The Thomson problem is to determine the stable equilibrium positions of classical electrons constrained to move on the surface of
a sphere and repelling each other by an inverse square
law. Exact solutions for to 8 are known, but and 11 are still unknown.
This problem is related to spherical codes, which are arrangements of points on a sphere such that the minimum distance between any
pair of points is maximized.
In reality, Earnshaw's theorem guarantees that no system of discrete electric charges can be held in stable equilibrium under the influence of their electrical interaction alone (Aspden 1987).
Altschuler, E. L.; Williams, T. J.; Ratner, E. R.; Dowla, F.; and Wooten, F. "Method of Constrained Global Optimization."
Phys. Rev. Let.72, 2671-2674, 1994.Altschuler, E. L.;
Williams, T. J.; Ratner, E. R.; Dowla, F.; and Wooten, F. "Method
of Constrained Global Optimization--Reply." Phys. Rev. Let.74,
1483, 1995.Ashby, N. and Brittin, W. E. "Thomson's Problem."
Amer. J. Phys.54, 776-777, 1986.Aspden, H. "Earnshaw's
Theorem." Amer. J. Phys.55, 199-200, 1987.Berezin,
A. A. "Spontaneous Symmetry Breaking in Classical Systems." Amer.
J. Phys.53, 1036-1037, 1985.Calkin, M. G.; Kiang, D.;
and Tindall, D. A. "Minimum Energy Configurations." Nature319,
454, 1986.Erber, T. and Hockney, G. M. "Comment on 'Method
of Constrained Global Optimization.' " Phys. Rev. Let.74, 1482-1483,
1995.Marx, E. "Five Charges on a Sphere." J. Franklin Inst.290,
71-74, Jul. 1970.Melnyk, T. W.; Knop, O.; and Smith, W. R.
"Extremal Arrangements of Points and Unit Charges on a Sphere: Equilibrium Configurations
Revisited." Canad. J. Chem.55, 1745-1761, 1977.Whyte,
L. L. "Unique Arrangement of Points on a Sphere." Amer. Math. Monthly59,
606-611, 1952.
classical electrons constrained to move on the surface of
a sphere and repelling each other by an inverse square
law. Exact solutions for 
to 8 are known, but 
and 11 are still unknown.
such as 
, 4, 6, and 12, where the equilibrium distributions are the
vertices of an equilateral triangle circumscribed
about a great circle, a regular
tetrahedron, a regular octahedron, and
a regular icosahedron, respectively.
