## Abstract

Consider n points on the unit 2-sphere. The potential energy of the interaction of two points is a function f (r ) of the distance r between the points. The total energy E of n points is the sum of the pairwise energies. The question is how to place the points on the sphere to minimize the energy E. For the Coulomb potential f (r ) = 1/r, the problem goes back to Thomson (1904). The results for n < 5 are simple and well known; we focus on the case n = 5, which turns out to be difficult. Dragnev, Legg, and Townsend [2] give a solution of the problem for f (r ) = -log r (known as Whyte's problem). Hou and Shao give a rigorous computer-aided solution for f (r ) = -r, while Schwartz [4] gives one for Thomson's problem. Finally, we give a solution for biquadratic potentials.

Original language | English (US) |
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Pages (from-to) | 1717-1731 |

Number of pages | 15 |

Journal | Indiana University Mathematics Journal |

Volume | 62 |

Issue number | 6 |

DOIs | |

State | Published - 2013 |

## Keywords

- Cauchy matrix
- Discrete energy
- Thomson's problem

## ASJC Scopus subject areas

- General Mathematics