2011/03/01 by P. K. Aravind · 1 citation
Mathematics · #Mathematics and Applications #Sphericity #Dual polyhedron #Polyhedron #Mathematics #Isoperimetric inequality #Quotient #Regular polygon #Combinatorics #Vertex (graph theory) #Pure mathematics #Algebraic geometry #Geometry #Graph
paper · doi:10.4169/college.math.j.42.2.098
openalex publication_date 2011/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
SummaryThe Isoperimetric Quotient, or IQ, introduced by G. Polya, characterizes the degree of sphericity of a convex solid. This paper obtains closed form expressions for the surface area and volume of any Archimedean polyhedron in terms of the integers specifying the type and number of regular polygons occurring around each vertex. Similar results are obtained for the Catalan solids, which are the duals of the Archimedeans. These results are used to compute the IQs of the Archimedean and Catalan solids and it is found that nine of them have greater sphericity than the truncated icosahedron, the solid which serves as the geometric framework for a molecule of C-60, or “Buckyball.”