Talk:Uniform polyhedron

Initial comment
The article could be expanded with brief sections on Platonic, Archimedan, and Kepler solids, the 53 non-convex uniform polyhedra and the prisms and antiprisms. But it seems to me this is not the place for an exhaustive discussion of any of these subjects, but just a place to define and describe this class of polyhedra and refer the readers to other in-depth articles.

the definition of uniformity
Technically, "identical vertices" is not enough; a famous counterexample is J37. Uniformity requires that the vertices all belong to the same equivalence class under some rotation group; is there a more concise way to say that? --Anton Sherwood 08:19, 3 January 2006 (UTC)


 * Compare to Semiregular polyhedra
 * A semiregular polyhedron is a geometric shape constructed from a finite number of regular polygon faces with every face edge shared by one other face, and with every vertex containing the same sequence of faces, and, moreover, for every two vertices there is an isometry mapping one into the other.
 * Tom Ruen 08:47, 3 January 2006 (UTC)


 * So now I'm wondering why Semiregular and Uniform are two separate articles. Is there a difference that I've failed to spot?  (And what's the word for polyhedra whose vertices and edges are alike, such as the cuboctahedron?  I've seen "quasiregular" used both for those and for the Catalans.)
 * Anton Sherwood 21:21, 3 January 2006 (UTC)


 * I added the semiregular polyhedra article originally because that's what I had always called them. Then I found out about the uniform polyhedra which include nonconvex forms. I had never heard of uniform polyhedra before. I'm perfectly okay with removing off on the semireg. article and putting it all under uniform polyhedron. Quasiregular is given in the polyhedron article Polyhedron, as the two Catalans with rhombic faces. This would be "face-uniform" and "edge-uniform". Tom Ruen 05:11, 4 January 2006 (UTC)

Nonconvex Archimedeans, now? Does that mean those with convex vertex figures? &mdash;Tamfang 00:14, 18 January 2006 (UTC)


 * Actually reverse I figured - polyhedra with all convex faces and nonconvex vertex figures. Tom Ruen 00:17, 18 January 2006 (UTC)


 * ... Alright, I don't know what I'm talking about, just thought 'Stellated Archimedeans made no sense. Obviously this page need a lot of work! Tom Ruen 00:24, 18 January 2006 (UTC)


 * I'd divide the nonconvex polytopes into orientable and non, mainly because I find the former set prettier! I took "nonconvex Archimedeans" to be a subset of orientables.  &mdash;Tamfang 04:51, 18 January 2006 (UTC)


 * Sounds like a useful division, but then it should state as such, "Orientable Nonconvex Uniform Polyhedra". Tom Ruen 06:13, 18 January 2006 (UTC)

I'd like to keep seperate articles for the various families of uniform polyhedra. Mainly as a way of grouping polyhedra with similar properties. Here the Wythoff symbol is very useful the familes are I've created a page grouping the polyhedra by their wythoff symbom.
 * regular: p|q2 vertex figure qp
 * quasi-regular: p|qr (q!=2,r!=2) vertex figure (q.r)p two sub familes
 * 2|qr: v.f. (q.r)2 cubeoctohedron etc. (semi-regular)
 * 3|qr: v.f. (q.r)3 ditrogonal-semi-regulars
 * Wythoff p q|r vertex figure p.2r.q.2r
 * p q|2: p.4.q.4 rhombic (p,q integer) quasi-rhombic (p or q fractional)
 * 2 q|r: 2r.q.2r truncated and quasitruncated (r fractional)
 * p/m p/n|r: 2r.p.2r.q hemi-hedra (versi-regular)
 * Wythoff p q r| vertex figure 2p.2q.2r
 * 2 q r| quasi-truncated and some rhombic forms
 * Wythoff |p q r snub-polyhedron

So semi-regular is a special form of quasi-regular. I'd avoid the term "nonconvex Archimedeans" as its not a standard term used in the litrature. --Salix alba (talk) 15:15, 18 January 2006 (UTC) (formally pfafrich)

How can there be nonconvex polyhedra among the uniform? how do you map a convex vertex into a nonconvex vertex with an isometry? Gbnogkfs 24 August 2006, 5:46 (UT)


 * You don't need to. A uniform polyhedra will always have all convex verticies or all non convex vertices. The isometries will map the non-convext verticies onto each other, and never to a convex vertex. --Salix alba (talk) 10:52, 24 August 2006 (UTC)


 * there can't be a polyhedron with all nonconvex vertices.
 * However, eventually I got it: the "nonconvex vertices" are not vertices of the lattice defining the polyhedron: only the convex vertices are. That is: not all the 0-dimensional intersections between the faces are in fact vertices. That is not clear at all in many ployhedra-related articles: is there anyone able to better clarify this? Gbnogkfs 20:44, 24 August 2006 (UT)


 * Good point. Yes we do need to make clear the differenence between points of intersection of 3 or more faces are not all verticies. Likewise not all the lines intersections of faces are what are classed as edges. To find the edges find the faces (excluding some of the snubs) a face will be the largest flat regions, the edges will be the boundary of the faces, and the verticies the corners of the faces, i.e. the boundary points of the edges. --Salix alba (talk) 23:00, 24 August 2006 (UTC)

On merge with polyhedron section
I reduced this article (back) to a simple description and summary. I agree further merging is desireable. Specifically Polyhedron is too long. I'd actually suggest moving content there to here.

