A # quasi-regular polyhedron

A quasi-regular polyhedron is a polyhedron that consists of two sets of regular polygons, m-sided and n-sided respectively, and is constructed so that each polygon in one set is surrounded by members of the other set. There are three convex quasiregular solids: the cuboctahedron (m = 3, n = 4), the icosidodecahedron (m = 3, n = 5), and the octahedron (m = n = 3). In each case four faces meet at each vertex in the cyclic order (m, n, m, n). Because of this, these polyhedra have some special properties, one of which is that their edges form a system of great circles. The edges of the octahedron form three squares; the edges of the cuboctahedron form four hexagons, and the edges of the icosidodecahedron form six decagons. Among the nonconvex polyhedra are two examples of type (m, n, m, n): the dodecadodecahedron (m = 5, n = 5/2) and the great icosidodecahedron (m = 3, n = 5/2), which can be made by truncating the Kepler-Poinsot polyhedra at their edge midpoints. There are also three nonconvex examples of type (m, n, m, n, m, n): the small triambic icosidodecahedron (m=3, n=5/2), the triambic dodecadodecahedron (m = 5/3, n = 5), and the great triambic icosidodecahedron (m = 3, n = 5). Finally, there is a group of nine "hemihedra," in which some faces pass through the polyhedron's center. These hemi faces each cut a sphere into two hemispheres.