Shape of electric charges on sphere in equilibrium state

When electric charges of equal magnitude and sign are released on a regular sphere (and assume that they stick to the surface of the sphere, but they are free to move along its surface), what is the shape of the figure made by the charges as vertexes when they come to a state of equilibrium?

Case 1 - Only one charge is there:

Already in equilibrium as there are no other charges.

Case 2 - Two charges:

Two charges are on opposite points of one diameter of the sphere.

Case 3 - Three charges:

Three charges make a shape of an equilateral triangle.

Case 4 - Four charges:

A regular tetrahedron comes up when they reach the state of equilibrium.

While extrapolating these cases to higher number of charges, one roadblock comes up: how to adapt a more-than-three-dimensional figure to the three-dimensional sphere?

Any hint taking to the correct answer or better approach to the problem is welcome.

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Related MO.SE question: mathoverflow.net/q/187063/13917 – Qmechanic Oct 8 '12 at 15:20
@Qmechanic I'll just add that the main important result from that question was $N^2/2$ is the $1/r$ summation for uniformly distributed, infinite, points. I'm having fun with some of the links posted here, they do indeed limit to the N^2/2, but there are some interesting lower order terms, not to mention some deviations that can't be approximated with calculus approaches, which is fascinating. – Alan Rominger Oct 8 '12 at 15:58

This problem with $N$ point charges on a sphere is a famous problem in electrostatics known as the Thomson problem. For large $N$, it is in general an open problem still under active research.
This has been a problem since Thomson proposed the arrangement of electrons and positive charges (nucleus was not known at that time) in rigid electron shells of atom which is what called Plum-pudding model of atom. He suggested that electrons are arranged in a symmetrical pattern with respect to the center of sphere which is applicable only to smaller elements in periodic table (Old-timer wandered a lot after discovering the $e/m$ ratio).