I'm yet in process in a compact complete list of uniform polyhedra and associated some 80+ individual object articles and images! I'll leave it up to someone else for now if anyone wants to merge this with polyhedron.

Tom Ruen 06:18, 16 October 2005 (UTC)
 * I'd actually suggest moving content there to here. Yes, and may I make some more suggestions for the merge:
 * Wikify each of the named objects (don't forget the duals!) into individual articles (as Tomruen started doing). This way, anyone can add unique quirks of each.
 * Subclassify the named objects as far as they can go, and give each named group an article, like, say, Quasi-regular non-convex polyhedron. See how they did it in Wikispecies.  This way, visitors can drill down as they please, and group properties are just in one place.
 * Mirror this categorisation unto Wikipedia's Category feature. That is, the Cuboctahedron article is in Category:Convex polyhedra and in Category:Quasi-regular polyhedra and Category:Archimedean solid (and therefore sub to Category:Uniform polyhedra and sub further to Category:Polyhedra).
 * Compile all the Schlafi, Wythoff, etc. symbols, in a table called list of polyhedra. IMO they're hard to absorb in list or paragraph form, and list of uniform polyhedra is limited to uniform forms.
 * Retain the list of characteristics in the main polyhedron article, and wikifying each characteristic. I'll also be adding Schlafi symbol, Wynthoff symbol, etc., to this list.
 * What do you guys think?
 * --Perfecto 17:08, 16 October 2005 (UTC)


 * I think we're in agreement, even in regards to moving more content back here from polyhedron.
 * I don't understand categories in wiki, if this is something special.
 * For uniformity I do plan to REPLACE my first test template
 * Template:Infobox Polyhedron with vertfig
 * That I made from:
 * Template:Infobox Polyhedron
 * with a new one that more closely matches entries in list of uniform polyhedra
 * Template:Infobox Uniform Polyhedron
 * But I also need to add some columns to list of uniform polyhedra at least including Symmetry Group, and perhaps other information.
 * Overall I'm overwhelmed just completing the uniform polyhedra stubs and pictures, and yet I see that does encourage a little more care to do things right a first time if possible.
 * We might try a more orderly coordinated approach.
 * My primary concern now is getting an agreement on data to include in:
 * Template:Infobox Uniform Polyhedron
 * There's actually somewhat of a mess now, between at least three template versions and direct coding. I'm really not sure if all uniform polyhedra need identical templates, or if it isn't better to keep a set of them between different variations. Certainly the duals can use a different template, as can the planar tilings.
 * Perhaps I'll suggest some formats and get some feedback on sample articles using templates before jumping back into article creation.
 * Tom Ruen 01:55, 17 October 2005 (UTC)

"...see that does encourage a little more care to do things right a first time if possible." You nailed it. I'm asking you to step back and consider a template that fits all named polyhedrons and tilings, not just the uniform ones. Please correct me if I'm wrong: are there characteristics that regular polyhedra have that Catalan solids don't? are there characteristics that tilings have that the tetrahemihexahedron doesn't? Only one template is needed for all polyhedra.

Again I suggest that we broaden List of uniform polyhedra into a List of polyhedra. If we don't do it now, then I won't be surprised someday someone will make a completely different stab at organising the Catalan and Johnson solids. That's another mess someday someone will need to fix.

You mentioned "Symmetry Group" -- that's another one! Please exhaust the table first, then the one template will be clearer to you. :)

"I don't understand categories in wiki, if this is something special." Please look at Category:Polyhedra and the mess it is in. Along the way, it'd be wonderful to turn Category:Polyhedra into a beautiful hierarchy (no overlaps!) of polyhedra and polyhedron articles. If you don't know how Wiki categories work, peek inside an article there. :) --Perfecto 22:32, 17 October 2005 (UTC)

P.S. What do you guys think of wikifying all named polyhedra, whether they have existing articles or not? There'll be a lot of reds now, but less work later!


