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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. |
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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). |
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Incidentally, you may be interested in the article by
Joseph Malkevitch,
"Milestones in the history of polyhedra,"
which appeared in
Shaping Space: A Polyhedral Approach,
Marjorie Senechal and George Fleck, editors,
pages 80-92. 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.
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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. |
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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. |
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My proof can be found here: ywhmaths.webs.com/Geometry/ArchimedeanSolids.pdf |
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