ABSTRACT
By their nature, facts do not change. However, our interpretation of facts changes quite often, frequently due to changes in definitions. Uniform polyhedra (also called Archimedean by some-but we shall come back to this later), that is, polyhedra with regular polygons as faces, and with all vertices in a single orbit under symmetries of the polyhedron, have been studied for a long time. The fact that the family of convex uniform polyhedra consists-besides the regular polyhedra-of the infinite families of prisms and antiprisms together with thirteen individual polyhedra, has been established countless times. In contrast, the enumeration of all uniform polyhedra, convex and nonconvex, has been carried out only gradually, and much more recently. Only in 1953 was the complete list published [1], without a claim of completeness. In fact, that enumeration was proved to be complete in [15] and [14]; a different approach to the enumeration and a proof of completeness is reported to be contained in [16]—unfortunately, I have not had the opportunity to see this work, and probably would not have been able to overcome the language barrier in any case. Illustrations and data can be found in [1], [10] and [17]. However, these “facts” should be replaced by new ones as soon as more inclusive definitions are accepted for regular polygons, for polyhedra and for their symmetries.
