# Platonic Solids

33 32 the archImedean SolIdS thirteen semiregular polyhedra The thirteen Archimedean Solids opposite are the subject of much of the rest of this book. Also known as the semiregular polyhedra, they have regular faces of more than one type, and identical vertices. They all fit perfectly within a sphere, with tetrahedral, octahedral or icosahedral symmetry. Although their earliest attribution is to Archimedes 287 BC 212 BC, Kepler seems to have been the first since antiquity to describe the whole set of thirteen in his Harmonices Mundi. He further noted the two infinite sets of regular prisms and antiprisms examples below which also have identical vertices and regular faces. Turn one octagonal cap of the rhombicuboctahedron by an eighth of a turn to obtain the pseudorhombicuboctahedron below. Its vertices, while surrounded by the same regular polygons, are of two types relative to the polyhedron as a whole. There are fiftythree semiregular nonconvex polyhedra, one example being the dodecadodecahedron below. Together with the Platonic and Archimedean Solids, and the KeplerPoinsot Polyhedra, they form the set of seventyfive Uniform Polyhedra.