Most Regularity of a Polygon 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.
 A: The on-the-surface instance of your problem
is sometimes known as the Thomson problem, after J.J. who posed the problem
over a century ago.
There is a detailed Wikipedia page
on the topic.
The $n=5$ case was only settled in 2010—"a rigorous computer-assisted proof"
established that the triangular bipyramid is the unique minimizer of
the Coulomb potential.

Richard Evan Schwartz.
  "The 5 Electron Case of Thomson's Problem."
  2010.
  arXiv link
"In the [triangular bipyramid], two points are antipodal points on [the sphere] and the remaining
  3 points form an equilateral triangle on the equator midway between the
  two antipodal points":


 
 
 
 
 


 
 
 
 
 (Image from brakke @this link)

Many other candidates for minimal energy configurations have been identified but
not rigorously proven.
For example, for $n=14$, it seems that the gyroelongated hexagonal dipyramid is
the minimal energy configuration (combinatorially).
For a deeper look:

Cohn, Henry, and Abhinav Kumar. "Universally optimal distribution of points on spheres." Journal of the American Mathematical Society. 20.1 (2007): 99-148.

