An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex

The dihedral angle of a regular n-dimensional simplex is the angle formed by a pair of intersecting faces. In the case n = 2 we are describing the angle at the vertex of an equilateral triangle, while in the case n = 3 we are describing the angle formed by the faces of the regular tetrahedron. (This angle is cos-'(1/3), or approximately 70.5?.) In the general n-dimensional case, the dihedral angle is known to equal cos-'(l/n) (see Coxeter [1]). This fact has been proved elegantly by R. Krasnodebski in [3]. Krasnodebski bases his work on a construction used by Coxeter in proving Gosset's theorem (see Coxeter [2]), so Krasnodebski's proof is neither self-contained nor elementary. As far as we know, there is no simple, elementary proof in print. om the sual generati g function fo the Stirling numbers of the second kind,