 * I respect your desire for order and a beautiful hierarchy, but I'm not convinced it can be done. At least not as a tree, even as relations exist between various groupings.
 * I'm happy to support an article List of polyhedra which is would attempt a comprehensive listing of polyhedra, categorically, and individually. I would say such an article would be best done simply with: A thumbnail picture, and 1+ names below that link to a individual article. I don't know if the names can be done systematically, but this would allow different names to be listed under the same object.
 * Equally valuable would be an Index of polyhedra which would be alphabetical and possibly list the same object link multiple times under different names.
 * And even Wythoff symbol index of polyhedra and tessellations which would group polyhedra ordered by their systematic symbolic names.
 * And so on. :
 * You can see I also added another list List_of_Wenninger_polyhedron_models which lists 119 polyhedra in the numbering system used by Wenninger in his 1971 book. I included a full table like List of uniform polyhedra, mostly because I'm using it to cross reference data for correctness.
 * In the long run, such a list might be better off with minimal information because duplicates increase the likelihood of errors, and worse, partially corrected errors! (Unless there's a way of making a wikitable row reference an external description which would provide the column data(??)
 * You can put the data of each row in a separate template, for use in multiple tables. It is easiest if the format for each table is the same. If data have to be extracted and rearranged, somewhat complicated template techniques are needed.--Patrick 01:05, 18 October 2005 (UTC)
 * I also created List of uniform planar tilings as a short listing of the 11 uniform tilings and their duals. I appreciate compact list articles that can independently demonstrate different relations, rather than attempting a hierarchical structure which would be only able to relate a single hierarchy which I don't think necessarily exists.
 * On characteristic differences, definite differences between uniform polyhedra and duals at least.
 * My proposal would start by an article Index of polyhedra which would summarize the structural groups offered under polyhedron.
 * On all named polyhedra, I'd say great. Ideally I'm hoping for a scripting system to generate stubs, but maybe hand-building blank articles that say "In geometry, XXX is a polyhedron." is worth something?
 * Tom Ruen 23:07, 17 October 2005 (UTC)


 * For organizational plans, how about we create:WikiProject_Geometric_objects
 * See: Category:WikiProjects
 * Example: WikiProject_Astronomical_objects
 * It would attempt to standardize explicit named geometric objects of any dimensions.
 * Tom Ruen 23:40, 17 October 2005 (UTC)


 * I looked over the wiki geometric categories..
 * Found these, and linked them more extensively, although not comprehensively
 * Category:Polyhedra
 * Category:Platonic solids
 * Category:Archimedean solids
 * Category:Chiral polyhedra
 * Category:Catalan solids
 * Category:Johnson solids
 * Category:Deltahedra
 * Category:Tiling
 * Added
 * Category:Uniform polyhedra
 * Category:Prismatic polyhedron
 * Category:Quasiregular polyhedra
 * Category:Polyhedral net - for images
 * Category:Polyhedral image - for images
 * Seems a useful starting place to get these categories applied for further organizational work

REMOVED SECTIONS BELOW
Okay, given these text below was incomplete (bad links), I moved it below until it can be sorted out or safely deleted.

Tom Ruen 06:20, 22 February 2006 (UTC)

Also removed "Mathematics" section (moved below). It seems too general to be in this article since it applies to all convex polyhedra.

Tom Ruen 05:18, 23 February 2006 (UTC)

Non-convex quasi-regular polyhedra
Quasi-regular means vertex- and edge-uniform but not face-uniform, and every face is a regular polygon. This implies that there are two kinds of faces, and that at every edge one of each meet; and that the two kinds alternate around every vertex.

The quasi-regular polyhedra include the two convex polyhedra and 14 non-convex polyhedra (Hart): The Small dodecicosahedron has a ditrogonal vertex figure but is not edge uniform.
 * Cuboctahedron
 * Icosidodecahedron
 * 2 with a rhombic vertex figure (convex vertices with 4 adjacent faces):
 * Dodecadodecahedron
 * Great icosidodecahedron
 * 9 Hemihedra: ditrigonal vertex figures, (non-convex verticies with 4 adjacent faces):
 * Tetrahemihexahedron
 * Octahemioctahedron
 * Cubohemioctahedron
 * Small icosihemidodecahedron
 * Small dodecahemidodecahedron
 * Great dodecahemiicosahedron
 * Small dodecahemiicosahedron
 * Great dodecahemidodecahedron
 * Great icosihemidodecahedron
 * 3 triambic polyhedra (each vertex has 6 adjacent faces):
 * Small triambic icosidodecahedron (small ditrogonal icosidodecahedron)
 * Triambic dodecadodecahedron (ditrogonal dodecahedron)
 * Great triambic icosidodecahedron (great ditrogonal icosidodecahedron)

Non-convex semi-regular polyhedra
Main article Semiregular polyhedra.

