Suppose $n$ points lie on the sphere $S^2=\{x\in\mathbb{R}^3\mid \|x\|=1\}$ and are subjected to a repulsive acceleration that pushes away a point from each other point with an intensity proportional to the square of the mutual distance. Assume that the sphere itself poses no friction, so that points are free to move. You may think of them as $n$ electrons in a superconductive sphere. What are, if any, the configuration up to isometries of $S^2$ of stable equilibria?

My intuition suggests that there is at least one stable equilibrium for each $n$, and it is unique for small $n$. Assume up to isometries that one of the points is the north pole $N:=(0,0,1)$; then the $n$ points are in a stable equilibrium if the other $n-1$ are:

  • $n=2$: on $-N$;
  • $n=3$: forming an equilateral triangle with $N$ (with one vertex lying in $\{y=0\}$);
  • $n=4$: forming a regular tetrahedron with $N$ (with one vertex lying in $\{y=0\}$);
  • $n=5$: one on $-N$ and the others on an equilateral triangle on the equator $\{z=0\}$ (with one vertex lying on $(1,0,0)$);
  • $n=6$: forming a regular octahedron with $N$ (with one vertex lying in $\{y=0\}$);
  • $n=7$: ???, but I suspect three of them to lie close to a parallel in the northern hemisphere and other three close to a parallel in the southern hemisphere;
  • $n=8$: forming a regular cube with $N$ (with one vertex lieing in $\{y=0\}$);
  • $n=9$: ???, I suspect it to be similar to the case $n=7$, with just one more point on each parallel


  • $n=12$: forming a regular icosahedron with $N$ (with one vertex lying in $\{y=0\}$);


  • $n=20$: forming a regular dodecahedron with $N$ (with one vertex lying in $\{y=0\}$);

Is there any pattern going on here? Is there any proof of existence/uniqueness of stable equilibria? Do the polyhedra given by the convex hulls of these configurations go under some particular name?


This is the famous Thomson problem. You can find a list of optimal configurations and many references on the Wikipedia page. Your intuitions for $n=7, 8, 9, 20$ are wrong, and $n=5$ is not that obvious. By the way, your description of the interaction between electrons is not correct.

Thanks to jeq for providing another manuscript in the comment. Indeed, there are many other notions of well distributed points, minimizing some other potential function or maximizing the least distance.

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