Given a sphere in R3, the problem is to figure out how a collection of n electrons would distribute themselves so as to minimize the total potential energy. The potential energy between any pair of electrons is the reciprocal of the distance between them. The total potential energy is the sum over all pairs. For n = 2, the optimal solution consists of two antipodal points. For n = 3, the solution is at the vertices of an equilateral triangle circumscribed in a great circle. For n = 4, the solution is at the vertices of a regular inscribed tetrahedron. Most other cases have no simple answer. The one shown above is for n = 10. To get a better feel for it, set it in motion by dragging and releasing the globe.
This problem is a nonconvex optimization problem. To see the AMPL file that was used to generate the distribution shown above, click here. LOQO was used to solve the nonconvex optimization problem.
For the distribution of 5 electrons, click here.
For the distribution of 200 electrons, click here.