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The uniform polychora lists are moved here fromPolychora. Some things that need to be done:

• Reorganize this page so thatregular polychora andsemiregular 4-polytopes appear somewhere sensible. Currently only a subset of regular polychora is described, underconvex regular 4-polytope.
• At least some general description of non-convex uniform polychora should be here, under its own heading (does anyone knows what are the 29 categories referred to in the 2nd paragraph? I copied that from the polychoron page; I'm not sure who wrote it nor where I can look it up).
• Some explanation for the names of various polychora:
  • What does "cantellated" mean, for example?
  • Or "runcinated" or "runcitruncated"?
  • What precisely is the difference between "truncated" and "bitruncated"?

—Tetracube07:15, 10 January 2006 (UTC)[reply]

I've made a stab at defining some of the operations, as well as some other changes suggested by George Olshevsky. —Anton Sherwood00:27, 13 January 2006 (UTC)[reply]

Thanks! That helps a lot.—Tetracube07:08, 13 January 2006 (UTC)[reply]

How do you like my first table? (I must dash now, will make the others later.) Someone besides me should check it against[1]. --Anton Sherwood18:46, 10 January 2006 (UTC)[reply]

I like it! Although, it seems that you didn't link each polychoron to its own page. I'll add in those links (some of them already exist as separate pages).—Tetracube18:59, 10 January 2006 (UTC)[reply]

Hi, I saw the new tables you put in. I see that you've put in all 46 uniform polychora. Cool!

However, I'm not sure about the paragraph on the antiprisms. I question putting the antiprisms here... as far as I know, they arenot vertex-uniform, because the vertices on one cell are not congruent to the vertices on the dual cell. I suspect they may belong to a more general category of polychora, but I don't see how they satisfy the requirements of being uniform polychora.Actually, scratch that. I didn't read the paragraph carefully. Sorry :-)—Tetracube06:07, 11 January 2006 (UTC)[reply]

See the paragraph I've just added to "grand antiprism". I'm guessing that analogous forms can be constructed, whose cells are 4n 'n'-antiprisms and 12n² tetrahedra, which meet part of the definition of uniformity – a symmetry group on the vertices – but whose facets are not uniform. I'd like to add that if someone can confirm it (my 4d geometry is still very weak). —Tamfang04:06, 13 February 2006 (UTC)[reply]

To see why the facets are non-uniform, consider the grand antiprism's vertex figure. If the long edge is anything but τ (representing a pentagon), either the figure is not inscribed in a sphere (as a vertex figure must be) or the triangles are not equilateral. —Tamfang20:00, 19 February 2006 (UTC)[reply]

It sounds plausible to me; only the tetrahedra would become non-uniform, but I don't see any need for the antiprismic cells to be non-uniform. Keep in mind, though, that this construction, although resembling the 3D antiprism, isn't the only possible analogue. For one thing, the two cycles of antiprisms lie along two mutually perpendicular rings which are only possible in 4D and above (being related to the Hopf fibration of the 3-sphere); they could hardly be considered the equivalents of the "top" and "bottom" faces of a 3D antiprism. I'd say they are more "snub"-like. Another possible construction, which is closer to the structure of the 3D antiprism, is to have two dual cells connected by pyramids (like you've alluded to some time ago, such as a cube-octahedron antiprism, with square pyramids and tetrahedra joining the cube and octahedron), or perhaps alternating triangular prisms. But this would not be uniform unless the "top" and "bottom" cells were self-dual, which could only be the tetrahedron, which gives rise to a 16-cell. Now, I don't know if there are any other self-dual polyhedra (with non-regular facets); if there were, they might form vertex-uniform polychora under this construction as well.—Tetracube01:49, 16 February 2006 (UTC)[reply]

No other self-dualuniform polyhedra, no. Pyramids and elongated pyramids are topologically self-dual and can be distorted into geometric self-duals (the "canonical" form). —Tamfang04:23, 16 February 2006 (UTC)[reply]

Good point, I didn't think about that. I wonder if there are other more spherical (non-uniform) polyhedra that are also self-dual? Just a personal curiosity.

I would bet there are many. —Tamfang07:48, 16 February 2006 (UTC)[reply]

I also wonder, although this is outside the scope of this article, whether there is a consistent way to extend duality to non-polytopic objects. E. g., acone seems to be self-dual in some sense (generalizing fromn-gonal pyramids asn approaches ∞), and a cylinder's dual seems to be a di-cone (two cones joined at the base), generalizing from then-gonal prisms. I don't know what the dual of a sphere might be... perhaps a point?—Tetracube04:53, 16 February 2006 (UTC)[reply]

A sphere's dual is a sphere: an infinite number of vertices corresponds to an infinite number of faces. —Tamfang07:48, 16 February 2006 (UTC)[reply]

I noticed that the wikipedia style guidelines suggest that article names be singular where possible, rather than plural. Should we rename this page touniform polychoron instead?—Tetracube18:51, 27 January 2006 (UTC)[reply]

I agree - more consistent withUniform polyhedron andPolychoron too. In contrast I'd keepList of uniform polyhedra as plural, while eventually I'd like a parallel tabular version asList of convex uniform polychora.Tom Ruen22:04, 27 January 2006 (UTC)[reply]

OK, I've put up a request inWikipedia:requested moves 'cosuniform polychoron already has an edit history. What's the process for getting this resolved? Just wait for an admin to notice that we (more or less) have consensus here?—Tetracube07:58, 2 February 2006 (UTC)[reply]

I'd just be bold and use theMove this page option. I mean I've done it for other pages and it's quick and easy, automatically create a REDIRECT from the old as well, alhought you can look at "what links here" and make links all go to the new name. I don't know of any other problems. I admit it's good to be cautious, but no one has objected in almost a week! So I say you're free!Tom Ruen08:03, 2 February 2006 (UTC)[reply]

Oh I know I can just do it, but Wikipedia refuses to move the page because the target page has a non-trivial edit history, so the server thinks that manual intervention is necessary to merge the pages. (I've just tried it; I get an error message.)—Tetracube08:10, 2 February 2006 (UTC)[reply]

Ah, guess we'll have to wait. We can start our "consensus process" below at least....

