apeirotopes here, then for consistency we need skew polytopes, like I suggested above, and by your logic, if I can document it, I should add it and no one has the right to tell me it can go elsewhere, no matter how big the article gets, or how easily we could split into multiple articles for more clarity for readers.
: By question of a sense of ownership and rights to shape an article, I believe there are reasonable positions that a long article should be focused, and I don't see your work as fitting, no matter how important it is. I'm open to moving compounds elsewhere for instance. I'm open to moving honeycombs elsewhere, except seeing the finite polytopes are honeycombs too gives me reason to keep them together. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 02:46, 11 March 2015 (UTC)
::My work? Here? I try to present reliably sourced material here, that is all. You seem to betray such a strong sense of [[WP:OWNERSHIP]] here that you cannot conceive that I do not. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 11:10, 11 March 2015 (UTC)
::: Seeing how important completeness is to you, I added the honeycombs as apeirotopes up to 5 dimensions, with references. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 12:48, 11 March 2015 (UTC)
::::Don't forget to move down and merge in all the tessellation material in the previous sections. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 13:23, 11 March 2015 (UTC)
::::: Wait, you want me to move all the tessellation sections?! You can call (''n''-1)-honeycombs as special cases of ''n''-apeirotopes (1-honeycomb as 2-apeirotope, 2-honeycomb as 3-apeirotope, etc), but that doesn't make them the same things. I'll thinking about supporting merging sections IF anyone is interested in adding [[skew polygon]]s, [[skew polyhedra]], etc... here as well, which I don't support. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 13:54, 11 March 2015 (UTC)
::::::You have it back to front. ''n''-honeycombs ''are'' a class of ''n''-apeirotope, they are one of the main classes of figure that the term "apeirotope" was coined to accommodate. It is wholly inappropriate to classify them according to the dimensionality of their finite analogues. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 14:26, 11 March 2015 (UTC)
:::::: Aren't opinions fun? [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 14:42, 11 March 2015 (UTC)
:::::::I wouldn't know. There are reliable sources to back the statements I make, there are none to back your personal opinions. You want we should take this to the Mathematics wikiproject? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 15:22, 11 March 2015 (UTC)

== {∞,p} apeirohedra in Euclidean space==
: Please explain what {∞,3}, {∞,4}, {∞,6} are as "apeirohedra in Euclidean space". To me these symbols imply regular tilings of H<sup>2</sup>. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 04:52, 11 March 2015 (UTC)

{| class=wikitable
|+ Tilings of H<sup>2</sup>
!{∞,3}
!{∞,4}
!{∞,6}
|-
|[[File:H2_tiling_23i-1.png|160px]]
|[[File:H2_tiling_24i-1.png|160px]]
|[[File:H2_tiling_26i-1.png|160px]]
|}

I removed this claim until it can be explained more clearly. References are great, but confusing claims from references are less useful. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 05:56, 12 March 2015 (UTC)
<blockquote>
There are thirty regular apeirohedra in Euclidean space.<ref>{{Harvtxt |McMullen |Schulte |2002 |loc=Section 7E}}</ref> These include those listed above, as well as (in the plane) polytopes of type: {∞,3}, {∞,4}, {∞,6} and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)
</blockquote>
{{reflist}}
:The material is cited. A clarification tag is more appropriate than summarily removing it and burying it among the discussions. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 09:46, 12 March 2015 (UTC)
:: Got it, although a clarification tag looks more buried to me. Maybe I'm more likely to believe they are real if we get pictures. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 11:23, 12 March 2015 (UTC)

Here's the chart on p.224 on the 12 pure forms, if you can decode it! [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 11:37, 12 March 2015 (UTC)
: [[File:Pure_3-dimensional_apeirohedra_chart.png|480px]]
::Not a clue. I'd have to spend time with the rest of the book and I have neither. I don't know who added the material originally, do you? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 11:42, 12 March 2015 (UTC)
: Ah, I assumed it was you. So even more reason to me to remove the section, if neither of us can verify. Also you are WRONG in dimensions. They are apeirohedra! [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 11:44, 12 March 2015 (UTC)
: I traced the section back to [[User:Mike40033]] from 2006 [https://en.wikipedia.org/w/index.php?title=List_of_regular_polytopes_and_compounds&diff=76706974&oldid=76294915]
: 20:47, September 19, 2006‎ Mike40033 (talk | contribs)‎ . . (30,436 bytes) (+2,001)‎ . . (reorganisation)
: p.224 says the 12 "pure" forms (chart above) are derived from {4,3,4} which is Euclidean. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 12:06, 12 March 2015 (UTC)
::The cited text says they exist in 3-space. We cannot second-guess which apeirohedra until we have fathomed the cited source. Nor can we treat another editor with any less respect than we do ourselves, how disrespectful to try and sneak deletions in behind someone's back when you can't do it to someone's face. Also, there is no need to constantly repeat the message about what apeirohedra are - you have not explained why the text as originally written and cited is unacceptable and why a clumsy repetition is necessary. Your "opinion"-driven content building again? &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 12:34, 12 March 2015 (UTC)
:: I have the book. The are 2D surfaces in 3D. I'm not second guessing. I read it! And my "clumsy repetition" isn't clearly unnecesary if you're confused by dimensions. It is confusing to be mixing dimensions of a skew surface and dimensions in space. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 12:54, 12 March 2015 (UTC)
::Got there now. Your "3-apeirohedra" was misleading me because it was wrong: it explicitly specifies three dimensions when in many cases there are only two. Delete that mistake and things become much clearer. And after all my comments on the need to see the book, you finally admit you had a copy all along. Still, I am glad you finally thought to read it. However you posting of the diagram suggest that you did not understand what you were reading. Ah well, one lives in hope. &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 13:23, 12 March 2015 (UTC)
:: Got it. I never said I didn't have it, but do read in bits, not quite enough to live up to [[Cliff Clavin]] standards of confusion. I would prefer to make pictures as I can and let others write incorrect details that can last 9 years without questioning. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 13:55, 12 March 2015 (UTC)
:: And you're still wrong on dimensions somewhere. McMullen writes on p.25 "An infinite n-polytope is also called an n-apeirotope; when n=2 [2-apeirotope], we also refer to it as an apeirogon, and when n=3 [3-apeirotope] as an apeirohedron." [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 14:01, 12 March 2015 (UTC)
:::Heyy - you ''are'' reading it at last. Something I said has actually made a difference. :) &mdash; Cheers, [[User:Steelpillow|Steelpillow]] ([[User Talk:Steelpillow|Talk]]) 14:19, 12 March 2015 (UTC)

