Talk:Uniform polychoron

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Nice article, but rather peripheral subject. Arcfrk 10:38, 26 May 2007 (UTC)

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

• Reorganize this page so that regular polychora and semiregular 4-polytopes appear somewhere sensible. Currently only a subset of regular polychora is described, under convex 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"?

—Tetracube 07:15, 10 January 2006 (UTC)

I've made a stab at defining some of the operations, as well as some other changes suggested by George Olshevsky. —Anton Sherwood 00:27, 13 January 2006 (UTC)

Thanks! That helps a lot.—Tetracube 07:08, 13 January 2006 (UTC)

Contents 1 table of corresponding elements 2 Antiprisms 3 Rename to singular form? 4 Move consensus question 5 New category? 6 Geometric derivations 7 Grand antiprism 8 New arrangement 9 Sources? 9.1 Who discovered the uniform polychora? And when? 9.2 Some references 10 Expanding data tables 11 Data completion 12 Rename? 13 Polychoron image projections 14 Coxeter-Dynkin diagrams 15 New image notes

[edit] table of corresponding elements

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 Sherwood 18:46, 10 January 2006 (UTC)

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).—Tetracube 18:59, 10 January 2006 (UTC)

[edit] Antiprisms

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 are not 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 :-) —Tetracube 06:07, 11 January 2006 (UTC)

---

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). —Tamfang 04:06, 13 February 2006 (UTC)

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. —Tamfang 20:00, 19 February 2006 (UTC)

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.—Tetracube 01:49, 16 February 2006 (UTC)

No other self-dual uniform polyhedra, no. Pyramids and elongated pyramids are topologically self-dual and can be distorted into geometric self-duals (the "canonical" form). —Tamfang 04:23, 16 February 2006 (UTC)

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. —Tamfang 07:48, 16 February 2006 (UTC)

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., a cone seems to be self-dual in some sense (generalizing from n-gonal pyramids as n approaches ∞), and a cylinder's dual seems to be a di-cone (two cones joined at the base), generalizing from the n-gonal prisms. I don't know what the dual of a sphere might be... perhaps a point?—Tetracube 04:53, 16 February 2006 (UTC)

A sphere's dual is a sphere: an infinite number of vertices corresponds to an infinite number of faces. —Tamfang 07:48, 16 February 2006 (UTC)

[edit] Rename to singular form?

I noticed that the wikipedia style guidelines suggest that article names be singular where possible, rather than plural. Should we rename this page to uniform polychoron instead?—Tetracube 18:51, 27 January 2006 (UTC)

I agree - more consistent with Uniform polyhedron and Polychoron too. In contrast I'd keep List of uniform polyhedra as plural, while eventually I'd like a parallel tabular version as List of convex uniform polychora. Tom Ruen 22:04, 27 January 2006 (UTC)

OK, I've put up a request in Wikipedia:requested moves 'cos uniform 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?—Tetracube 07:58, 2 February 2006 (UTC)

I'd just be bold and use the Move 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 Ruen 08:03, 2 February 2006 (UTC)

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.)—Tetracube 08:10, 2 February 2006 (UTC)

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

[edit] Move consensus question

[edit] New category?

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

Currently they are under Category:Polytopes which isn't bad, except for not specifying 4D objects, although not much going on above 4D yet! Tom Ruen 06:30, 7 February 2006 (UTC)

I noticed that Category:Polytopes already has sub-categories for polygons and polyhedra. Maybe we should move the current polychora into Category:Polychora, at the very least?—Tetracube 08:20, 8 February 2006 (UTC)

[edit] Geometric derivations

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

I added a definition for 'snub which seems to be correct for uniform polyhedra, but don't know well how it applies to polychora like snub 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 Ruen 02:04, 13 February 2006 (UTC)

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 with n-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.—Tetracube 02:35, 13 February 2006 (UTC)

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 term snub 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?

—Tamfang 02:59, 13 February 2006 (UTC)

Marek Čtrnáct gave me a surprising definition of snub. 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. —Tamfang 18:22, 15 February 2006 (UTC)

Wow, this is very interesting. I wonder if anyone else recognizes this definition of snub? 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.—Tetracube 18:55, 15 February 2006 (UTC)

Since each edge has exactly two ends, I can't see a problem offhand in generalizing to n dimensions. But I lack intuition. —Tamfang 20:25, 15 February 2006 (UTC)

[edit] Grand antiprism

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 the duocylinder, 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. —Tamfang 05:27, 13 February 2006 (UTC)

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:

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

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.—Tetracube 06:01, 13 February 2006 (UTC)

That's pretty much what I thought you meant, though somehow I had the impression that "manifold" meant something unbounded (though not necessarily infinite). —Tamfang 06:30, 13 February 2006 (UTC)

Whoops, look who last edited duocylinder ;) —Tamfang 06:34, 13 February 2006 (UTC)

Anyway, I'm thinking of moving the info about the grand antiprism into its own page. What do you think?—Tetracube 04:16, 13 February 2006 (UTC)

Yeah, sure. The bit about an almost-uniform family that I suggested above belongs here, though, more than it belongs at grand 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. —Tamfang 05:27, 13 February 2006 (UTC)

Which 20 vertices, though? I assume it can't be any random set of 20 vertices.—Tetracube 06:01, 13 February 2006 (UTC)

Well, they must be on two great circles. Does that help? —Tamfang 06:30, 13 February 2006 (UTC)

OK, I've made a draft of the grand antiprism article. Comments?—Tetracube 06:09, 13 February 2006 (UTC)