I saw polychora articles updated for new name here. I was thinking, do we want aCategory:Polychora orCategory:Polychoron orCategory:Uniform polychora orCategory:Uniform polychoron?

Currently they are underCategory:Polytopes which isn't bad, except for not specifying 4D objects, although not much going on above 4D yet!Tom Ruen06:30, 7 February 2006 (UTC)[reply]

I noticed thatCategory:Polytopes already has sub-categories for polygons and polyhedra. Maybe we should move the current polychora intoCategory:Polychora, at the very least?—Tetracube08:20, 8 February 2006 (UTC)[reply]

Not sure about this header title for a section declaring terminology, but good enough.

I added a definition forsnub which seems to be correct for uniform polyhedra, but don't know well how it applies to polychora likesnub 24-cell.

Obviously it would be good to have some sequential images or even animations to show these operations. Maybe I can add something sometime, but I don't think I can do all of them.Tom Ruen02:04, 13 February 2006 (UTC)[reply]

Hmm, maybe 'Nomenclature' might be a better section heading?

About snub polytopes... I believe the word 'snub' means to surround with triangles, in which case it would only apply to polyhedra. It would have to be generalized to surrounding withn-simplices for higher polytopes. Now, a gyration of faces in a polyhedron is easy enough to see, as there is only 1 possible plane of rotation and it's easy enough to find the appropriate angle that would yield a uniform polyhedron. With polychoron cells, though, I'm not so sure. There are 3 possible planes of rotation, and it isn't obvious which one is being applied. Also, Tamfang did discuss the snub 24-cell a bit (in the 24-cell section), and it doesn't seem to be quite the same as a snub polyhedron.—Tetracube02:35, 13 February 2006 (UTC)[reply]

The title is good enough for me. A "Nomenclature" section should address the different systems of names (represented in Olshevsky's list) used by Manning, Johnson, Conway, Sloane. The "operations" section should be distinct from that: these different operations are applicable whatever they may be called.

Snubs: yeah, how come the cube's facets are preserved (though rotated) in the snub cube, and the {3,4,3}'s facets are not preserved in s{3,4,3}? I can almost imagine a definition that covers both, but it's so involved that most readers are better off left with something vague like "The termsnub is applied to different kinds of operations that slightly reduce the figure's symmetry," or no definition at all since we only use it once (it might as well be called "strange 24-cell"). We should ask the experts (Bowers et al.) whether there are other (nonconvex) 4D "snubs", and what they have in common.

Diagrams: The projections I have in mind should help, if I ever get around to doing them. (Wikipedia is so addicting!) I have in mind stereographic projections of S3 to E3, built in Povray. Is there a source for vertex coordinates for all of these?

—Tamfang02:59, 13 February 2006 (UTC)[reply]

Marek Čtrnáct gave me a surprising definition ofsnub. Start with a polytope whose faces all have even degree, such as an omnitruncate; then you can remove alternate vertices, inserting vertex figures (like rectification but twice as deep). Deform the result as necessary to make it uniform:

• cube →tetrahedron
• 2n-prism →n-antiprism
• truncated octahedron →icosahedron (as "snub tetratetrahedron")
• truncated cuboctahedron →snub cube
• truncated icosidodecahedron →snub dodecahedron
• tesseract →16-cell
• truncated 24-cell →snub 24-cell

Several other convex polychora can be given this treatment but the result cannot be made uniform. Marek didn't go into nonconvex examples. —Tamfang18:22, 15 February 2006 (UTC)[reply]

Wow, this is very interesting. I wonder if anyone else recognizes this definition ofsnub? Also, how well does it generalize to higher dimensions? "Remove alternate vertices" may not be that trivial once you get beyond 4 dimensions. Nevertheless, it is a very useful definition, as far as I can tell, and very intriguing as well.—Tetracube18:55, 15 February 2006 (UTC)[reply]

Since each edge has exactly two ends, I can't see a problem offhand in generalizing to n dimensions. But I lack intuition. —Tamfang20:25, 15 February 2006 (UTC)[reply]

The problem is that alternated polytopes in higher dimensions usually aren't uniform. An alternated polyhedron has three kinds of edge, so making them equal is two equations, and there are two degrees of freedom in placing the vertex, so the constraints can always be met. An alternated polytope inn+1 dimensions (or alternated tiling inn dimensions) hasn(n+1)/2 kinds of edge, andn degrees of freedom; for polychora this is 5 constraints and 3 degrees of freedom, and it keeps getting worse in higher dimensions. —Tamfang (talk)

That's for alternatedomnitruncates. The constraints may be different for snubs derived from other even-faced figures (but they're rarer). —Tamfang (talk)04:12, 23 July 2013 (UTC)[reply]

Yes, snubbing works in higher dimensions too.

There are in fact quite a few 4D snubs, most of them non-Wythoffian. Looking at the non-Wythoffian snubs, the sadsadox regiment (sidtaps) is formed by blending 10 roxes (rectified 600-cells), while the gadsadox regiment (gidtaps) is formed by blending 10 raggixes (rectified grand 600-cells). The idcossids are formed from the 10-padohi compound (padohi is one possible 4D raded), while the dircospids are formed from the 10-gidipthi compound (gidipthi is one possible 4D gaddid).

If you're thinking ofWythoffian snubs, there's a retrosnub version of the snub 24-cell. There are four others (rappisdi, sirhapsippady, girhapsippady, and rapsady), three having 24-cell symmetry, and the last having 120-cell symmetry.Double sharp (talk)10:40, 6 August 2012 (UTC)[reply]

Hey Tamfang, I just saw your latest edit to the grand antiprism. From the description, it seems that the girthing band of tetrahedra is topologically equivalent to the ridge of theduocylinder, which is topologically isomorphic to the 2-torus; and the two rings of pentagonal antiprisms are topologically equivalent to the duocylinder's two bounding 3-manifolds. This is very interesting. I should like to get hold of the vertices of the grand antiprism so that I can plot its projections into 3-space, to confirm my theory.

