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Note: failed FLC can be found at Wikipedia:Featured list candidates/List of regular polytopes. Double sharp ( talk) 04:37, 8 March 2015 (UTC)
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)
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)
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.
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)
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)
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
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)
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 polygons. 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? Kevin Lamoreau ( talk) 00:47, 4 April 2010 (UTC)
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)
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)
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)
"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)
Just to nail the coffin, here's a calculator with said ego, rendering further non-standardised coinage unimpressive: Polytope Name Calculator 86.152.102.2 ( talk) 11:42, 4 January 2013 (UTC)
What is the basis for the idea that every bounded line segment must always be a polytope? — Cheers, Steelpillow ( Talk) 23:18, 12 January 2015 (UTC)
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)
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? 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. Double sharp ( talk) 13:58, 25 January 2015 (UTC)
Finally, is it known if this list is complete? 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? — Cheers, Steelpillow ( Talk) 15:16, 25 January 2015 (UTC)
{2,∞} is not mentioned; is there a reason why it doesn't work? Double sharp ( talk) 08:42, 25 January 2015 (UTC)
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.) Double sharp ( talk) 11:56, 1 February 2015 (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,∞} would seem to need to have circumradius 0i, looking at the circumradius of {3,7}, {3,8}, etc.: this seems strange to me. And it would get even weirder for noncompact honeycombs... Double sharp ( talk) 15:25, 1 February 2015 (UTC)
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)
What does {p,2,r} end up like? Double sharp ( talk) 14:48, 1 February 2015 (UTC)
What about {∞,2,2} and {2,2,∞}? Double sharp ( talk) 15:30, 2 February 2015 (UTC)
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.
{{
Regular tetracomb table}}
Double sharp (
talk)
08:00, 5 February 2015 (UTC)The trouble comes for S5/E5/H5: 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 αn (simplices), βn (cross-polytopes), γn (hypercubes), and δn (hypercubic honeycombs). (They cannot form hyperbolic honeycombs for n ≥ 6: attempting to use them as facets or vertex figures only results in αn+1, βn+1, γn+1, and δn+1.) Double sharp ( talk) 15:23, 2 February 2015 (UTC)
It seems like the 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
3D
4D
5D
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. Tom Ruen ( talk) 15:05, 5 March 2015 (UTC)
Ah, found it: Tom Ruen ( 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 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. Tom Ruen ( talk) 11:14, 6 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? — Cheers, Steelpillow ( Talk) 16:01, 7 March 2015 (UTC)
Feel free to add or edit pro/con reasons below. Tom Ruen ( talk) 18:26, 7 March 2015 (UTC)
Reasons to focus this list article to flat polytopes and honeycombs:
Reasons to include regular skew polytopes and apeirogons:
Reasons to exclude regular skew polytopes and apeirogons:
@ 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. — Cheers, Steelpillow ( Talk) 10:53, 10 March 2015 (UTC)
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.
The abstract polytopes 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 manifolds, 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 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.
{∞,3} | {∞,4} | {∞,6} |
---|---|---|
I removed this claim until it can be explained more clearly. References are great, but confusing claims from references are less useful. Tom Ruen ( talk) 05:56, 12 March 2015 (UTC)
There are thirty regular apeirohedra in Euclidean space. [1] 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)
Here's the chart on p.224 on the 12 pure forms, if you can decode it! Tom Ruen ( talk) 11:37, 12 March 2015 (UTC)
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)
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? Jumpow ( talk) 14:03, 10 January 2016 (UTC)
Sorry, found above: Euclidean plane tiling Jumpow ( talk) 14:53, 10 January 2016 (UTC)
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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)
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.
whut Ayen2022-3 ( talk) 12:04, 31 March 2022 (UTC)
McMullen paper from 2007. Double sharp ( talk) 23:22, 31 January 2023 (UTC)
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)
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)
Why not, for example, {10/3, 7} ? —Tamfang ( talk) 15:53, 26 March 2024 (UTC)
This article is rated List-class on Wikipedia's
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This article is a former featured list candidate. Please view its sub-page to see why the nomination failed. Once the objections have been addressed you may resubmit the article for featured list status. |
Note: failed FLC can be found at Wikipedia:Featured list candidates/List of regular polytopes. Double sharp ( talk) 04:37, 8 March 2015 (UTC)
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)
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)
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.
