# 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.