I'm not sure what you mean by "two bounding 3-manifolds", but otherwise my intuition matches yours. —Tamfang05:27, 13 February 2006 (UTC)[reply]

They are the two 3-manifolds that together form the surface of the duocylinder. It's rather hard to describe, because this enclosure can only happen in 4-space and above. Basically, you can think of the exterior of the duocylinder as consisting of two mutually perpendicular (and identical) pieces. Each piece is the volume of revolution of a disk in the XY plane about the ZW plane:

${\displaystyle V=\{(x,y,z,w)|x^2+y^2\le r^2;z^2+w^2=r^2\}}$

The boundary of the piece may be described as the set of points:

${\displaystyle B=\{(x,y,z,w)|x^2+y^2=r^2;z^2+w^2=r^2\}}$

Notice that you can rotate one piece in such a way that it is perpendicular to the other piece (i. e., by exchanging coordinates), but has precisely the same boundary. This is how both pieces together enclose the inside of the duocylinder. Intuitively, you can think of a piece as the result of bending a 3D cylinder around the ZW plane such that the top and bottom lids meet. As a result of this bending, the curved side of the cylinder deforms into the duocylinder's ridge, forming a torus-shaped hole which is exactly the same shape as the bent cylinder itself. Filling this hole with a second bent cylinder produces the duocylinder.—Tetracube06:01, 13 February 2006 (UTC)[reply]

That's pretty much what I thought you meant, though somehow I had the impression that "manifold" meant something unbounded (though not necessarily infinite). —Tamfang06:30, 13 February 2006 (UTC)[reply]

Whoops, look who last editedduocylinder ;) —Tamfang06:34, 13 February 2006 (UTC)[reply]

Anyway, I'm thinking of moving the info about the grand antiprism into its own page. What do you think?—Tetracube04:16, 13 February 2006 (UTC)[reply]

Yeah, sure. The bit about an almost-uniform family that I suggested above belongs here, though, more than it belongs atgrand antiprism. Another thing to add (if I'm not mistaken): the grand antiprism (like the snub-24) can be constructed by diminishing the 600, that is, by removing 20 of the 120 vertices and taking the convex hull of the remainder. —Tamfang05:27, 13 February 2006 (UTC)[reply]

Which 20 vertices, though? I assume it can't be any random set of 20 vertices.—Tetracube06:01, 13 February 2006 (UTC)[reply]

Well, they must be on two great circles. Does that help? —Tamfang06:30, 13 February 2006 (UTC)[reply]

OK, I've made adraft of the grand antiprism article. Comments?—Tetracube06:09, 13 February 2006 (UTC)[reply]

Hey TamFang... I just noticed that you shortened many of the polychoron names (e.g., "runcinated tesseract" → "runcinated") to "save space". I'm not sure I understand the rationale behind this, since this makes the entry ambiguous and hard to understand. (In the 24-cell section, it can perhaps be inferred; but in the other sections, I'm not sure this is a good idea.) I reverted the 120-cell/600-cell section before I realized what was going on, but I'd like to discuss this before either one of us edits it either way.—Tetracube15:56, 11 July 2006 (UTC)[reply]

I'm not wedded to the idea. The point is partly to save space and partly to emphasize that runcination, bitruncation and omnitruncation produce the same result from either parent. It becomes problematic in the 5-cell family where some have independent names (decachoron). —Tamfang20:21, 12 July 2006 (UTC)[reply]

I decided to try some editing, just on the 8/16-cell table, expanding all names, and adding line-breaks on second names. Thoughts?

"also" does not immediately suggest "also called" to me; I see "also" and my first thought is of an isomer – a different object with some shared properties. For whatever it's worth, I still like "or" better. —Tamfang06:12, 13 July 2006 (UTC)[reply]

I've been interested in a more compact table with pictures andvertex configuration names. Example (parallel copy) I made a while ago:User:Tomruen/uniform_polychoron#The_8-cell.2F16-cell_family_.7B4.2C3.2C3.7D_and_.7B3.2C3.2C4.7D

Tom Ruen22:51, 12 July 2006 (UTC)[reply]

Not bad. It would help to have several versions of some of the images: ideally corresponding faces should have the same color. —Tamfang06:12, 13 July 2006 (UTC)[reply]

I like this idea. It's compact, and immediately conveys the necessary information.—Tetracube01:39, 14 July 2006 (UTC)[reply]

Okay, a brave edit tonight - replacing table entries with pictures and vertex configuration links. I agree with Anton (Tamfang) that face-color consistency on polyhedra pictures would be good, but can be improved later if new pictures are uploaded.

I also made all the tables consistent in order, listing regular and dual forms as pairs, and putting symmetric forms after (and one snub last).

Tom Ruen06:24, 14 July 2006 (UTC)[reply]

Okay, one more possibly annoying change. I tried my idea of changing background color to signify which regular form is the generator - red=original, blue=dual, green=symmetric operation for both.Tom Ruen23:00, 12 July 2006 (UTC)[reply]

I'm not in love with it. Both can be considered generators; all forms in a family (except the snub) are built from the same mirrors. —Tamfang06:12, 13 July 2006 (UTC)[reply]

At least I reordered so the dual operated pairs are given together f{p,q,r} and f{r,q,p}, and the last three f{p,q,r}=f{r,q,p}.Tom Ruen22:07, 13 July 2006 (UTC)[reply]

In case anyone wanted to know, my previous arrangement was simplest first, i.e. by number of cells, then by number of faces and so on. —Tamfang18:35, 15 July 2006 (UTC)[reply]

Tom, the {3,3,4} and {3,4,3} families sharethree members, not two, but that makes the total come out wrong ... —Tamfang18:34, 15 July 2006 (UTC)[reply]

Fixed, forgot to count snub 24-cell!Tom Ruen18:44, 15 July 2006 (UTC)[reply]

In regards to the nonconvex polychora and theUniform Polychora Project, I'd have to judge it is "unpublished ongoing research" and not clearly defendable within the context of an encyclopedia, ALTHOUGH may be worthy to include on a TALK page, like here!

I might include a statement on nonconvex forms like:

Like the set ofuniform polyhedra, there are many more nonconvex uniform polychora than convex ones. Defining and enumerating this list is an active area of research now with an amateur polyhedronists including ....