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)
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)
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
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)
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 polygons. 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? Kevin Lamoreau ( talk) 00:47, 4 April 2010 (UTC)
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)
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)
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)
"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)
Just to nail the coffin, here's a calculator with said ego, rendering further non-standardised coinage unimpressive: Polytope Name Calculator 86.152.102.2 ( talk) 11:42, 4 January 2013 (UTC)
What is the basis for the idea that every bounded line segment must always be a polytope? — Cheers, Steelpillow ( Talk) 23:18, 12 January 2015 (UTC)
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)
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? 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. Double sharp ( talk) 13:58, 25 January 2015 (UTC)
Finally, is it known if this list is complete? 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? — Cheers, Steelpillow ( Talk) 15:16, 25 January 2015 (UTC)
{2,∞} is not mentioned; is there a reason why it doesn't work? Double sharp ( talk) 08:42, 25 January 2015 (UTC)
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.) Double sharp ( talk) 11:56, 1 February 2015 (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,∞} would seem to need to have circumradius 0i, looking at the circumradius of {3,7}, {3,8}, etc.: this seems strange to me. And it would get even weirder for noncompact honeycombs... Double sharp ( talk) 15:25, 1 February 2015 (UTC)
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)
What does {p,2,r} end up like? Double sharp ( talk) 14:48, 1 February 2015 (UTC)
What about {∞,2,2} and {2,2,∞}? Double sharp ( talk) 15:30, 2 February 2015 (UTC)
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.
{{
Regular tetracomb table}}
Double sharp (
talk)
08:00, 5 February 2015 (UTC)The trouble comes for S5/E5/H5: 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 αn (simplices), βn (cross-polytopes), γn (hypercubes), and δn (hypercubic honeycombs). (They cannot form hyperbolic honeycombs for n ≥ 6: attempting to use them as facets or vertex figures only results in αn+1, βn+1, γn+1, and δn+1.) Double sharp ( talk) 15:23, 2 February 2015 (UTC)
It seems like the 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
3D
4D
5D
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. Tom Ruen ( talk) 15:05, 5 March 2015 (UTC)
Ah, found it: Tom Ruen ( 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 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. Tom Ruen ( talk) 11:14, 6 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? — Cheers, Steelpillow ( Talk) 16:01, 7 March 2015 (UTC)
Feel free to add or edit pro/con reasons below. Tom Ruen ( talk) 18:26, 7 March 2015 (UTC)
Reasons to focus this list article to flat polytopes and honeycombs:
Reasons to include regular skew polytopes and apeirogons:
Reasons to exclude regular skew polytopes and apeirogons:
@ 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. — Cheers, Steelpillow ( Talk) 10:53, 10 March 2015 (UTC)
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.
The abstract polytopes 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 manifolds, 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 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.
{∞,3} | {∞,4} | {∞,6} |
---|---|---|
I removed this claim until it can be explained more clearly. References are great, but confusing claims from references are less useful. Tom Ruen ( talk) 05:56, 12 March 2015 (UTC)
There are thirty regular apeirohedra in Euclidean space. [1] 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)
Here's the chart on p.224 on the 12 pure forms, if you can decode it! Tom Ruen ( talk) 11:37, 12 March 2015 (UTC)
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)
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? Jumpow ( talk) 14:03, 10 January 2016 (UTC)
Sorry, found above: Euclidean plane tiling Jumpow ( talk) 14:53, 10 January 2016 (UTC)
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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)
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.
whut Ayen2022-3 ( talk) 12:04, 31 March 2022 (UTC)
McMullen paper from 2007. Double sharp ( talk) 23:22, 31 January 2023 (UTC)
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)
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)
Why not, for example, {10/3, 7} ? —Tamfang ( talk) 15:53, 26 March 2024 (UTC)