The remaining uniform polyhedra are all semi-regular non-convex polyhedra and include the 17 nonconvex Archimedean solids:


 * 3 have cubic symmetry:
 * Great cubicuboctahedron
 * Cuboctatruncated cuboctahedron or cubitruncated cuboctahedron
 * Quasitruncated hexahedron or stellated truncated cube
 * 11 have icosahedral symmetry:
 * Quasitruncated small stellated dodecahedron or small stellated truncated dodecahedron
 * Quasitruncated great stellated dodecahedron or great stellated truncated dodecahedron
 * Great truncated dodecahedron
 * Great truncated icosahedron
 * Rhombidodecadodecahedron
 * Icosidodecatruncated icosidodecahedron **
 * Small ditrigonal icosidodecahedron **
 * Great ditrigonal icosidodecahedron **
 * Great quasitruncated icosidodecahedron or great truncated icosidodecahedron
 * Great dodecicosidodecahedron
 * Great ditrigonal dodecicosidodecahedron **
 * 3 have the symmetry of the snub dodecahedron:
 * Snub dodecadodecahedron
 * Great inverted snub icosidodecahedron or great vertisnub icosidodecahedron
 * Great inverted retrosnub icosidodecahedron or great retrosnub icosidodecahedron

There are 23 more semi-regular non-convex polyhedra:


 * ? have octahedral symmetry:
 * Small cubicuboctahedron
 * Small rhombicube
 * Great rhombicube
 * Great truncated cuboctahedron
 * ? have icosahedral symmetry:
 * Rhombicosahedron
 * Small rhombidodecahedron
 * Great rhombidodecahedron
 * Small dodecicosahedron
 * Great dodecicosahedron
 * Small dodecicosidodecahedron
 * Icositruncated dodecadodecahedron
 * Great truncated dodecadodecahedron
 * Vertisnub dodecadodecahedron
 * Icosidodecadodecahedron
 * Snub icosidodecadodecahedron
 * Small dodekic icosidodecahedron ??
 * Great dodekic icosidodecahedron ??
 * Small icosic icosidodecahedron ??
 * Great icosic icosidodecahedron ??
 * Great snub icosidodecahedron
 * Snub disicosidodecahedron
 * Retrosnub disicosidodecahedron
 * Great snub icosidisdodecahedron
 * Great snub disicosidisdodecahedron

TODO: Check this list for duplicates/alternate names

Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieces which can be reassembled to yield the second polyhedron. This is a version of Hilbert's third problem; the answer is "no", as was shown by Dehn in 1900.

Mathematics
Euclid was the first to show that for a convex polyhedron the vertex angles of the polygons at each vertex must add up to less than 360°. For example the angles at each vertex of a cube are 90°+90°+90°=270°<360°.

The angle defect at each vertex 360° less the angles of the adjacent polygons, for a cube this is 90°. Descartes proved that for convex polyhedron the total angular deficit for all the vertices is 720°. For a cube this is 8 × 90° = 720°.

Euler's theorem shows that for convex polyhedron V-E+F=2.

Cleanup (Feb 06)
I added the cleanup tag because this article needs more work. I'm mostly content with the listing by symmetry groups, but perhaps even that is better moved to list of polyhedra by symmetry groups, if this article was better defined.

As of now, the sections are not overly complete or rationally included. I'm always more content at simple "lists" than definitions, but if no one else wants to help here, I'll see what I can do too.

Also new stubs: I've added quick stub articles for all polyhedra as named in list of uniform polyhedra. I just have images and one line indexing them. I hope to have a table format added via User:Salix alba's templates - User:Pfafrich/test to prevent unnecessary duplication of data. Well, I've not taken time to understand it, but his tests looked promising!

Also new images: I've also got a FULL BATCH of replacement images for all 75 and prismatic forms to upload (replacing my PNG images), new ones created by Robert Webb and his Great Stella software. So far just uploaded Skilling's figure, and still wondering what licensing statement to offer for his images. (hopefully I'll have some time to upload all 90 some images this weekend.)

Tom Ruen 09:35, 22 February 2006 (UTC)

Don't know if you've noticed the change to Small dodecahemicosahedron I've now included some details on it using a template. I'm using this as a test example to check the template inclusion. You may want to have a look at it.

You may well be right on Skilling's figure. Wenninger listed it in the chapter on non-convex snub, and I always assumed it was. However looking at the image there does appear to be reflection symmetry.

In Point_groups_in_three_dimensions they only have T, Td, Th, O, Oh, I, Ih. So the other snub polyhedron should be just I and not Ih.

p.s. I've now changed my user name to User:Salix alba.


 * The uncrossed pentagram antiprism is D5h, not D5d. My intuition isn't quite good enough to assign the other star antiprisms but – if it has a reflexion plane perpendicular to the highest rotation axis it's Dnh, if all reflexion planes contain the highest rotation axis it's Dnd (or Dnv, I can never remember), if it's chiral it's Dn.  The full tetrahedral group is Td, the pyrite group is Th.  &mdash;Tamfang 17:20, 22 February 2006 (UTC)


 * User:Salix alba and Tamfang, thanks for the corrections on symmetry groups. I get stupid when I get more ambitious than I ought to be, well and they are a little confusing too!


 * Looks like star antiprisms need to be rethought on symmetry - I'm confused, but agreed crossed/not are different. I'll have to construct a few more of them to convince myself what I know or don't know.


