## Distribution of Electrons on a Sphere

### Example of a NonConvex Optimization Problem

Drag mouse to rotate 3D model. Hold *shift* key to zoom in and out.
Given a sphere in *R*^{3}, the problem is to figure out how a
collection of *n* electrons would distribute themselves so as to
minimize the total potential energy. The potential energy between any
pair of electrons is the reciprocal of the distance between them. The total
potential energy is the sum over all pairs. For *n* = 2, the optimal
solution consists of two antipodal points. For *n* = 3, the solution is
at the vertices of an equilateral triangle circumscribed in a
great circle. For *n* = 4, the solution is at
the vertices of a regular inscribed tetrahedron. Most other cases have
no simple answer. The one shown above is for *n* = 10.
To get a better feel for it,
set it in motion by dragging and releasing the globe.

This problem is a
nonconvex optimization problem. To see the AMPL file that was used to generate
the distribution shown above, click here.
LOQO was used to solve the nonconvex optimization problem.

For the distribution of 5 electrons, click
here.

For the distribution of 200 electrons, click
here.

For the distribution of 1000 electrons, click
here.

*Updated* 2011 Apr 20