Quasiregular Polyhedra

Quasiregular Polyhedra

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 step closer to regularity than the semiregular which are merely vertex-transitive.

There are only two 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 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).

More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing r (2 or more) instances of the faces around the vertex.

Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)2. Or more generally, (p.q)2, with 1/p+1/q<1/2.

Some regular polyhedra and tilings (those with an even number of faces at each vertex) can also be considered quasiregular by differentiating between faces of the same number of sides, but representing them differently, like having different colors, but no surface features defining their orientation. A regular figure with Schläfli symbol {p,q} can be quasiregular, with vertex configuration (p.p)q/2, if q is even.

The octahedron can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), (3_a.3_b)2, alternating two colors of triangular faces. Similarly the square tiling (4_a.4_b)2 can be considered quasiregular, colored as a checkerboard. Also the triangular tiling can have alternately colored triangle faces, (3_a.3_b)3.