Quasiregular polyhedron
A polyhedron which has regular faces and is transitive on its edges is said to be quasiregular.
A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex. There are three convex quasiregular polyhedra:
- The cuboctahedron
- The icosidodecahedron
- The octahedron, which is also regular
Each of these forms the common core of a dual pair of regular polyhedra. The names of the first two listed give clues to the associated dual pair, respectively the cube + octahedron and the icosahedron + dodecahedron. The octahedron is the core of a dual pair of tetrahedra (an arrangement known as the stella octangula).
Coxeter, H.S.M. et.al. (1954) also classify certain star polyhedra having the same characteristics as being quasiregular.
Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals must be quasiregular too. But not everybody accepts this view. These duals have regular vertices and are transitive on their edges. They are, in corresponding order as above:
- The rhombic dodecahedron
- The rhombic triacontahedron
- The cube, which is also regular
References
- Coxeter, H.S.M. Longuet-Higgins, M.S. and Miller, J.C.P. Uniform Polyhedra, Philosophical Transactions of the Royal Society of London 246 A (1954), pp. 401-450.
- Cromwell, P. Polyhedra, Cambridge University Press (1977).