Gravitational collapse and moment of inertia of regular polyhedral configurations

Abstract

This paper proposes and solves two problems in classical mechanics that are tied closely to the geometry of the Platonic solids. The problems are: (1) to calculate the gravitational collapse time of a set of identical point masses released from rest from the vertices of a regular polyhedron; and (2) to calculate the moment of inertia (about any axis through the center) of a uniform body in the shape of one of the Platonic solids. Dimensionless figures of merit are introduced to gauge the relative performance of the Platonic solids in each of the problems studied. These figures of merit are similar to the isoperimetric quotient introduced by George Polya a long time ago. The rhombic dodecahedron and triacontahedron are also considered, and it is shown that the triacontahedron surpasses the performance of the Platonic solids in two of the problems examined.