A zeta function approach to the relation between the numbers of symmetry planes and axes of a polytope

Abstract

A derivation of the Cesàro–Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-smooth boundaries. Several results are obtained that relate the coefficients, bi, in the Shephard–Todd polynomial to the geometry of the fundamental domain. For the 3-sphere, it is shown that b4 is given by the ratio of the volume of the fundamental tetrahedron to its Schläfli reciprocal.