REMOVED TEXT

TheUniform Polychora Project has classified the 1,845 currently known uniform polychora into 29 groups. There may be more. Most of these arenon-convex polychora, and the count does not include the prismatic uniform polychora (see below). Previously, the count of uniform polychora had reached 8190; however, this was becauseJonathan Bowers had used a laxer definition of 'uniform polychoron' that allowed many degenerate and exotic combinations of uniform polyhedra. The more traditional definition used byNorman Johnson was adopted recently, and the number of 'proper' uniform polychora was reduced to the current figure. The remaining 6345 objects that no longer fall under this new definition are now known aspolychoroids.

In regards to the 47 uniform polychora, 18 convex prismatic forms, and infinite set of duoprisms, this appears solid to me, and I'd just like to see more history, and sources. As far as I know the entire content has been extracted fromGeorge Olshevsky's website, and his website doesn't contain clear referenced sources. From this article we don't even know WHO discovered these and when!

Well, I hope this opens the door to getting the sources we want here!Tom Ruen03:47, 15 July 2006 (UTC)[reply]

I hope so as well. I placed the unsourced tag on the page bacause the entire uniform polyhedra project appears to be unpublished research. I don't see why it would be difficult to set up a uniform polyhedra wiki to put this research on, instead of wikipedia, or to put it in print if it is meant to be more than amateur research. I plan to prod the page in a few days unless some printed, peer reviewed sources can be found.CMummert19:01, 15 July 2006 (UTC)[reply]

• [2]
  1. Thorold Gosset - halfregular Polytope was compound in n dimensions from different, but for even regular Polytopen lower dimension.
  2. Alicia Boole Stott - permitted the Archimedean polyhedrons as parts of the Polytope.
  3. Norman Woodason Johnson
  4. Willem Abraham Wythoff
  5. John Horton Conway and Mike J.T. Guy
     • Only Conway and Guy worked in the early 1960's on a first proof; however only a bilateral summary was published.
  6. Branko Grünbaum

I removed reference to the open research articleUniform Polychora Project and added a draft history section using information above. Unfortunately fuzzy in details, but a start.Tom Ruen19:32, 15 July 2006 (UTC)[reply]

I expanded the new history section and a reference section, selected from the [www.polytope.de] website. That's all I can do now. I'm happy if anyone can expand or improve.Tom Ruen20:19, 15 July 2006 (UTC)[reply]

Has someone actually held these sources in their hands and verified that the material in this article appears in the book?CMummert01:16, 16 July 2006 (UTC)[reply]

Ah, yeah - one disaster at a time! So far, I've only seen the 1954 "Uniform Polyhedra" paper reference which only covers thelist of uniform polyhedra, but I'm in process of seeing what I can get access to by a friend at the University. As a last resort I've been in past email contact with Johnson, Grumbaum, and Conway.Tom Ruen01:22, 16 July 2006 (UTC)[reply]

For what it's worth, the list of alternate names given by George Olshevsky for each convex uniform polychoron demonstrates that more than one mathematician – includingJohn Horton Conway,Norman Johnson,Neil Sloane – has taken an interest. ;) —Tamfang03:52, 16 July 2006 (UTC)[reply]

The question is not interest. The subject is, I agree, interesting. The question is the existence of printed, reliable sources for the information. That is the criteria that has been established.CMummert03:59, 16 July 2006 (UTC)[reply]

The University of MN library has a copy of the book by B. Grünbaum, Convex polytopes, 2003[3] (1st and 2nd editions). I'll try to stop over there adn look at it in the new few weeks.Tom Ruen22:15, 17 July 2006 (UTC)[reply]

Hi all, I've finally found a reference to the paper that describes the uniform polychora (at least, the convex ones). Unfortunately, I don't currently have access to a university library to actually get a copy of this paper, but maybe somebody can do it. The reference is:

J. H. Conway, "Four-dimensional Archimedean polytopes", Proc. Colloquium on Convexity, Copenhagen 1965, Kobenhavns Univ. Mat. Institut (1967) 38–39.

This page describes some aspects of the paper, including some references to how the uniform polychora are constructed.

Now, with respect to the more specificsemiregular 4-polytopes, the references (also listed on the above site) are:

G. Blind and R. Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150–154.

T. Gosset, "On the regular and semiregular figures in spaces ofn dimensions", Messenger of Mathematics 29 (1900) 43–48.

Somebody with access to these journals can help us look up these articles and check against the material on this page. I hope this helps to ground this article on reliable sources. :-) —Tetracube06:07, 9 August 2006 (UTC)[reply]

Hmm, I just realized that Conway's paper has already been referenced in the article. Never mind. :-) Nevertheless, somebody should take a look at some of the other references listed on the page linked above.—Tetracube06:19, 9 August 2006 (UTC)[reply]

I have the 1900Gosset paper as a PDF, and I can share. Terminology is a bit confusing, and semiregular forms are now listed from it atTalk:Semiregular 4-polytope.Tom Ruen06:23, 9 August 2006 (UTC)[reply]

That would be nice. How do I get a copy of it?—Tetracube07:07, 9 August 2006 (UTC)[reply]

Send me an email via[4], and I'll send it to you.

As a first test, I added cell/face/edge/vertex counts to the 5-cell family. I'm also interested in adding columns for face counts by type, and cells per vertex, but don't want the table too wide, so I'll leave out for now.

Tom Ruen21:17, 20 July 2006 (UTC)[reply]

There are now stub articles for all of the first 48 forms. If anyone wants to help fill in data, I've put a summary data table at:User:Tomruen/uniform_polychoron_table. This independent source should agree with George O's data at[5][6][7][8][9], etc, although I've not compared all of them. My spare time is pretty much gone for the rest of August. Thanks!Tom Ruen04:48, 14 August 2006 (UTC)[reply]

I wonder if this should be renamed toConvex uniform polychoron, since there is actually zero content on nonconvex forms?

This would parallel names forconvex regular polychoron andconvex uniform honeycomb.Polychoron could reference the existence of nonconvex forms.

Tom Ruen03:02, 18 September 2006 (UTC)[reply]

Discussion?

