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Added Tammes problem history, as recounted by Musin & Tarasov.
Joseph O'Rourke
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There is considerable literature on this question, and closely related variations. See:

  • The Thompson problem: Which configurations of $n$ electrons on a sphere minimize the electrostatic potential energy?
  • The Tammes problem: Which configurations of $n$ points on a sphere maximizes the smallest distance between any two points? Sometimes phrased as packing $n$ congruent circles on a sphere:

          ![DiskPacking][1]
          (Image from [Paul Sutcliffe](http://www.maths.dur.ac.uk/~dma0pms/my-projects/proj4_1213.html).)
According to

Musin, Oleg R., and Alexey S. Tarasov. "The Tammes problem for $N=14$." arXiv:1410.2536 Abstract (2014).

the Tammes problem is solved exactly for

  • For $n=3,4,6,12$ by L. Fejes Toth (1943).
  • For $n=5,7,8,9$ by Schütte and van der Waerden (1951).
  • For $n=10,11$ by Danzer (1963).
  • For $n=24$ by Robinson (1961).
  • For $n=13, 14$ by Musin and Tarsov (2014).

          [![N=14.][2]][2]
          Fig.1 from Musin and Tarsov: $n=14$.
Joseph O'Rourke
  • 150.9k
  • 36
  • 358
  • 958