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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 largest number $d_N$ such that there are $N$ points on $S^2$ with pairwise distances at least $d_N$.\nFind also the configurations of $N$ points that attain it.\n\n## Remarks\n\nEquivalently, find the largest radius of $N$ equal non-overlapping circular caps on a sphere, or the best [[spherical code|SDCX4WDP4F]] of size $N$ in $\\R^3$.\nThe botanist Tammes posed it in 1930 while studying the distribution of pores on pollen grains.\nFor $N = 4$, $6$ and $12$ the answers are the vertices of the regular tetrahedron, octahedron and icosahedron.\nThe problem is solved for $N \\le 14$ and for $N = 24$.\nOleg Musin and Alexey Tarasov solved the case $N = 13$.\nIt is linked with the [[kissing number problem in three dimensions|Q1H6BCKTBE]]: thirteen unit balls cannot touch a central one, so the optimal angle for $13$ points is below $60^\\circ$.\n\n## Explanations\n\n### L1 human\n\nHow should $N$ people stand on a round planet so that the two closest people are as far apart as possible?\nFor small $N$ the answer is often a familiar solid, such as the corners of a tetrahedron for four people.\nFor most $N$ the answer is unknown, and even small cases needed long proofs.\n",
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      "first_stated": "1930",
      "posed_by": "Pieter Merkus Lambertus Tammes"
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    "name": "Tammes problem",
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