Quasiregular polytope
In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence a step closer to regular polyhedra than the semiregular which are merely vertex-transitive. The dual figures is also sometimes considered quasiregular, except that they are edge-transitive, face-transitive, and alternate between two regular vertex figures.
There are only two regular convex quasiregular polyhedra, the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing their faces contain all the faces of the dual-pair cube and octahedron, in the first, and the dual-pair icosahedron and dodecahedron in the second case.
These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol or r{p,q} to represent their containing the faces of both the regular {p,q} and dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)2).
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