 * I saw the added template paragraphs in Small dodecahemicosahedron - very impressive. I have no suggestions for improvement, even if wording can always be tweaked.
 * Perhaps Kaleido should be externally relinked to either of these?


 * Tom Ruen 21:39, 22 February 2006 (UTC)

By the way, I reckon that classifying by symmetry group is as good a way as any, since the presence of a symmetry group is part of the definition of uniform. &mdash;Tamfang 05:44, 23 February 2006 (UTC)

I removed the cleanup notice.

With only two sections history/list-by-symmetry, it could use a little expansion, but I'm content here. (Symmetry header linked articles are a bit messy yet with images)

You'll also notice I uploaded a full set of new images from Robert Webb and his Great Stella software. I replaced most of my png from nonconvex forms, and a few unused older convex images under png, but unreferenced.

A few of the image names were changed, but mostly just moving from jpg images. I also updated links on list of uniform polyhedra, but other pages also could be updated on newer images. (Green transparent jpgs are fine to keep as well, just a matter whether a page wants to use a consistent set.)

I also fixed up the symmetry a bit more, a number of snub forms are achiral.

Tom Ruen 07:19, 25 February 2006 (UTC)

Skilling Figure. We state this was discovered by Skilling in 1975. However my copy of Polyhedron models asserts this to Coxeter et al, being number 92 in Coxters numbering. Skillings paper, did not find the new polyhedon, instead it showed that the list was complete. --Salix alba (talk) 11:02, 25 February 2006 (UTC)

Octahedral symmetry
As an experiment, I regrouped the octahedral polyhedra by "convex hull" vertex arrangements, and named groups like polychoron defined terms.

I'm not sure if the terminology applies like this, but seems consistent anyway.

By "convex hull", I mean taking the vertex set and adding faces by the convex hull. This mostly defines a single vertex geometry, but can also define topologically similar polyhedra with nonregular faces.

Example (Common 4.6.8 topology)

Obviously the same thing can be done with the Icosahedral symmetry, although takes a bit of work!

Tom Ruen 01:06, 26 February 2006 (UTC)


 * Okay, I regrouped Icosahedral as well. I also subgrouped like "small truncated" for smaller truncations, although visually I can't tell how many categories could be created like this. ALSO I expanded the tetrahedral section similarly, including octahedron/tetrahedron with higher symmety, but also tetrahedral symmetry if different symmetry face coloring used (and created image:snub tetrahedron.png for icosahedron]] as an example too. Tom Ruen 02:59, 26 February 2006 (UTC)


 * Maybe I'm done for the day, with dihedrals as well, and two new images Image:tetragonal prism.png and image:trigonal antiprism.png. I also tried term ''irregular' for vertex arrangements with convex hull faces which are not regular polygons. Seems reasonable, even if there's DIFFERENT types/poroportions of irregulars! Tom Ruen 04:15, 26 February 2006 (UTC)

undecim, hendeca
oh the embarrassment, to get caught confusing Greek with Latin in that way, when I'm usually the pedantry police! &mdash;Tamfang 06:47, 9 August 2006 (UTC)

Subsections
I'm not sure moving the symmetry subsections "up" one level was an improvement. The major section heading indicates a subsequent listing by symmetry, so it seems reasonable for those listings to be subsections of that section. I won't revert the change, however, unless further discussion here indicates that others share me feelings. Paul D. Anderson 22:55, 3 September 2006 (UTC)


 * I agree – and like you, I wasn't gonna bother moving it back merely to please myself. &mdash;Tamfang 23:48, 3 September 2006 (UTC)


 * I agree also, held off reverting, but since I made the sections, I'll please myself and reverted it ! :) Tom Ruen 01:03, 4 September 2006 (UTC) (Not to say other improvements aren't needed!)

Cleanup/merge
All of our articles on polyhedra could use a comprehensive reworking someday. I propose that List of uniform polyhedra be merged here; its ordering doesn't match anything else anyway. The way, the truth, and the light 04:59, 22 April 2007 (UTC)


 * I definitely disagree on a merge, whatever work is needed. A list is a list, not intended for more detailed information about theory or symmetry relations. Tom Ruen 05:48, 22 April 2007 (UTC)


 * Well all the uniform polyhedra are shown in this article anyway. The distinction between 'nonconvex with convex faces' and 'nonconvex with nonconvex faces' has no justification, and I've never seen it anywhere else. The way, the truth, and the light 06:01, 22 April 2007 (UTC)


 * The ordering of the list is somewhat arbitrary. My goal was to use the template databases to make header-sorttable tables (clicking on headers jumped to different list articles), some tests at: Tom Ruen 06:51, 22 April 2007 (UTC)
 * User:Tomruen/List_of_uniform_polyhedra_and_tilings
 * User:Tomruen/List_of_uniform_polyhedra_and_tilings2

User:The way, the truth, and the light, Why did you delete forms from the prism listing? Tom Ruen 06:48, 25 April 2007 (UTC)


