Quasiregular Polyhedron - Quasiregular Duals

Quasiregular Duals

Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals must be quasiregular too. But not everybody believes this to be true. These duals are transitive on their edges and faces (but not on their vertices); they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above:

1. The rhombic dodecahedron, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
2. The rhombic triacontahedron, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.

In addition, by duality with the octahedron, the cube, which is usually regular, can be made quasiregular if alternate vertices are given different colors.

Their face configuration are of the form V3.n.3.n, and Coxeter-Dynkin diagram

These three quasiregular duals are also characterised by having rhombic faces.

This rhombic-faced pattern continues as V3.6.3.6, the rhombille tiling.