Find the arrangement of $N$ identical point charges on a sphere. For uniqueness, assume one charge sits on the north pole and another one lies on a fixed latitude of the sphere
Given a circumference in 2D, the charges will distribute to form an N-gon. Whereas in 3D and for $N=4,6,8,12,20$ one gets the charges to sit at the vertices of regular polyhedra.
I am using the electrostatic potential to find these distributions: $\sum_{i \neq j} \frac{1}{|r_i-r_j|}$. Intuitively, I expect the same equilibrium positions to show up at the minimal value of any repelling potential of the form $\frac{1}{|r_i-r_j|^\alpha}$, or even by maximizing the function $\sum_{i \neq j} d(r_i, r_j)$ for the Euclidean distance $d$. However, I'm not sure about this, but hope it helps in finding the easiest way to answer my question.
I can't be the first one wondering about this, but I am not able to find any relevant source of information, analytical or numerical, maybe this is known by other words?
This is not a generalization of the regular polyhedra, because at $n=8$ and $n=20$ you do not get the cube or the dodecahedron. The cube is easiest to see - just rotate one of the square faces 45 degrees so as to produce an antiprism, and it's easy to check that both the sum of the pairwise distances and the sum of the reciprocals of the distances improve relative to the cube.
Related questions here are the Thomson problem and the Tammes problem.