Conseider $n$ electrons in an empty sphere. What structure do they make?
This question have two cases: (i) if electrons should be sit on the boundry of sphere (one can suppose that the boundry of sphere has so much positive electronic charge; (ii) if electrons can occur in everywhere of sphere.
It seems that if there is a Platonic solid (see for example http://en.wikipedia.org/wiki/Platonic_solid) with $n$ vertices, then the electrons make this solid in case (ii) of question.But we know there is Platonic solids just for $n = 4, 6, 8, 12, 20$. What is the structure of electrons, when for example $n = 7$ or $11$ and other numbers?
As a generalization, I ask the similar question for spheres in higher dimensions.