On the number of Archimedean solids Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)?
I have seen a couple of algebraic discussions but no true proof. Also, I am looking more at trying to prove it topologically, but for now, any resource will help.*
*I worked on this project a bit as an undergraduate and am just now getting back into it.
 A: I use a slightly different approach than Cromwell.  Please see the Exercises at the end of Chapter 5 here:  http://staff.imsa.edu/~vmatsko/pgsCh1-5.pdf.
This is a draft of a textbook I am writing, and currently using to teach a course on polyhedra.  The level of the text is mid-level undergraduate, so strictly speaking, the Exercises are really an outline of a rigorous enumeration.  Symmetry considerations are glossed over.
A: My proof can be found here: http://ywhmaths.webs.com/Geometry/ArchimedeanSolids.pdf 
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A: A proof of the enumeration theorem for the Archimedean solids (which basically dates back to Kepler) can be found in the beautiful book "Polyhedra" by P.R. Cromwell (Cambridge University Press 1997, pp. 162-167).
A: Incidentally, you may be interested in the article by 
Joseph Malkevitch,
"Milestones in the history of polyhedra,"
which appeared in

Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination,
  Marjorie Senechal, editor,
  pages 53-63. Springer, 2013. (Earlier edition: Birkhauser, Boston, 1988).

There he makes the case (following Grünbaum) that there should be 14 Archimedean
solids rather than 13, including the pseudorhombicuboctahedron as the 14th.


A: Following up on Joseph's comment:  Branko Grünbaum and others have pointed out that besides the 13 or 14, there are also two infinite families of polyhedra meeting the definition of Archimedean, although generally not considered to be Archimedean.  Why prisms and antiprisms are excluded from the list has never been clear to me.
In any case, this is not just a historical curiosity --- in any attempt you make to classify them, you should run into these two infinite families.
If you use a modern definition, i.e. vertex-transitive, then you will also get 13 others.  And a little group theory can help in the classification.  If you use a more classical definition, i.e. "locally vertex-regular," you will indeed find a 14th.
A: I write a program to find all of the sets of regular polygons whose angles sum to less than 360 degrees. I use the fact that there are between 3 and 5 such polygons. The program has nested loops and they run like a speedometer. This spits out all such sets including the antiprisms and prisms. Then using the local conditions on these results. e.g.  3.m.n is only possible at a vertex if m equals n. This knocks out all but the constructable ones. But existence proofs are necessary for these. One can use truncations and snubbing arguments to finish the proofs. I have some nice proofs for some of these using the principle of continuity.
This is a rough outline. If you would like to see the output of the program, I would be happy to send it. Also a copy of the program written in qb64 can be made available.
Gerson Sparer PhD 
