Talk:Uniform polyhedron

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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.

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[edit] 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#Quasi-regular_duals , 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? —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. —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

  • 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

I've created a page grouping the polyhedra by their wythoff symbom.

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)

[edit] 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)


[edit] 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)

[edit] 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.

[edit] 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:

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

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.

[edit] 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.

[edit] 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#The_seven_remaining_point_groups 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. —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? [1] [2]
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. —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)

[edit] 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)

[edit] undecim, hendeca

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

[edit] 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. —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!)
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