Self-Intersecting Quasi-Regular Duals

The dual of a quasi-regular polyhedron is face-transitive and edge-transitive but not vertex-transitive. In this context, transitivity means that for any two faces (edges, vertices) of the polyhedron, there exists a translation, rotation, and/or reflection that leaves the outward appearance of the polyhedron unchanged yet moves one face (edge, vertex) to the other. This definition implies that a quasi-regular dual must have two kinds of regular vertices, where each vertex of one type is connected along all of its edges to vertices of the other type. There are only two quasi-regular duals that are not self-intersecting, namely the Rhombic Dodecahedron and the Rhombic Triacontahedron. When self-intersection is allowed, there are 14 other quasi-regular duals, but only five of these are finite. The nine versi-regular polyhedra, due to their faces that pass through their centers, produce duals with vertices at infinity.