 * I wanted to restrict to forms having 3,4,5,6,8,10,12 faces for mathematical reasons - we don't have any pages on 9,11 anyway. This page is already long, and will be even longer if merged with the other one. Finally, I wanted to be able to have a column in the table for retroprisms, which is only possible if 7,9,11 are excluded - though as you can see, I couldn't really figure out the formatting. The way, the truth, and the light 06:53, 25 April 2007 (UTC)

I corrected the index of uniform polyhedra in this article as noted in the edit summary. I also moved the pentagrammic antiprism to D5d - while it's technically incorrect, it's probably what readers expect. Revert that part of my changes if you want. The way, the truth, and the light 06:57, 25 April 2007 (UTC)


 * I restored the original prism table. If the table is by symmetry, there's no question on where they should be, unsure why you'd say anything else. Tom Ruen 07:05, 25 April 2007 (UTC)


 * Fixed the symmetry. Still can't figure out how tables work. The way, the truth, and the light 07:11, 25 April 2007 (UTC)


 * How is it FIXED if you've still removed content I want there? Tom Ruen 07:12, 25 April 2007 (UTC)


 * I seriously don't understand why you insist on having the longer list of prisms. Also the 'retrograde' category is important, at least in my thinking. The way, the truth, and the light 07:20, 25 April 2007 (UTC)


 * It's DECEPTIVE/UNIFORMITIVE to skip inconvenient forms because they don't fit your desires. Goodnight. Tom Ruen 07:37, 25 April 2007 (UTC)


 * Ridiculous. There are an infinite number of prisms, and we can only list finitely many. So obviously we have to skip almost all of them. Calling that 'deceptive' is just wrong. The way, the truth, and the light 23:25, 25 April 2007 (UTC)


 * It's deceptive because on a first look on your table, looks like there's always only a few. A complete listing of images up to 12 is reasonable to show the variations and progressions, even/odd symmetry, etc. Tom Ruen 00:22, 26 April 2007 (UTC)

Bowers-style acronyms
I second this sentiment. Despite several attempts to find them, there appear to be no reliable sources to the words. A mailing list is not a reliable source. Wikipedia itself is not a reliable source, so the Bowers column in List of uniform polyhedra is not a source (and it is doubtful the column should exist in the first place.) Weregerbil 17:11, 18 May 2007 (UTC)


 * The websites and mailing list, I say, are enough to show notability in this field. They constitute reliable source for this purpose - showing that the terms are in use. WP:RS specifically says that common sense must be used, and that there are no absolute guidelines (except BLP issues) for what is a reliable source. Since this material is not contentious or extraordinary - and is clearly useful - the bar to inclusion should be rather low. The way, the truth, and the light 19:36, 18 May 2007 (UTC)


 * Hi, I'm sorry but the comment above, which you posted on my talk page makes no sense to me. I deleted 5 redirects to a nonexistant part of an article, which previously had had zero sources. In a general case, a mailing list isn't a reliable source, since anyone can post in one. I've no clue on what website you're talking about, so I can't comment on that. Please take the time to explain what you are talking about if you're going to cross-post things. Thanks. - Bobet 12:56, 20 May 2007 (UTC)


 * No, a person making up words and posting them on a mailing list is not "clearly useful", reliably sourced, or notable. Weregerbil 20:54, 20 May 2007 (UTC)


 * Whether a concept is 'useful' has nothing to do with the origin of the concept. Also, this concept has been used by many other people besides Bowers; that was my point. The way, the truth, and the light 04:31, 21 May 2007 (UTC)


 * No reliable sources = not on Wikipedia. Weregerbil 05:00, 21 May 2007 (UTC)


 * That claim is absurd, of course, as much of Wikipedia has no sources at all (which is not a bad thing). In any case, what is a reliable source is relative to what claim is being discussed. The key is whether it is reasonably verifiable. Here, no factual claims are being asserted (and as I stated above, BSAs are neither controversial not extraordinary) so verifiability can consist only of verifying that the things are actually used, which is what I have done.


 * There is only one Google hit other than Wikipedia and his own site, but that's simply because they are not usually referred to as 'Bowers style acronyms' when they are used. The way, the truth, and the light 05:11, 21 May 2007 (UTC)


 * That claim is policy. WP:OTHERCRAPEXISTS is not an effective argument. Self-published sources are not reliable sources, please see WP:ATT and WP:RS. Weregerbil 05:16, 21 May 2007 (UTC)


 * No, it isn't policy. WP:V doesn't actually cover nomenclature, but the closest thing it does say is 'Material from self-published sources ... may be used in articles about themselves' - in the case of naming conventions, every source is writing about themselves, as the only claim is to the use of the names itself. WP:RS specifically says that common sense should override it. As for WP:OTHERSTUFFEXISTS, I think that it may often be valid of specific articles, but when used as I did, to apply to all of this encyclopedia, it's absurd. The way, the truth, and the light 05:32, 21 May 2007 (UTC)


