Archimedean solid

In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the semi-regular convex polyhedrons composed of regular polygons meeting in identical vertices, excluding the 5 Platonic solids (which are composed of only one type of polygon) and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

"Identical vertices" means that for any two vertices, there is a global isometry of the entire solid that takes one vertex to the other. Branko Grünbaum (2009) pointed out a widespread error in the literature on Archimedan solids: some authors only require that the faces that meet at one vertex be related by a local isometry to the faces that meet at any other vertex, and incorrectly claim that there are 13 solids satisfying this definition. There are 14, because the elongated square gyrobicupola (pseudo-rhombicuboctahedron) is the unique convex polyhedron that has regular polygons meeting in the same way at each vertex, but that does not have a global symmetry taking any vertex to any other vertex. Because of this, Grünbaum suggested that the elongated square gyrobicupola should be counted as an Archimedean solid, which would give 14 Archimedean solids, but most authors (including Archimedes himself) do not include it in their lists of 13 Archimedean solids.

Prisms and antiprisms, whose symmetry groups are the dihedral groups, are generally not considered to be Archimedean solids, even though their faces are regular polygons and their symmetry groups act transitively on their vertices. Excluding these two infinite families, there are 13 Archimedean solids. All the Archimedan solids (but not the elongated square gyrobicupola) can be made via Wythoff constructions from the Platonic solids with tetrahedral, octahedral and icosahedral symmetry.

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