The complete set of uniform polyhedra

A definitive enumeration of all uniform polyhedra is presented (a uniform polyhedron has all vertices equivalent and all its faces regular polygons). It is shown that the set of uniform polyhedra presented by Coxeter, Longuet-Higgins & Miller (1953) is essentially complete. However, after a natural extension of their definition of a polyhedron to allow the coincidence of two or more edges, one extra polyhedron is found, namely the great disnub dirhombidodecahedron.

Uniform polyhedra

Uniform polyhedra have regular faces meeting in the same manner at every vertex. Besides the five Platonic solids, the thirteen Archimedean solids, the four regular star-polyhedra of Kepler (1619) and Poinsot (1810), and the infinite families of prisms and antiprisms, there are at least fifty-three others, forty-one of which were discovered by Badoureau (1881) and Pitsch (1881). The remaining twelve were discovered by two of the present authors (H.S.M.C. and J.C.P.M.) between 1930 and 1932, but publication was postponed in the hope of obtaining a proof that there are no more. Independently, between 1942 and 1944, the third author (M.S.L.-H.) in collaboration with H. C. Longuet-Higgins, rediscovered eleven of the twelve. We now believe that further delay is pointless; we have temporarily abandoned our hope of obtaining a proof that our enumeration is complete, but we shall be much surprised if any new uniform polyhedron is found in the future. We have classified the known figures with the aid of a systematic notation and we publish drawings (by J.C.P.M.) and photographs of models (by M .S.L.-H.) which include all those not previously constructed. One remarkable new polyhedron is contained in the present list, having eight edges at a vertex. This is the only one which cannot be derived immediately from a spherical triangle by Wythoff's construction.