 * There has been no evidence presented that these "acronyms" are used outside of Bowers' sites and Wikipedia (and Wikipedia mirrors).
 * At best, 'Material from self-published sources ... may be used in articles about themselves' means that Bowers' acronyms could be used in the Bowers article, if they were sufficiently notable there.
 * "Clearly useful", although false, is not relevent. If the acronyms were used in a peer-reviewed published paper, or by a recognized expert in the field of higher-dimensional polytopes, such as Coxeter (but not Jonathan Bowers), there might be a case to be made for notability.
 * &mdash; Arthur Rubin | (talk) 06:45, 21 May 2007 (UTC)


 * Bowers is not himself notable, so his acronyms wouldn't be if he were the only one using them. Examples of use by others were given in the AfD discussion, and Tom Ruen has given them below.


 * I don't see how you can judge that they aren't useful when the people that work with them find them so. To be fair, I did say 'clearly' useful, and that may be an overstatement. The way, the truth, and the light 18:50, 21 May 2007 (UTC)


 * Also, by retracting your argument that it's not 'clearly useful' or notable in its field, you are conceding those claims. The way, the truth, and the light 05:32, 21 May 2007 (UTC)


 * Re: Also, by retracting your argument: no I'm not retracting anything. I am merely avoiding re-re-re-restating the same thing over and over again. Please do not make up unhelpful rules of discussion, or misrepresent the position of others. Thank you. Weregerbil 17:00, 21 May 2007 (UTC)


 * I'm not making anything up. You, in fact, ceased to defend those claims. I don't think you are qualified to evaluate their being useful or notable if you have no experience with polytopes. The way, the truth, and the light 18:50, 21 May 2007 (UTC)


 * I ceased to repeat the same thing. If you interpret that as anything other than me getting bored repeating the same thing, you are simply wrong. As wrong as I would be if I claimed that you have stopped believing everything you have written previously but did not defend in your 18:50, 21 May 2007 message. This is getting pretty silly, huh...? Weregerbil 19:07, 22 May 2007 (UTC)


 * I can't read your mind. I talked about usefulness and notability and you replied with 'no reliable sources'. That's what I saw and that's what I had to reply to. The way, the truth, and the light 00:54, 23 May 2007 (UTC)


 * My 2 cents - I created the table List of uniform polyhedra, and someday(?!) will substitute a template/database version like User:Tomruen/List_of_uniform_polyhedra_and_tilings2 which will make it easier to add/remove column data. I removed references to BSA from everywhere else when the original article was deleted. I hope Bowers or another will be interested in publishing their usage. They get more useful in the higher uniform polytopes when trying to describe them. My PRIMARY reason for including them at all was that I wanted a quick translation table when reading about polytopes from people who use them on a polyhedron email list. I tried to include ALL the different names/indexing I could, partly just for cross-referencing and error correction. (For my own use, I also have an excel list from Richard Klitzing of all the names defined up to 8 dimensions, only cross referenced to email dates from the list.) Anyway, I won't fight here, although I think Wikipedia is somewhat inconsistent - popular culture can quote gossip rags as gospel, while harmless useful terminology is neglected in mathematics. YES, I understand Wikipedia isn't intended to help make terminology popular, although it seems a worthy goal as I sort through books all using different names and notations.


 * Tom Ruen 18:33, 21 May 2007 (UTC)

For reference, online sources I know include:
 * 1)  Roger Kaufman
 * 2)  Richard Klitzing
 * 3)  Jim McNeill
 * 4)  Robert Webb, Stella (software)
 * 5)  George Olshevsky - graphically lists the convex uniform polyhedra with Bowers names
 * 6) Symmetry: Culture and Science, Vol. 11, Nos. 1-4, 139-181, 2000


 * Tom Ruen 21:43, 23 May 2007 (UTC), Tom Ruen (talk) 22:01, 29 August 2008 (UTC)

grammar cleanup
I saw some minor errors with number (singular/plural) and possessive in the large table under Convex forms and fundamental vertex arrangements, and what seemed to be a cut-and-paste error, and I fixed them; but I'd forgotten to sign in. Some geometer, please make sure my changes didn't introduce errors. -- Thnidu (talk) 00:57, 15 June 2008 (UTC)

Reworking sectioning
I wanted to expand the Wythoff construction of the convex forms so I moved the nonconvex forms to a subarticle. This is reasonable since there's very limited explanation of what the nonconvex forms are at all so far.

Currently it's a bit of a mess while I rework it. It might be better to move the convex forms also to a subarticle and make this article a shorter summary. So, I'll see what I can do in the coming days to get it cleaned up in a better way.

