User talk:Tetracube

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Welcome!

Welcome to Wikipedia, Tetracube! My name is Ryan, aka Acetic Acid. I noticed that you were new and haven't received any messages yet. I just wanted to see how you were doing. Wikipedia can be a little intimidating at first, since it uses different formatting than other sites that use HTML and CSS. In the long run, though, you'll find that the WikiSyntax is a lot easier and faster than those other ways. Here are a few links to get you started:

There are a lot of policies and guides to read, but I highly recommend reading over those first. If you have any questions, feel free to leave me a message on my talk page. Please be sure to sign your name on Talk Pages using four tildes (~~~~) to produce your name and the current date, along with a link to your user page. This way, others know when you left a message and how to find you. It's easier than having to type out your name, right? :)

I hope you enjoy contributing to Wikipedia. We can use all the help we can get! Have a nice day. Sincerely, Ryan 06:11, August 5, 2005 (UTC)

No problem. We target blank (red) talk pages. We have no other way of pointing out new users, which is why we often end up welcoming experienced users. Ryan 21:46, August 5, 2005 (UTC)

Contents

[edit] 24-cell

Thanks for the complements on the animated 24-cell. I just finished the 16-cell, and I'm working on the 5-cell now. I would love to tackle the 120 and 600, but I don't think I can build them vertex by vertex like I did for the others without going insane. Is there any algorithm that can generate a list of vertices and connecting edges for these shapes?

Here's the 120-cell VEF data - pretty clear format I hope User:Tomruen/120celldata. Tom Ruen 22:52, 20 February 2007 (UTC)
Thanks, that looks like exactly what I need. Now to bury myself in the mel documentation :) JasonHise 04:15, 21 February 2007 (UTC)
You built the 24-cell vertex-by-vertex?? Wow... and I thought I was the only one crazy enough to hand-code the face lattice of the 24-cell. :-) But yeah, for the 120-cell and 600-cell, you don't wanna be doing this by hand. 600 vertices and 1200 edges will do some serious injury to your wrists. I did write some Perl scripts for generating n-dimensional simplices, crosses, and cubes (full face lattice), although they are in a peculiar format I use for an experimental program I'm writing. I have a script for generating the 24-cell, but none for the 120-cell or 600-cell yet.
Oh, and BTW, are you planning on doing animations of the uniform polychora as well? I'm dying to see the bitruncated 24-cell in action. (Mainly to confirm whether it looks like what I see in my mind's eye.)—Tetracube 04:49, 22 February 2007 (UTC)
I figured out that mel scripting isn't powerful enough to handle the data structures I need for object construction, so I am currently planning to write a maya plugin capable of loading files in the VEF format you showed me to generate complete animated scenes automatically. This means that when it is done, I should be able to generate any 4D figure that I can find a VEF file for. Including the bitruncated 24-cell if you have it. — JasonHise 06:14, 22 February 2007 (UTC)
I can generate VEF data for all uniform polychora with reflection symmetry from vertex figure data, which I have. Maybe only the grand antiprism is outside my list? So anyway, just ask when you're ready. Tom Ruen 07:19, 22 February 2007 (UTC)

[edit] My User Page

Hey, thanks a million for this! --Tony (Talk), Vandalism Ninja 23:13, 16 February 2006 (UTC)

[edit] Bios theory and User:Lakinekaki

Hi, Tetracube, I see we both use Linux (go Tux!), share an interest in polytopes (Branko Grünbaum was on my thesis committee), and also share concerns about the claims made in Bios theory. I just wanted to let you know that in Talk:Bios theory I have enumerated almost a dozen serious problems with just the first paragraph. I have also listed evidence of a troubling conflict of interest on the part of User:Lakinekaki. Even worse, I have listed evidence of an apparent hidden agenda on the part of the organization which appears not only to employ Lakinekaki but also to be apparently the sole sponsor of "research" on "bios theory" (sic). What to do? ---CH 06:53, 12 May 2006 (UTC)

[edit] polychora

As a major contributor to the Polychoron articles, I need to ask you: Can you supply a reference to the term being used in a peer-reviewed journal? If not, it's got to go. 16:38, 14 July 2006 (UTC)

See Talk:Polychoron. The term is coined by Norman Johnson (the mathematician after whom the Johnson solids are named). According to Jonathan Bowers, Norman is currently writing a book on this subject. If this is not notable enough, then we might as well remove all the math articles. Personally, I couldn't care less whether or not the term itself is used in these articles, but I submit that it would make the wikipedia entries so much less readable if we are forced to use the cumbersome "4-dimensional polytope" every time we want to talk about these things. As for the notability of the polychora themselves, the convex uniform polychora have been known at least since the 19th century (IIRC—it may well be earlier, since the regular 4-polytopes have been known for a lot longer than that). I'll have to look up the reference for this—there is actually a paper that describes these things.—Tetracube 16:47, 22 July 2006 (UTC)
Acknowledged. I may comment further, there &mdash and, I really was only asking about the name. Uniform polytopes seem sufficiently notable to me, (although self-intersecting polytopes, in general, need a better definition) even if I cannot find a reference at the moment. I understand that 4-space has unique properties in regard non-prismatic regular polytopes. — Arthur Rubin | (talk) 19:02, 22 July 2006 (UTC)

[edit] Uniform polytopes

I'm just curious; why must the 2-facets (faces?) be regular. One could construct the following recursive definition:

If n ≥ 2, then an n-dimensional polytope is said to be uniform if

  1. All facets of dimension 2 through n-1 are uniform, and
  2. For any two vertices, there is an isometry mapping one to the other.