• It sounds like a good idea, although the question lingers as to what we should do withuniform polychoron: should it redirect to convex uniform polychoron? That seems to defeat the purpose of the rename. If not, should it be a (stub?) article that links to the convex/nonconvex pages? Right now, we don't seem to have any solid references for listing nonconvex uniforms, so this seems redundant (uniform polychoron would only point toconvex uniform polychoron). Or maybe it can redirect to simplypolychoron?—Tetracube04:41, 18 September 2006 (UTC)[reply]
• I think you could redirect toPolychoron#Categories.Tom Ruen06:36, 18 September 2006 (UTC)[reply]
• For nonconvex forms, the "easy" cases are the some 52 nonconvexuniform polyhedrons duplicated as hyperprisms, as well as infinite sets ofstar prisms hyperprisms, but overall, I don't find them very interesting, especially knowing there's thousands. I am considering a similar split asconvex uniform polyhedron as well, although someday would like to consider regrouping the nonconvex list in their Wythoff constructions.Tom Ruen06:53, 18 September 2006 (UTC)[reply]

Vote?

1. Yes -Tom Ruen03:02, 18 September 2006 (UTC)[reply]
2. Unsure.Tetracube04:41, 18 September 2006 (UTC)[reply]
3. Ambivalent. —Tamfang05:43, 19 September 2006 (UTC)[reply]

SeeTalk:Cantellated 5-cell if you have interest or opinions on projection terminology. Thanks!Tom Ruen22:26, 3 January 2007 (UTC)[reply]

I addedCoxeter-Dynkin diagram for most of the figures, and expanded tables for the prismatic forms. New tables need elements cross-checked for correctness. I'll try to expand some more sample tables at the end for smallest duoprisms. I'm done for the weekend, except maybe a bit of cross-checking...

This article is getting a bit long, but I like them all in one article. Perhaps useful to consider at some point aList of uniform polychora parallel toList of uniform polyhedra. Then this article could have tables reduced to less information perhaps.

Tom Ruen14:34, 20 January 2007 (UTC)[reply]

Okay, I was too curious so I added the B4 family polychora (from7. Uniform polychora derived from glomeric tetrahedron B4) , all repeats, but I was curious about how they were related. I added new Y-graph Coxeter-Dynkin diagrams, but had to leave the cells grouped like the {4,3,3} and {3,4,3} families, since I didn't know how they are related. I'll look for more information about these.Tom Ruen05:21, 21 January 2007 (UTC)[reply]

Last effort - added Coxeter group names to every family - need to explain more, but seemed useful to include.Tom Ruen05:43, 21 January 2007 (UTC)[reply]

I'm trying out some new images, making some consistent sets between truncated forms within each class. First test run done on simplex family, centering on cell pos. 3, and showing cells at pos. 0. Seems a good start.Tom Ruen03:01, 16 March 2007 (UTC)[reply]

5-cell image set:

t0{3,3,3}

t0,1

t1

t0,2

t0,3

t1,2

t0,1,2

t0,1,3

t0,1,2,3

Thought on 8/16-cell and 120/600-cell families. Probably best to end up splitting the tables into two by each regular generator (2 tables of 9, rather than one 1 table of 15), duplicating the symmetric forms (bitruncated/runcinated/omnitruncated). This is needed so I can show the truncations from each direction and the middle forms will have two images with different cells visible.Tom Ruen03:57, 16 March 2007 (UTC)[reply]

8-cell/16-cell image set:

t0{4,3,3}	t0,1	t1	t0,2	t0,3	t1,2	t0,1,2	t0,1,3	t0,1,2,3
t0{3,3,4}	t0,1	t1	t0,2	t0,3	t1,2	t0,1,2	t0,1,3	t0,1,2,3	sr

[This paragraph is ambiguous in the extreme. I THINK this is what is meant:]

There are three families of uniform polychora that are considered prismatic:

Polyhedral prisms

Duoprisms

Polygonal prismatic prisms

These prisms generalize the properties ...

[Under each of these 3 heading belong the following lead sentences:]

Polyhedral prisms
The first family contains three subsets: Tetrahedral prisms, Octahedral prisms, and Icosahedral prisms. They are the most obvious family of prismatic polychora, i.e. products of a polyhedron ...

Duoprisms: ...
The second family is the infinite set of duoprisms–products of two regular polygons. The snubbed form–prisms of anti-prisms–is excluded.
Their Coxeter-Dynkin diagram ...

Polygonal prismatic prisms: ...
The third family contains two subsets: Prismatic prisms and Anti-prismatic prisms. Both subsets are infinite.
The prismatic prisms overlap with ...
The anti-prismatic prisms are constructed ...

[However, I do not want to make edit changes, if I am wrong. Further, I believe "prisms of anti-prisms" are the same as "Anti-prismatic prisms" and should be cross referenced--i.e. "excluded" replaced by "included in the third family".]
18:52, 29 January 2011 Colin.campbell.27

I agree that paragraph needs improvement. I'll try my hand. —Tamfang (talk)21:17, 29 January 2011 (UTC)[reply]

The numbers don't add up.
Where does the "75" come from? Where are they? I can't get to "75 nonprismatic uniform polyhedra" from "64 convex uniform polychora".
Do the two references to "18 convex" refer to the same set of prisms? (I assume they do.)
22 prisms are listed. 5 of them are duplicates (within "[ ]"). 1 of these is the cube-prism/tesseract. Does 18 = 22 – 5 + 1?
Clarification is needed.
18:52, 29 January 2011 Colin.campbell.27

(Next time use the template {{cn}}.)

There's no reason to expect the number of nonprismaticuniform polyhedra to relate to the number of convex polychora.

Yes, the 18 convex polyhedron-prisms are made from 18 convex polyhedra, four of which are listed twice. (I thinkthis article would not suffer much if the "rectified-tetrahedral prism" and so on were dropped.) —Tamfang (talk)21:14, 29 January 2011 (UTC)[reply]

Right. There's nononconvex polychora listed here.Tom Ruen (talk)22:39, 29 January 2011 (UTC)[reply]

I'm still checking, but here's a first pass list of 3-sphere symmetry groups, in Coxeter's notation (FromRegular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591], mostly from a single table). There's definitely more lower symmetry groups unlisted!

