This question is inspired by projects like the National Ignition Facility (NIF), which have to arrange a fixed number of points on a sphere in as uniform a way as possible. In NIF's case, the points correspond to laser beams incident on a pellet of fusion fuel (for the direct drive configuration). Obviously, it's not possible to arrange the points to be totally symmetric unless the number of points happens to be the number of vertices of a Platonic solid. So the soft question is: How best to do this?
One way to make this into a (more) rigorous question is as follows: How should one arrange N points on a sphere such that:
As many 'moments' of the points vanish as possible.
Subject to 1, the maximum distance on the sphere to the N point set is as small as possible.
I put 'moments' in quotes because I'm not quite sure of the most efficient way to formulate this in general. For the 'zeroth moment' to vanish, I mean just that the average position of the points should be zero. For higher moments, one way to proceed is by spherical multipole moments, but there may be a more elegant way to formulate this idea of the points not being ''clustered'' too much.
I presume some similar problems have been considered before (e.g. by the NIF guys), so references are welcome as well as solutions (to either the soft or question or the particular formulation I suggested above).