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A derivation of the Ces\`aro-Fedorov relation from the Selberg trace formulaon an orbifolded 2-sphere is elaborated and extended to higher dimensions usingthe known heat-kernel coefficients for manifolds with piecewise-linearboundaries. Several results are obtained that relate the coefficients, $b_i$,in the Shephard-Todd polynomial to the geometry of the fundamental domain. Forthe 3-sphere we show that $b_4$ is given by the ratio of the volume of thefundamental tetrahedron to its Schl\"afli reciprocal.Comment: Plain TeX, 26 pages (eqn. (86) corrected

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