Closed-form expressions for uniform polyhedra and their duals
This article contains a nice account of the uniform polyhedra and their duals. A uniform polyhedron is one whose faces are regular polygons, and whose symmetry group acts transitively on its vertices. There are 77 kinds of uniform polyhedra, being the 5 Platonic solids, the 13 Archimedian solids, the 4 Kepler-Poinsot star polyhedra, 53 non-regular star polyhedra, and 2 infinite families, the prisms and the antiprisms. With one exception, the uniform polyhedra can be described by their Wythoff symbols. The Wythoff symbol is a permutation of three rational numbers and a vertical bar `\(|\)' which divides the rationals into two groups. The rational numbers are chosen from \(\{2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, 5/4\}\) (except in the cases of the prisms and antiprisms), with the numerators 4 and 5 never appearing in the same Wythoff symbol. Thus 2~3 \(|\) 5/4 and \(|\) 3/2 4 4 are valid Wythoff symbols corresponding to particular uniform polyhedra. Often, different Wythoff symbols lead to the same uniform polyhedra. The article under review gives numerous new formulas for the metrical properties (angles, radii, etc.) of the uniform polyhedra and their duals in terms of their Wythoff symbols, including the single non-Wythoff uniform polyhedron. The appendices include a complete table for important metrics of snub polyhedra, and a table listing all uniform polyhedra along with the types of their faces. There are 42 references to other works, making this article a good starting point for anyone interested in performing an in-depth study on the uniform polyhedra (or some subset of them), and especially useful to those interested to visualise or construct models of these polyhedra.
scientific article; zbMATH DE number 1747946