1954/05/13 by H. S. M. Coxeter, M. S. Longuet‐Higgins, J. C. P. Miller · 15 citations
Mathematics · Physics and Astronomy · #Mathematics and Applications #History and Theory of Mathematics #Advanced Mathematical Theories and Applications
paper · doi:10.1098/rsta.1954.0003
openalex publication_date 1954/05/13 · openalex created_date 2020/07/02 · openalex updated_date 2026/07/22
Abstract 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.