[edit] New arrangement

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.—Tetracube 15:56, 11 July 2006 (UTC)

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). —Tamfang 20:21, 12 July 2006 (UTC)

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. —Tamfang 06:12, 13 July 2006 (UTC)

I've been interested in a more compact table with pictures and vertex 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 Ruen 22:51, 12 July 2006 (UTC)

Not bad. It would help to have several versions of some of the images: ideally corresponding faces should have the same color. —Tamfang 06:12, 13 July 2006 (UTC)

I like this idea. It's compact, and immediately conveys the necessary information.—Tetracube 01:39, 14 July 2006 (UTC)

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 Ruen 06:24, 14 July 2006 (UTC)

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 Ruen 23:00, 12 July 2006 (UTC)

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. —Tamfang 06:12, 13 July 2006 (UTC)

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 Ruen 22:07, 13 July 2006 (UTC)

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. —Tamfang 18:35, 15 July 2006 (UTC)

Tom, the {3,3,4} and {3,4,3} families share three members, not two, but that makes the total come out wrong ... —Tamfang 18:34, 15 July 2006 (UTC)

Fixed, forgot to count snub 24-cell! Tom Ruen 18:44, 15 July 2006 (UTC)

[edit] Sources?

In regards to the nonconvex polychora and the Uniform 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 of uniform 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

The Uniform Polychora Project has classified the 1,845 currently known uniform polychora into 29 groups. There may be more. Most of these are non-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 because Jonathan Bowers had used a laxer definition of 'uniform polychoron' that allowed many degenerate and exotic combinations of uniform polyhedra. The more traditional definition used by Norman 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 as polychoroids.

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 from George 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 Ruen 03:47, 15 July 2006 (UTC)

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. CMummert 19:01, 15 July 2006 (UTC)

[edit] Who discovered the uniform polychora? And when?

• [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 article Uniform Polychora Project and added a draft history section using information above. Unfortunately fuzzy in details, but a start. Tom Ruen 19:32, 15 July 2006 (UTC)

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 Ruen 20:19, 15 July 2006 (UTC)

Has someone actually held these sources in their hands and verified that the material in this article appears in the book? CMummert 01:16, 16 July 2006 (UTC)

Ah, yeah - one disaster at a time! So far, I've only seen the 1954 "Uniform Polyhedra" paper reference which only covers the list 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 Ruen 01:22, 16 July 2006 (UTC)

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 – including John Horton Conway, Norman Johnson, Neil Sloane – has taken an interest. ;) —Tamfang 03:52, 16 July 2006 (UTC)

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. CMummert 03:59, 16 July 2006 (UTC)

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 Ruen 22:15, 17 July 2006 (UTC)

[edit] Some references

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 specific semiregular 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 of n 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. :-) —Tetracube 06:07, 9 August 2006 (UTC)

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.—Tetracube 06:19, 9 August 2006 (UTC)

I have the 1900 Gosset paper as a PDF, and I can share. Terminology is a bit confusing, and semiregular forms are now listed from it at Talk:Semiregular 4-polytope. Tom Ruen 06:23, 9 August 2006 (UTC)

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

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

[edit] Expanding data tables

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 Ruen 21:17, 20 July 2006 (UTC)

[edit] Data completion

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 Ruen 04:48, 14 August 2006 (UTC)

[edit] Rename?

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

This would parallel names for convex regular polychoron and convex uniform honeycomb. Polychoron could reference the existence of nonconvex forms.

Tom Ruen 03:02, 18 September 2006 (UTC)

Discussion?

• It sounds like a good idea, although the question lingers as to what we should do with uniform 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 to convex uniform polychoron). Or maybe it can redirect to simply polychoron?—Tetracube 04:41, 18 September 2006 (UTC)
• I think you could redirect to Polychoron#Categories. Tom Ruen 06:36, 18 September 2006 (UTC)
• For nonconvex forms, the "easy" cases are the some 52 nonconvex uniform polyhedrons duplicated as hyperprisms, as well as infinite sets of star prisms hyperprisms, but overall, I don't find them very interesting, especially knowing there's thousands. I am considering a similar split as convex uniform polyhedron as well, although someday would like to consider regrouping the nonconvex list in their Wythoff constructions. Tom Ruen 06:53, 18 September 2006 (UTC)

---

Vote?

1. Yes - Tom Ruen 03:02, 18 September 2006 (UTC)
2. Unsure. Tetracube 04:41, 18 September 2006 (UTC)
3. Ambivalent. —Tamfang 05:43, 19 September 2006 (UTC)

---

[edit] Polychoron image projections

See Talk:Cantellated 5-cell if you have interest or opinions on projection terminology. Thanks! Tom Ruen 22:26, 3 January 2007 (UTC)

[edit] Coxeter-Dynkin diagrams

I added Coxeter-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 a List of uniform polychora parallel to List of uniform polyhedra. Then this article could have tables reduced to less information perhaps.

Tom Ruen 14:34, 20 January 2007 (UTC)

Okay, I was too curious so I added the B4 family polychora (from 7. 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 Ruen 05:21, 21 January 2007 (UTC)

Last effort - added Coxeter group names to every family - need to explain more, but seemed useful to include. Tom Ruen 05:43, 21 January 2007 (UTC)

[edit] New image notes

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 Ruen 03:01, 16 March 2007 (UTC)

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 Ruen 03:57, 16 March 2007 (UTC)

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	h0,1,2