A ZETA-FUNCTION APPROACH TO THE RELATION BETWEEN THE NUMBERS OF SYMMETRY PLANES AND AXES OF A POLYTOPE

A derivation of the Cesaro-Fedorov relation from the Selberg trace for mula on an orbifolded 2-sphere is elaborated and extended to higher di mensions using the known heat-kernel coefficients for manifolds with p iecewise-smooth boundaries. Several results are obtained that relate t he coefficients, b(i), in the Shephard-Todd polynomial to the geometry of the fundamental domain. For the 3-sphere, it is shown that b4 is g iven by the ratio of the volume of the fundamental tetrahedron to its Schlafli reciprocal.