Distributing Points Evenly on a Sphere

Question

I am looking for an algorithm to put $n$-points on a sphere, so that the minimum distance between any two points is as large as possible.

I have found some related questions on stackoverflow but those algorithms are not an exact solution more random distributions. For special number of points (inscribed Platonic solids) it is clear but how about 5 points for example. I would be grateful for hints to the literature.

Thank you everyone for your time.

Answer 1

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 maximize the smallest distance between any two points? Sometimes phrased as packing $n$ congruent circles on a sphere:

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          ^{(Image from Paul Sutcliffe.)}

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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). Added (8Sep15): Exact radius for $n=10$ by Sugimoto & Tanemura.
• For $n=24$ by Robinson (1961).
• For $n=13, 14$ by Musin and Tarasov (2014).

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            ^{Fig.1 from Musin & Tarasov: $n=14$.}

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  Added (8Sep15): The exact radius for $n=10$ was just found:

Teruhisa Sugimoto, Masaharu Tanemura. "Exact value of Tammes problem for N=10." Sep 2015. arXiv 1509.01768 Abstract.

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          ^{Fig.1b from Sugimoto & Tanemura.}

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Added (31Dec2017) in response to a question by @R_Berger: For $n=20$, the best arrangement for the Tammes problem is not the dodecahedron's vertices. The optimal is unknown, but this beats the dodecahedron:

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          ^{Coordinates from Neil Sloane link, due to R.H. Hardin, N.J.A. Sloane & W.D. Smith (1994).}

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