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# Thomson Problem

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 the the minimum distance between any pair of points is maximized.

The Thomson problem has been solved exactly for only a few values of 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.

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).

Sphere Point Picking, Spherical Code

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## References

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." Nature 319, 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. Monthly 59, 606-611, 1952.

Thomson Problem

## Cite this as:

Weisstein, Eric W. "Thomson Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ThomsonProblem.html