(0- and 1- polytopes are trivially uniform, under almost any definition, so I'm restricting the definition to dimensions 2 and higher.)

The uniform polygons would then still have equal angles at each vertex, but might have two different (alternating) edge lengths.

It seems to me to be more geometrically intrinsic than the definition used. — Arthur Rubin | (talk) 22:03, 8 August 2006 (UTC)

I've actually thought about this before, too. In fact, I'd be interested to see the consequences of allowing non-regular 2-faces on polytopes: there'd be a lot more variety! However, that is not the current understanding of the term "uniform", at least as it applies to 3D. Otherwise, you'd have an uncountable number of, say, truncated tetrahedra (truncate the vertices of the tetrahedron by some arbitrary amount up to, but not including, 1/2 edge length). Many of the current Archimedean polyhedra would have infinitely many variants. Allowing 3-faces to be non-regular doesn't quite have the same effect as long as they are still Archimedean, since there are only a countable number of Archimedean polyhedra. (Not saying this is good or bad, but it does give us a lot of "redundant" polytopes that are almost the same.)
Now, as far as it relates to 4D polytopes, I suppose you could argue that allowing Archimedean cells seems rather arbitrary, and maybe we should stick with regular cells, in which case we get the semiregular 4-polytopes, of which there are only 3 (assuming convexity, of course). So it appears that allowing Archimedean cells makes for a much more interesting set of "uniform" polytopes, without making the possibilities too unbounded. I have no experience with 5D polytopes or above, so I don't know how the situation pans out there, but I suspect the situation would be more straightforward.—Tetracube 03:39, 9 August 2006 (UTC)
Well, thinking about it further, I can understand the reason for the definition, as any n-"box" (rectangular parallelepiped, hypercuboid) is uniform under my definition. For polyhedra, "clearly" an arbitrary truncation or cantellation of a regular polyhedron is "uniform". Still, it would be interesting to see if a further analysis could be done with this definition.
The new faces for the truncated {p,q} have each vertex becoming a regular q-gon and each face becoming a uniform 2p-gon.
The new faces for the cantellated {p,q} have each vertex becoming a regular q-gon, each edge become a rectangle (uniform 4-gon), and each face remaining a regular p-gon.
So there are at least those 10 1-dimensionally infinite families, and the 2-dimensionally infinite family of cuboids. — Arthur Rubin | (talk) 02:15, 11 August 2006 (UTC)
These infinite families are topologically identical to the archetypical Archimedeans that we have under the current definition. I don't find that quite as interesting... unless having unequal edge lengths can somehow give rise to (topologically) new polytopes that are excluded by enforcing equal edge lengths. I'd love to know if this is actually possible.
On another note, I'm quite interested in polytopes that have congruent facets but with no other constraints on the shape of the facets (except perhaps convexity). Catalan solids would fall under this category. (Has anybody studied 4D Catalans yet?) In 4D, interestingly enough, there are two non-regular, non-Catalan polychora that fall under this definition: the bitruncated 24-cell and the bitruncated 5-cell, which happen to also be uniform (under the existing definition). I'm curious to know if there are any non-uniform, non-Catalan examples of such polytopes.—Tetracube 05:10, 11 August 2006 (UTC)
The symmetry groups may depend on the precise edge lengths; for example, n-boxes have a symmetry group isomorphic to (Z2)n Wr G, where G is the permutation group acting on the n dimensions which preserve the corresponding edge length, so the classification by permutation group becomes difficult.
But ... for (convex, anyway) uniform polyhedra, the vertex configuration must still have the same properties, as the angle defect is not affected by whether the polygons are regular or "uniform", and the topological requirements (in p.q.r(. ...), if q is odd, then p=r) still holds. Now, when that maps up to 4 dimensions, more complex combinations might be possible. — Arthur Rubin | (talk) 19:06, 11 August 2006 (UTC)
Hmm, I'm not sure I see why it might be different in 4D. As far as I can tell, the angle defect is still unaffected in 4D, so putting the polyhedra together isn't going to produce anything new. I may be missing something obvious, of course.—Tetracube 22:49, 11 August 2006 (UTC)

[edit] Polyhedron club?

Hi Tetracube,

User:RobertAustin asked me to make a user box template for people who like polyhedra. You can add it to your main user page if you like it. You can see others who use it at (Special:Whatlinkshere/Template:User_Polyhedron). Tom Ruen 05:53, 7 January 2007 (UTC) ADD

This user is interested in polyhedra.


Cool, thanks.—Tetracube 03:42, 25 January 2007 (UTC)


[edit] New uniform polyhoron images

Hey Tetracube! I'm on wikibreak for a week, but thought I'd ask your opinion. Check out images at [1] for possible candidates to replace uniform polychora images with a consistent set. They're from the new Stella (software) beta prerelease. Still trying to evaluate what's best but these orthogonal projections with solid cells are interesting, and useful, including the Schläfli-Hess polychoron. I also uploaded some new images for the duoprisms. AND I can make vertex figures and nets for all! Hardest thing is to make easy consistent coloring between, but can be done with patience. Also getting identical projection orientations takes some work. Anyway, thought I'd ask if you'd approve simpler orthogonal images. (OH, there is no "4D export" now, although I'm trying to encourage the developer to add one. He does have a 4D import, extending OFF format but including 4 vertex coordinates and listing cells by indced faces.) Tom Ruen 14:53, 25 February 2007 (UTC)

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