When it looks solid, I'll make an article listList_of_spherical_symmetry_groups (for 2-sphere groups), and link to this under the stat tables for the uniform 4-polytopes. Currently it links toCoxeter group which is really only the pure reflectional symmetries of the Coxeter-Dynkin graphs.Tom Ruen (talk)03:13, 24 May 2011 (UTC)[reply]

Where you write "3-sphere symmetry groups", do you mean "finite subgroups of O(4)" ??? (Where of course O(4) denotes the full isometry group of the 3-sphere.) Or something else?50.205.142.50 (talk)17:20, 26 June 2020 (UTC)[reply]

Finite groups

[ ]: Symbol Order [1]+ 1.1 [1] = [ ] 2.1 [2,1,1]: Symbol Order [2+,1,1]+ 1.1 [2,1,1]+ 2.1 [2,1,1] 4.1 [2,2,1]: Symbol Order [2+,2+,1]+ = [(2+,2+,2+)] 1.1 [2+,2+,1] 2.1 [2,2,1]+ 4.1 [2+,2,1] 4.1 [2,2,1] 8.1 [2,2,2]: Symbol Order [(2+,2+,2+,2+)] = [2+,2+,2+]+ 1.1 [2+,2+,2+] 2.1 [2+,2,2+] 4.1 [(2,2)+,2+] 4 [[2+,2+,2+]] 4 [2,2,2]+ 8 [2+,2,2] 8.1 [(2,2)+,2] 8 [[2+,2,2+]] 8.1 [2,2,2] 16.1 [[2,2,2]]+ 16 [[2,2+,2]] 16 [[2,2,2]] 32 [p,1,1]: Symbol Order [3,1,1]+ 3.1 [4,1,1]+ 4.2 [5,1,1]+ 5.1 [6,1,1]+ 6.1 [p,1,1]+ p [3,1,1] 6.2 [4,1,1] 8.4 [5,1,1] 10.2 [6,1,1] 12.3 [p,1,1] 2p [p,2,1]: Symbol Order [3,2,1]+ 6.1 [4,2,1]+ 8.3 [5,2,1]+ 10.2 [6,2,1]+ 12.3 [p,2,1]+ 2p [3,2,1] 12.3 [4,2,1] 16.6 [5,2,1] 20.3 [6,2,1] 24.6 [p,2,1] 4p [2p,2+,1]: Symbol Order [6,2+,1] 12.3 [8,2+,1] 16.12 [10,2+,1] 20.3 [12,2+,1] 24.12 [2p,2+,1] 4p

[p,2,2]: Symbol Order [3+,2,2+] 6 [4+,2,2+] 8 [5+,2,2+] 10 [6+,2,2+] 12 [p+,2,2+] 2p [(3,2)+,2+] 6 [(4,2)+,2+] 8 [(5,2)+,2+] 10 [(6,2)+,2+] 12 [(p,2)+,2+] 2p [3,2,2]+ 12 [4,2,2]+ 16 [5,2,2]+ 20 [6,2,2]+ 24 [p,2,2]+ 4p [3,2,2+] 12 [4,2,2+] 16 [5,2,2+] 20 [6,2,2+] 24 [p,2,2+] 4p [3+,2,2] 12 [4+,2,2] 16 [5+,2,2] 20 [6+,2,2] 24 [p+,2,2] 4p [(3,2)+,2] 12 [(4,2)+,2] 16 [(5,2)+,2] 20 [(6,2)+,2] 24 [(p,2)+,2] 4p [3,2,2] 24 [4,2,2] 32 [5,2,2] 40 [6,2,2] 48 [p,2,2] 8p [2p,2+,2]: Symbol Order [6+,2+,2+] 6 [8+,2+,2+] 8 [10+,2+,2+] 10 [12+,2+,2+] 12 [2p+,2+,2+] 2p [6+,2+,2] 12 [8+,2+,2] 16 [10+,2+,2] 20 [12+,2+,2] 24 [2p+,2+,2] 4p [6+,(2,2)+] 12 [8+,(2,2)+] 16 [10+,(2,2)+] 20 [12+,(2,2)+] 24 [2p+,(2,2)+] 4p [6,(2,2)+] 24 [8,(2,2)+] 32 [10,(2,2)+] 40 [12,(2,2)+] 48 [2p,(2,2)+] 8p [6,2+,2] 24 [8,2+,2] 32 [10,2+,2] 40 [12,2+,2] 48 [2p,2+,2] 8p

[p,2,q]: Symbol Order [3+,2,3+] 9 [4+,2,3+] 12 [5+,2,3+] 15 [6+,2,3+] 18 [4+,2,4+] 16 [5+,2,4+] 20 [6+,2,4+] 24 [5+,2,5+] 25 [6+,2,5+] 30 [6+,2,6+] 36 [p+,2,q+] pq [3,2,3]+ 18 [4,2,3]+ 24 [5,2,3]+ 30 [6,2,3]+ 36 [4,2,4]+ 32 [5,2,4]+ 40 [6,2,4]+ 48 [5,2,5]+ 50 [6,2,5]+ 60 [6,2,6]+ 72 [p,2,q]+ 2pq [3+,2,3] 18 [4+,2,3] 24 [5+,2,3] 30 [6+,2,3] 36 [4+,2,4] 32 [5+,2,4] 40 [6+,2,4] 48 [5+,2,5] 50 [6+,2,5] 60 [6+,2,6] 72 [p+,2,q] 2pq [3,2,3] 36 [4,2,3] 48 [5,2,3] 60 [6,2,3] 72 [4,2,4] 64 [5,2,4] 80 [6,2,4] 96 [5,2,5] 100 [6,2,5] 120 [6,2,6] 144 [p,2,q] 4pq

[(p,2)+,2q]: Symbol Order [(3,2)+,6+] 18 [(4,2)+,6+] 24 [(5,2)+,6+] 30 [(6,2)+,6+] 36 [(4,2)+,8+] 32 [(5,2)+,8+] 40 [(6,2)+,8+] 48 [(5,2)+,10+] 50 [(6,2)+,10+] 60 [(6,2)+,12+] 72 [(p,2)+,2q+] 2pq [(3,2)+,6] 36 [(4,2)+,6] 48 [(5,2)+,6] 60 [(6,2)+,6] 72 [(4,2)+,8] 64 [(5,2)+,8] 80 [(6,2)+,8] 96 [(5,2)+,10] 100 [(6,2)+,10] 120 [(6,2)+,12] 144 [(p,2)+,2q] 4pq [2p,2+,2q]: Symbol Order [6+,2+,6+] 18 [8+,2+,6+] 24 [10+,2+,6+] 30 [12+,2+,6+] 36 [8+,2+,8+] 32 [10+,2+,8+] 40 [12+,2+,8+] 48 [10+,2+,10+] 50 [12+,2+,10+] 60 [12+,2+,12+] 72 [2p+,2+,2q+] 2pq [6,2+,6+] 36 [8,2+,6+] 48 [10,2+,6+] 60 [12,2+,6+] 72 [8,2+,8+] 64 [10,2+,8+] 80 [12,2+,8+] 96 [10,2+,10+] 100 [12,2+,10+] 120 [12,2+,12+] 144 [2p,2+,2q+] 4pq [6,2+,6] 72 [8,2+,6] 96 [10,2+,6] 120 [12,2+,6] 144 [8,2+,8] 128 [10,2+,8] 160 [12,2+,8] 192 [10,2+,10] 200 [12,2+,10] 240 [12,2+,12] 288 [2p,2+,2q] 8pq