== Difficult language ==

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)'''? [[User:Jumpow|Jumpow]] ([[User talk:Jumpow|talk]]) 21:27, 9 January 2016 (UTC)
: It could be worded better. It's saying all higher integer values not listed give noncompact hyperbolic honeycombs, like [[Heptagonal_tiling_honeycomb|{7,3,3}]] for example. [[User:Tomruen|Tom Ruen]] ([[User talk:Tomruen|talk]]) 21:36, 9 January 2016 (UTC)

Thanks. Also it is not completely clear sentence

'''There are many enumerations that fit in the plane (1/''p'' + 1/''q'' = 1/2)'''

Where from 1/''p'' + 1/''q'' = 1/2 appears? [[User:Jumpow|Jumpow]] ([[User talk:Jumpow|talk]]) 14:03, 10 January 2016 (UTC)

Sorry, found above: Euclidean plane tiling [[User:Jumpow|Jumpow]] ([[User talk:Jumpow|talk]]) 14:53, 10 January 2016 (UTC)

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== No regular star-honeycombs in H<sup>3</sup>? ==

[https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.361.251&rep=rep1&type=pdf 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".) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 09:48, 14 August 2021 (UTC)
:(Answering myself late: yes you can, but I've not found an RS for it.) [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 23:22, 31 January 2023 (UTC)

== Terminology problems ==

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 space|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. [[User:AquitaneHungerForce|AquitaneHungerForce]] ([[User talk:AquitaneHungerForce|talk]]) 14:36, 24 January 2024 (UTC)

== [a] ==

whut [[User:Ayen2022-3|Ayen2022-3]] ([[User talk:Ayen2022-3|talk]]) 12:04, 31 March 2022 (UTC)

== More regular polyhedra in 4-space ==

[https://link.springer.com/article/10.1007/s00454-007-1342-7 McMullen paper from 2007]. [[User:Double sharp|Double sharp]] ([[User talk:Double sharp|talk]]) 23:22, 31 January 2023 (UTC)

== Inaccurate compound definition ==

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 [[Talk:Polytope_compound#Regular_compound_definition_does_not_match_source|thread I started on that page]] to read my issue and discuss. [[User:AquitaneHungerForce|AquitaneHungerForce]] ([[User talk:AquitaneHungerForce|talk]]) 04:58, 20 January 2024 (UTC)

== Proposal to split of "Regular compounds" from this article ==

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. [[User:AquitaneHungerForce|AquitaneHungerForce]] ([[User talk:AquitaneHungerForce|talk]]) 15:51, 22 January 2024 (UTC)
: Since no discussion happened I just went ahead and did it. [[WP:BEBOLD]]. [[User:AquitaneHungerForce|AquitaneHungerForce]] ([[User talk:AquitaneHungerForce|talk]]) 18:44, 15 February 2024 (UTC)

== hyperbolic star tilings ==

:There are 2 infinite forms of hyperbolic tilings whose [[Face (geometry)|faces]] or [[Vertex figure|vertex figures]] are star polygons: {''m''/2, ''m''} and their duals {''m'', ''m''/2} with ''m'' = 7, 9, 11, ....

Why not, for example, {10/3, 7}&nbsp;? [[User:Tamfang|—Tamfang]] ([[User talk:Tamfang|talk]]) 15:53, 26 March 2024 (UTC)

: With the specific example, {{math|{{mset|10/3, 7}}}} nominally exists, however unlike the listed examples their vertices become arbitrarily close and thus they are "non-discrete". Similar things happen  on the plane, {{abbr|e.g.|for example}} {{math|{{mset|8/3, 8}}}} is dense on the Euclidean plane, and the sphere, {{abbr|e.g.|for example}} {{math|{{mset|5/2, 5/2}}}} is dense on a sphere in 3-space. These sorts of polytopes are almost always implicitly or explicitly not counted.
: However the section does not cite a source, so I have no way of verifying these are the only discrete hyperbolic star tilings (ignoring skew tilings), and we can't be sure that the remainder of them are all eliminated because they are non-discrete. [[User:AquitaneHungerForce|AquitaneHungerForce]] ([[User talk:AquitaneHungerForce|talk]]) 16:11, 26 March 2024 (UTC)

== what's a Petrial? ==

No definition, no link … [[User:Tamfang|—Tamfang]] ([[User talk:Tamfang|talk]]) 06:52, 25 January 2025 (UTC)