Talk:Semiregular 4-polytope

Untitled

 * Moved this content - originally added to Convex_regular_4-polytope.


 * The content of this page has been made parallel to Semiregular polyhedra article which I also created recently. Both articles enumerate surfaces defined by regular polygons/polyhedra with identical vertex configurations.

Expand existing content:
 * Complete vertex/edge/face/cell counts for the finite semiregular polytopes.
 * Pictures!
 * Standardized names for the polytopes? Hyperprisms and Hyper-antiprisms are perfectly reasonable extensions of the 3D prisms and antiprism, but there may be other names I don't know about.


 * Expanded content?
 * Regular stellated or intersecting polytopes
 * Dual polytopes
 * Hyperbolic polytopes (Edge figures greater than 360 degrees)
 * There's also a larger class of (semiregular) polytopes constructed by regular or semiregular polyhedra, but I don't have a full ennumeration of these. SEE: Uniform_polychora

-- Tom Ruen 19:53, 18 September 2005 (UTC) 

You might want to consider adding/merging the content from Polychoron. There is a lot of overlap here. There has been some debate about the use of the term polychoron (see the talk page on polychoron), but there is a lot of common material here.&mdash;Tetracube 02:16, 19 September 2005 (UTC)


 * Thanks for pointing out this Polychoron, whatever name you want to call it. It looks like what I'm calling Semiregular Polytopes are a special subset of the larger set of "Uniform polytopes" or Uniform_polychora. The Polychoron article certainly needs work.
 * Since I'm not overly qualified to talk about the bigger picture of Uniform_polychora, I'll leave this subset article as-is.
 * My primary interest now is to complete the Vertex/Edge/Face/Cell counts for these shapes and get some pictures up.
 * If anyone can connect the hyper-prisms and hyper-antiprisms to other more comprehensive lists or naming systems, I'm interested!
 * Tom Ruen 02:41, 19 September 2005 (UTC)

If you want information on semi-regular 4-polytopes, take a look at [George Olshevsky's uniform polytopes page]. It has an exhaustive (!) list of all convex uniform polytopes, their vertex/edge/face/cell counts, vertex figures, and cell polytopes. It also classifies their symmetry groups. The only thing it lacks is good diagrams; but other than that, it's an incredibly valuable resource on this topic.&mdash;Tetracube 22:49, 19 September 2005 (UTC)

Tom Ruen 03:00, 23 September 2005 (UTC)
 * Another good website with uniform polytopes is http://www.polytope.de/ - the result of a 2004 PhD paper. This website is in German, but it includes Java applets showing 3D rotation of vertex figures and cells. I linked each semiregular polytope to this website.
 * http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf

BTW, do you have clearer images for the diagrams at the bottom of the Regular Polytopes section? They are very good diagrams, but they appear really faint on my laptop. Maybe they need to be edited to have a higher contrast? Thanks.

Also, if you want information on 4D prisms, you might be interested in the article on duoprisms, which is a large, infinite subset of the possible prisms in 4D. The only other kind of semi-regular prisms possible are the polyhedral prisms (formed by taking any of the semi-regular polyhedra and extruding them).&mdash;Tetracube 05:34, 9 January 2006 (UTC)


 * Looks like I need to have wider line widths for the wireframes to show up well with reduced sizes.
 * On extending the 4D content, I know there's lots of other uniform 4D polyhedra we can add, including infinite sets. I was limited to generating "vertex-uniform" polychora with regular cells, so I can't yet generate images with semiregular cells.
 * I'm also interested in Andreini tessellation, getting clear table of these 28 3D tessellations as well as pictures.
 * Things are busy, but I hope to do more in the future. Tom Ruen 06:35, 9 January 2006 (UTC)

Votes for deletion
I started this article last October before I recognized uniform polychoron as a wider and more useful category which allows all convex uniform polyhedra cells. Now with other articles fleshed out better, I'm prepared to let this article be deleted.

Opinions and votes?

YES
 * Tom Ruen 09:23, 14 March 2006 (UTC)


 * First let's make sure all content is repeated somewhere. But yes, there's no clear reason for a separate article.  &mdash;Tamfang 23:45, 14 March 2006 (UTC)

NO

Expand and refocus to semiregular polytopes?
I've been looking at this paper: (Referenced in uniform polychoron history.)
 * 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.

This wiki-article includes all of of the forms he enumerated, but he gives ALL for higher polytopes as well. Rather than deleteing this category as being less useful than the uniform polychora, expanding to include his full list seems useful, both to honor his work AND because there's no other work I know that describes ALL the higher semiregular polytopes/tessellations.

I'll look into this expansion and include in talk, and when ready, rename perhaps to something like: List of regular and semregular polytopes and honeycombs. Well, this is CLOSE to the extensive list of regular polytopes article as well, so maybe worth MERGING to that in the end since there's not that many semiregulars added.

polytope list
To be a bit more clear specifically Gosset lists: (His names in bold, and translating names as I can, and ??? for now otherwise)

Regular
Regular n-polytopes:
 * 1) n-ic pyramid: Simplex {3,...,3}
 * 2) n-ic double pyramid: Cross polytope {3,...,3,4}
 * 3) n-ic cube: Measure polytope {4,3,...,3}
 * 4) (n-1)-ic check: Measure polytope honeycomb {4,3...3,4}

Regular 4-polytopes:
 * 1) Octahedric: 24-cell {3,4,3}
 * 2) Tetrahedric: 120-cell {5,3,3}
 * 3) Dodecahedric: 600-cell {3,3,5}

Regular 5-polytopes: (honeycombs)
 * 1) Octahedric-check: {3,3,4,3}
 * 2) Double pyramidal check: {3,4,3,3}

Semiregular
Semiregular n-polytopes (n>2)
 * 1) (n-1)-ic semicheck: ??? honeycomb

Semiregular 4-polytopes
 * 1) Tetroctahedric: rectified 5-cell = r{3,3,3}
 * 2) tetricosahedric: snub 24-cell = r{3,4,3}
 * 3) Octicosahedric: rectified 600-cell = r{3,3,5}
 * 4) Simple tetroctahedric check: Tetrahedral-octahedral honeycomb
 * 5) Complex tetroctahedric check: Gyrated alternated cubic honeycomb
 * 6) Tetroctahedric semicheck: ??? honeycomb

Semiregular 5-polytopes
 * 1) 5-ic semi-regular: ??? polytope

Semiregular 6-polytopes
 * 1) 6-ic semi-regular: ??? polytope

Semiregular 7-polytopes
 * 1) 7-ic semi-regular: ??? polytope

Semiregular 8-polytopes
 * 1) 8-ic semi-regular: ??? polytope

Semiregular 9-polytopes
 * 1) 9-ic semi-regular: ??? honeycomb

The intro claims this list is complete, although no modern references (so far!) to confirm this, alhough Coxeter's 1964 "Regular polytopes" talks about him and this paper.


 * On first and second pass I'm rather lost on these higher forms, and won't be extended this quickly! Tom Ruen 01:33, 9 August 2006 (UTC)

Links

 * Gosset's Figure in 8 Dimensions
 * Regular, SemiRegular, Regular faced and Archimedean polytopes
 * Gosset figures
 * Polytopes and optimal packing of p points in n dimensional spheres