[[p,2,p]]: Symbol Order [[3+,2,3+]] 18 [[4+,2,4+]] 32 [[5+,2,5+]] 50 [[6+,2,6+]] 72 [[p+,2,p+]] 2p2 [[3,2,3]]+ 36 [[4,2,4]]+ 64 [[5,2,5]]+ 100 [[6,2,6]]+ 144 [[p,2,p]]+ 4p2 [[3,2,3]] 72 [[4,2,4]] 128 [[5,2,5]] 200 [[6,2,6]] 288 [[p,2,p]] 8p2 [[2p,2+,2p]]: Symbol Order [[6+,2+,6+]] 36 [[8+,2+,8+]] 64 [[10+,2+,10+]] 100 [[12+,2+,12+]] 144 [[2p+,2+,2p+]] 4p2 [[6,2+,6]] 144 [[8,2+,8]] 256 [[10,2+,10]] 400 [[12,2+,12]] 576 [[2p,2+,2p]] 16p2

[3,3,2]: Symbol Order [(3,3)+,2,1+] =[3,3]+ 12.5 [3,3,2,1+] =[3,3] 24 [(3,3)+,2] 24.10 [3,3,2]+ 24.10 [3,3,2] 48.36 [4,3,2]: Symbol Order [(4,3)+,2,1+] =[4,3]+ 24.15 [3+,4,2,1+] =[3+,4] 24.10 [3+,4,2+] 24 [(3,4)+,2+] 24 [4,3,2,1+] =[4,3] 48.36 [3,4,2+] 48 [4,(3,2)+] 48 [(4,3)+,2] 48.36 [4,3,2]+ 48.36 [4,3+,2] 48.22 [4,3,2] 96.5 [5,3,2]: Symbol Order [(5,3)+,2,1+] =[5,3]+ 60.13 [5,3,2,1+] =[5,3] 120.2 [(5,3)+,2] 120.2 [5,3,2]+ 120.2 [5,3,2] 240 (nc) [31,1,1]: Symbol Order [31,1,1]+ = [1+,4,(3,3)+] 96.1 [31,1,1] = [1+,4,3,3] 192.2 <[3,31,1]> = [4,3,3] 384.1 [3[31,1,1]] = [3,4,3] 1152.1 [3,3,3]: Symbol Order [3,3,3]+ 60.13 [3,3,3] 120.1 [[3,3,3]]+ 120.2 [[3,3,3]+] 120.1 [[3,3,3]] 240.1 [4,3,3]: Symbol Order [1+,4,(3,3)+] = [31,1,1]+ 96.1 [1+,4,3,3] = [3,31,1] 192.2 [4,(3,3)+] 192.1 [4,3,3]+ 192.3 [4,3,3] 384.1 [3,4,3]: Symbol Order [3+,4,3+] 288.1 [3,4,3]+ 576.2 [3+,4,3] 576.1 [[3+,4,3+]] 576 (nc) [3,4,3] 1152.1 [[3,4,3]]+ 1152 (nc) [[3,4,3]+] 1152 (nc) [[3,4,3]] 2304 (nc) [5,3,3]: Symbol Order [5,3,3]+ 7200 (nc) [5,3,3] 14400 (nc)

Maybe someday I'll grok these notations. ‹sigh› —Tamfang (talk)06:12, 29 May 2011 (UTC)[reply]

Maybe not too bad on a soft view. [[]] is a doubling of the symmetry of symmetric linear groups, so [[3,3]] ~= [4,3]. + is a half symmetry by rotation [4,3]+ ~= [3,3]. And also [4,3+] is another half symmetry if next to an even number of reflections. Also [3[3^1,1,1]] makes all 3 branches identical with 6x symmetry order, ~= [3,4,3]!Tom Ruen (talk)06:37, 29 May 2011 (UTC)[reply]

Geometry was easier than this when I was in school :(jni (talk)18:09, 23 November 2013 (UTC)[reply]

Surely the 17 convex polyhedral prisms (#48 to #64) can't be considered non-prismatic?Double sharp (talk)10:05, 6 August 2012 (UTC)[reply]

Sometimes I considersix impossible things before breakfast. —Tamfang (talk)21:21, 7 August 2012 (UTC)[reply]

I'd prefer a four-way distinction, between:

1. The non-prismatic ones with 4D symmetries (#1 to #47);
2. The polyhedral prisms, where the polyhedron is not prismatic (3D × 1D products) (#48 to #64);
3. The antiprismatic prisms (3D × 1D products) (an infinite set);
4. The duoprisms (2D × 2D products) (an infinite set).

Double sharp (talk)03:24, 8 August 2012 (UTC)[reply]

Is the word "alterated" in the article an accepted technical term? I don't find it in dictionaries.

