Equilibrium configurations based on Platonic geometries

The problem of the stable configurations of N electrons on a sphere minimizing the potential energy of the system is related to the mathematical problem of the extremal configurations in the distance geometry and to the problem of the densest lattice packing of the congruent closed spheres. The arrangement of the points on a sphere in three-space leading to the equilibrium solutions has been of interest since 1904 when J. J. Thomson tried to obtain the stable equilibrium patterns of electrons moving on a sphere and subject to the electrostatic force inversely proportional to the square of the distance between them. Utilizing the theory of the point vortex motion on a sphere, Platonic polyhedral extremal configurations are obtained in this paper using numerical methods.