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  "body": "\n## Problem\n\nLet $N \\ge 2$ be an integer and let $S^2 \\subset \\R^3$ be the unit sphere.\nFind the minimum of the energy\n$$E(x_1, \\dots, x_N) = \\sum_{1 \\le i \u003c j \\le N} \\frac{1}{\\abs{x_i - x_j}}$$\nover all $N$-tuples of distinct points $x_1, \\dots, x_N \\in S^2$, where $\\abs{x_i - x_j}$ is the [[Euclidean distance|0NPPGHQNXQ]].\nFind also the configurations that attain it.\n\n## Remarks\n\nThe energy is the electrostatic potential energy of $N$ equal point charges that repel by Coulomb's law and are confined to the sphere.\nThomson posed the question in 1904 in connection with his model of the atom.\nBy compactness a minimizer exists for each $N$.\nRigorous solutions are known for $N = 2, 3, 4, 5, 6$ and $12$.\nThey are two antipodal points, an equilateral triangle on a great circle, the regular tetrahedron, the triangular bipyramid, the regular octahedron and the regular icosahedron.\nThe case $N = 5$ was proved by Richard Schwartz with computer assistance.\nFor larger $N$ only numerical candidates are known.\nThe minimizers need not maximize the minimal distance, so the Thomson problem differs from the [[Tammes problem|9WD2TZ4ZB5]] in general.\n\n## Explanations\n\n### L1 human\n\nPut $N$ electrons on the surface of a ball and let them push each other apart.\nThey settle in a pattern of lowest energy, and the problem asks which pattern this is.\nFor a few small numbers the answer is a regular shape, but in general it is unknown.\n",
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      "first_stated": "1904",
      "posed_by": "J. J. Thomson"
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