The correct term isalternated. I fixed one typo alterated, which was missing the n.Tom Ruen (talk)19:58, 7 August 2014 (UTC)[reply]

There is a relevant discussion atTalk:4-polytope#Unknown_total_number_of_nonconvex_uniform_4-polytopes — Cheers,Steelpillow (Talk)11:05, 19 December 2014 (UTC)[reply]

This article is about uniform figures. The section onnonuniform alternations: a) is not relevant, and b) relies on self-published and sparse web content for its citations. This also applies to the recently-added subsection on "scaliform" figures. Is there an established reliable source for all this stuff so it can go somewhere else rather than be summarily deleted? — Cheers,Steelpillow (Talk)10:58, 23 December 2014 (UTC)[reply]

The nonuniform alternations are collected for completeness as a small set of legal wythoff construction alternations which are valid 4-polytopes. The scaliforms are also a small set of known exceptional cases and examples, two being legal wythoff construction alternations. I'm still checking for other sourcing.Tom Ruen (talk)11:22, 23 December 2014 (UTC)[reply]

@Tomruen: It might save you some pain if you could stop adding "scaliform" everywhere unless and until you can verify it. But AFAIK it is one of the UPP's neologisms and as such is unverifiable. — Cheers,Steelpillow (Talk)16:12, 23 December 2014 (UTC)[reply]

I've limited the scaliforms to a few cases, all but two are direct constructions by Coxeter's notation. If I can't find any sources outside of websites, I'll comment out the non-Wythoffian forms (that have no simple relations to Coxeter alternation constructions or terminology.) Personally right now I'm interested in their symmetry, and the two nonwythoffians are closest in origin to the uniformgrand antiprism, as a diminished form of the uniform solutions, discovered by computer search by Conway, and whose symmetry was identified by Johnson. If the termscaliform is an unacceptable neologism we could always sayvertex-transitive honeycombs and polytopes with regular, semiregular and Johnson solid cells in the 3D/4D cases, and it doesn't greatly matter to me.Tom Ruen (talk)10:13, 24 December 2014 (UTC)[reply]

If you have no RS for their existence, it doesn't matter what you call them, they cannot remain on Wikipedia. — Cheers,Steelpillow (Talk)13:51, 24 December 2014 (UTC)[reply]

I restored the alternation and scaliform sections, without the offending terminology, and commented out the unreliably documented nonwythoffian scaliforms. The rest are all topologically valid polytopes, that are directly constructed from Coxeter's theory, and necessary for completeness, to show why they are not included on the list.

Tom, you are re-inserting material that falls foul of fundamental policies and refusing to engage with those policies. Without a reliable source this counts as original research - seeWP:OR - and you must surely know that you are not allowed to post that. And I am sure you are also aware that Wikipedia policy demands "verifiability not truth" - seeWP:VERIFICATION.If you persist, this will have to go to the Mathematics WikiProject. — Cheers,Steelpillow (Talk)20:41, 24 December 2014 (UTC)[reply]

Coxeter mentions the missing alternations in regards to the snub 24-cell construction. I reduced the section size, but don't see why they shouldn't be listed for completeness.Tom Ruen (talk)21:57, 24 December 2014 (UTC)[reply]

Fair enough. I think that better presentation is still needed, I'll try and take a look later. — Cheers,Steelpillow (Talk)23:00, 24 December 2014 (UTC)[reply]

We have them for all the nonprismatics, and I'm uploading some prismatic examples. Since these are rather prominent visualization aids for the polychora perhaps we could add them in their own column.Double sharp (talk)08:14, 8 February 2015 (UTC)[reply]

OK, uploaded pictures for the nets of all thep,q-duoprisms for 3 ≤ p,q ≤ 10.Double sharp (talk)08:24, 8 February 2015 (UTC)[reply]

...and all thep-antiprismatic prisms for 4 ≤ p ≤ 10.Double sharp (talk)08:28, 8 February 2015 (UTC)[reply]

I re-added a short description on the uniform star polychora. The fact that they're on a program (Stella4D) where they can visualized surely makes the claims about their existence credible? If some other mathematician has mentioned them during the last few years, a source would be helpful. –OfficialURL (talk)18:26, 21 February 2020 (UTC)[reply]

Wait, so this has in fact been topic of heated past discussion over atTalk:4-polytope, and the general consensus is that, since no published article exists on them, they don't belong here on Wikipedia. I'll comment out my edit, in case it ever becomes something that can be added. (I really hope theNorman Johnson manuscript comes out soon). –OfficialURL (talk)00:11, 22 February 2020 (UTC)[reply]

@OfficialURL: It appears thatJohnson left the manuscript almost complete on his death and that a publication is forthcoming. (Although it has been forthcomingsince 1996...)Double sharp (talk)17:55, 22 February 2020 (UTC)[reply]

Jonathan Bowers wrote up a presentation for the Bridges conference in 2000. Unfortunately, it is so old that there were then 8186 uniform polychora, mostly exotic-celled.Double sharp (talk)20:54, 21 October 2022 (UTC)[reply]

Much of this article is written in obfuscatory language than anyone not intimately familiar with the technical terminology of polytopes is certain will be certain to find confusing.

Additionally, many other things are extremely unclear.

This is very unfortunate, because some Wikipedia writers have made great efforts to fill Wikipedia with extremely comprehensive information about polytopes, and 4-dimensional polytopes in particular.

For example, the very first table of this article isfilled with information. Butnobody knows what this table is for, because it is entirelyunlabeled andhas no caption.

I hope anyone who might like to improve this article, and who is familiar with the subject matter and who knows how to write clearly for a non-expert audience, will do so.50.205.142.50 (talk)20:22, 14 June 2020 (UTC)[reply]

One sentence reads as follows:

"The 5-cell has diploid pentachoric [3,3,3] symmetry,[7] of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way."

But isn't it necessary for — not just all pairs of vertices — but all vertices, as well as pairs, triples, and quadruples of vertices to be each related in the same way ... in order for the symmetry group to consist of all permutations of the vertices?50.205.142.50 (talk)17:17, 26 June 2020 (UTC)[reply]

The following Wikimedia Commons files used on this page or its Wikidata item have been nominated for deletion:

• 3-3 duoprism net.png
• 4-3 duoprism net.png
• 4-antiprismatic prism net.png
• 5-3 duoprism net.png
• 5-4 duoprism net.png
• 5-5 duoprism net.png
• 5-antiprismatic prism net.png
• 6-3 duoprism net.png
• 6-4 duoprism net.png
• 6-5 duoprism net.png
• 6-6 duoprism net.png
• 6-antiprismatic prism net.png
• 7-4 duoprism net.png
• 7-antiprismatic prism net.png
• 8-4 duoprism net.png

Participate in the deletion discussion at thenomination page. —Community Tech bot (talk)17:24, 23 August 2022 (UTC)[reply]

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