Mean distances on a sphere

In this article we estimate such mean distances for various choices of m points, including vertex sets of the five regular polyhedra. Surprisingly, these appear to be new. For example, given a regular tetrahedron inscribed in the unit sphere, the mean distance to the nearest vertex is approximately 0.690737280538164. In the final section we consider distance as a probability random variable, of which “expected value” is a synonym for mean distance.