Convex Polyhedra with Regularity Conditions and Hilbert’s Third Problem

This book develops in a systematic way the various regularity conditions imposed on the faces and solid angles of a convex polyhedron, which give rise to the well known sets of solids such as the five bodies of Plato (the regular polyhedra), the thirteen bodies of Archimedes (the semi-regular polyhedra), the eight deltahedra, the ninety-two regular faced polyhedra, etc. Groups of rotations/reflections of the five regular solids, as also their metrical properties are briefly treated. Interesting applications to Pick’s theorem and plane tessellations are given. Finally, a solution to a classical problem of polyhedral decompositions (Hilbert’s third problem) is given in the last chapter.