Tom Ruen (talk) 20:46, 31 July 2008 (UTC)

Stereographic projection
I'm working on some stereographic projections of the convex uniform polyhedra (as spherical tilings), and got caught short on time for now. I still need to clean up and decide how to present, but thought I'd share a summary (quick&messy) compilation here since they are pretty:
 * [[Image:Uniform polyhedron stereographic projections.png|600px]]

Tom Ruen (talk) 18:19, 14 August 2008 (UTC)
 * Columns left to right are: Regulars, truncations, rectifications, cantellations, and omnitruncations.
 * The regular forms by row from top to bottom are: tetrahedron, cube, octahedron, dodecahedron, icosahedron.
 * The rectified, cantellated, omnitruncated forms are presented from two different face centers for the regular-dual pairs.
 * There's no snubs yet, or prisms. The faces are colored to match the uniform polyhedra at Uniform_polyhedron.
 * I'll also make some Schlegel diagrams too (linear edges).
 * It's funny to me I've not seen these complete projections anywhere. The duals are also pretty, but single colored by face symmetry.
 * A large part of my motivation is comparison with the uniform polychora which can only be displayed as 3d projections.
 * For example, I started a 3d/4d comparison at: 120-cell


 * Hm, some of the outermost arcs look noncircular; did you fudge them so as not to have to reduce the inner faces too much? &mdash;Tamfang (talk) 04:21, 22 August 2008 (UTC)


 * There's no fudging the lines, pure computation, although I did "visually center" the orientations of the projection so the symmetrical arcs will not be identical. I agree many of the arcing edges seem to be elliptical. I'll check the calculations to see if something is wrong, but pretty simple so I'm not sure. Tom Ruen (talk) 23:14, 23 August 2008 (UTC)


 * I looked at my code and found I DID have a "fudge factor". This perspective is actually slightly above the sphere (by 1% of the radius). Apparently this squishes the outer most edge arcs in the largest forms. I'll reduce the factor to 0.01% for my "real" uploads later. The tolerance is mainly for "vertex-first" projections, so everything is in front of the view. Tom Ruen (talk) 21:58, 24 August 2008 (UTC)

another complaint from the language purist
I disapprove of "[p,q]-hedron", which I guess is yet another example of using hedron ('face') as a synonym for polyhedron ('[thing with] many faces'). A {p,q} does not have [p,q] faces, whatever that would mean. —Tamfang (talk) 20:00, 24 September 2010 (UTC)

Okay, removed. I was trying to make a summary of terminology between Kepler's names and Johnson's, but wasn't clear enough as an abstract comparison. Tom Ruen (talk) 10:30, 26 September 2010 (UTC)

I'd like to find my polyhedron!
I've now spent something like two hours trying to find my specific polyhedron. It's *really* hard when you start with descriptive qualities. And I know this isn't really the place where I should start discussing the issue, this being far too specific. But please indulge me, and help the issue along, since I probably couldn't carry this much further myself.

As background, my problem was that I would have liked to find the precise polyhedron which you get by twisting the bottom of a cube 45 degrees against its top. I *know* it has a name, and quite a lot of theory associated with it as well. It's bound to be on Wikipedia, somewhere. Yet I don't seem to be finding it. Evenas it's pretty crucial wrt an audio technology post I'm right now trying to respond to.

As such, polytopes should in mind be more easily findable. Better indexed. I suggest all of the polytopes/commonly-known-simplex-complexes should be thoroughly and visibly indexed by a) vertex count, b) edge count, c) face count, d) all of the higher order counts for higher dimensional simplex-complexes, e) the usual topological integer invariants such as the Euler one, but also a dozen more, where applicable, f) by the name of the attached symmetry group, e) systematically crosslinked by known derivatives/duals/biduals/twisted forms/whatnot that we know from literature. And so on. I think the topics should be systematically tagged, so as to be systematically crosslinked using automatic means, when the time comes.

My point is that most of this sort of tagging could be done purely mechanically. Most of the crosslinking could be about the same thing, if computationally heavy. And yet, it could help countless people find their very basic, nicest mathematical abstraction in the end -- I mean, my rant started with a fucking platonian solid, and didn't go beyond its slightly twisted form. This is stuff that a first grader with a cubical piece of Play-Doh could want to know about... Decoy (talk) 03:11, 29 July 2011 (UTC)


 * Try square antiprism. It is listed in this article, section Uniform_polyhedron. SockPuppetForTomruen (talk) 01:30, 30 July 2011 (UTC)


 * What, by the way, is a bidual? —Tamfang (talk) 08:47, 23 August 2011 (UTC)

Animated gif
I have created and uploaded this 19 [sic]-second animation showing the convex uniform polyhedra: I had in mind that it could be used on this page. I'm afraid the dithering isn't nice (it was done by the encoder, ffmpeg). I could try re-rendering with a more restricted range of shades, or flat-shade it instead. It could take a different path through the graph of truncations/cantellations/snubifications, if anyone has an opinion on that. Buster79 (talk) 19:00, 10 March 2015 (UTC)