Talk:Semiregular polyhedron

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This article contains a summary description of convex semiregular polyhedra. Semiregular polyhedra are a subset of Uniform polyhedrons which include concave forms. I find it useful to keep this more classical "convex" only article.

I admit this article could use some work - cutting and adding.

Tom Ruen 06:28, 16 October 2005 (UTC)

[edit] Reworked

I made a bold reduction of this article. Since most of the content was repeated elsewhere, I figured a short opening with links was good. The unique content, enumerating existence by vertex figure was retained, even if I'm unsure if it belongs here (Since it applies to regular polyhedra, and uniform tilings as well.)

On the bad side, I also removed most of the picture! But easy to add back from either polyhedron or uniform polyhedron!

One 'alternative approach is to move symmetry information to here from the section Uniform_polyhedron#Convex_forms_and_fundamental_vertex_arrangments. This is more interesting than the vertex figure enumerations!

Any other ideas are welcome, including votes to delete (or move existence) section.

Tom Ruen 09:52, 14 March 2006 (UTC)

I agree there's a lot of unnecessary repetition in polytope articles. Kudos for taking that bull by the horns. —Tamfang 23:42, 14 March 2006 (UTC)

[edit] Article title

Is there any objection to moving this article (back) to Semiregular polyhedron? I see it was moved earlier this year by MathBot, but it now violates the "singular title" guideline and is now the only polyhedron article with "-hedra" in the title rather than "-hedron". If there was no express purpose in changing the name I suggest changing it back. Any objections?

I'll wait a few days and if no one objects I will make the change. andersonpd 01:22, 3 August 2006 (UTC)

Support. Tom Ruen 02:14, 3 August 2006 (UTC)

Yes, obviously. Tamfang 04:26, 3 August 2006 (UTC)

Sure, let me make the change, then tell me about the nine-bazillion links to "Semiregular polyhedra". Stupid prisms and antiprisms! But seriously, I apparently misrepresented the move by MathBot -- that move was from "Semi-regular polyhedra" to "Semiregular polyhedra", not from "Semiregular polyhedron". My bad. andersonpd 05:32, 4 August 2006 (UTC)

Appreciate it, and now I know better to singular naming, although I think list of uniform polyhedra ought to stay plural, right?! Tom Ruen 05:56, 4 August 2006 (UTC)

What you mean? "List" are singular. —Tamfang 06:10, 4 August 2006 (UTC)

You're probably making funs of me. Tom Ruen 07:00, 4 August 2006 (UTC)

[edit] Definition

I find the definition given here confusing, especially in light of comparison with the text of Uniform polyhedron and Regular polyhedron. Here it says

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.

Nothing is said about convexivity, but it goes on to enumerate only convex polyhedra and says

These 4 sets compose the convex polyhedra, along with a set of 53 nonconvex forms compose the larger set of uniform polyhedra.

(which aside from anything else is ungrammatical). The implication seems to be that convex polyhedra satisfying the above requirements are semiregular, and these plus the nonconvex polyhedra satisfying those requirements are uniform. (Convex) regular polyhedra are included among the semiregular polyhedra.

But in Uniform polyhedron it says

A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. Furthermore, for every two vertices there is an isometry mapping one into the other, so there is a high degree of reflectional and rotational symmetry. Uniform polyhedra are regular or semi-regular but the faces and vertices need not be convex.

The definition seems to be the same as for semiregular polyhedra, other than that nothing is said about how many faces may share an edge. The implication seems to be that semiregular polyhedra are those (convex or not) which are uniform but not regular although this is not stated categorically.

Meanwhile in Regular polyhedron, convexivity is not required, and they are not said to be a subset of the semiregular polyhedra.

This kind of confusion seems not limited to Wikipedia. MathWorld says

A polyhedron or plane tessellation is called semiregular if its faces are all regular polygons and its corners are alike (Walsh 1972; Coxeter 1973, pp. 4 and 58; Holden 1991, p. 41). The usual name for a semiregular polyhedron is an Archimedean solid, of which there are exactly 13. In addition, a prism or antiprism is considered semiregular if all its faces are regular polygons.

which first suggests the term is synonymous with Archimedean solid, and then undermines that suggestion by adding in prisms and antiprisms, but not nonconvex polyhedra. And here it says

The uniform polyhedra are polyhedra with identical polyhedron vertices. Badoureau discovered 37 nonconvex uniform polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55). The uniform polyhedra include the Platonic solids and Kepler-Poinsot solids.

Coxeter et al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meet at an polyhedron edge, and this was subsequently proven. In addition, there are two polyhedra in which four faces meet at an edge, the great complex icosidodecahedron and small complex icosidodecahedron (both of which are compounds of two uniform polyhedra). Finally, the five pentagonal prisms can also be considered uniform polyhedra.

which seems utterly garbled -- nothing is said about regularity of faces, and the pentagonal prisms, but no other prisms, are thrown in without explanation.

Many writers seem to regard "semiregular polyhedra" as a synonym for "Archimedean solid". Encyclopedia Britannica's article on Archimedes (subscription required) refers to

the 13 semiregular (Archimedean) polyhedra (those bodies bounded by regular polygons, not necessarily all of the same type, that can be inscribed in a sphere)

with no other references to semiregular or uniform polyhedra that I can find. Of course the above is wrong: the Archimedean polyhedra are not the only regular-faced polyhedra (even limiting to convex) that can be inscribed in a sphere.

In "Uniform Polyhedra" (JSTOR subscription required) (Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 246, No. 916. (May 13, 1954), pp. 401-450) Coxeter et al. say

A polyhedron is said to be uniform if its faces are regular while its vertices are all alike. By this we mean that one vertex can be transformed into any other by a symmetry operation.

and then, rather surprisingly, semiregular polyhedra seem to be defined as those with Wythoff symbol of the form p q | r, encompassing prisms but not antiprisms, only 7 of the Archimedean solids, and numerous nonconvex polyhedra. I don't think I've seen that definition used anywhere else, but one can't disregard Coxeter on the subject of polyhedra, can one?

In summary, from what I've read, it looks as though "semiregular polyhedra" does not have a commonly accepted definition, unless it's that it's a synonym for Archimedean solid. If I've gotten the wrong impression I'd be happy to learn otherwise, but I'm thinking this article at the very least should (1) clarify its definition and be consistent with the Uniform polyhedra article and (2) acknowledge other definitions. -- Rsholmes 15:22, 14 December 2006 (UTC)