Distributing points evenly on a sphere

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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Since you are asking about an algorithmic perspective, I wanted to point out that a closely related variation (with logarithmic potential instead of hard core potential) is the subject of one of Smale's "Mathematical Problems for the Next Century." In Problem 7, Smale asks if there is a polynomial time algorithm (in a particular model of computation over real numbers) to place $N$ points on the sphere so that the total energy is $O(\log N)$ above the minimum energy.

Now, for specific finite values of $N$, Neil Sloane has a table of best known spherical codes with references: http://neilsloane.com/packings/

There is a rather comprehensive list of research articles about points on spheres and manifolds at https://my.vanderbilt.edu/edsaff/spheres-manifolds/

A nice paper is E. B. Saff, A. B. J. Kuijlaars, "Distributing Many Points on a Sphere", The Mathematical Intelligencer, Winter 1997, Volume 19, Issue 1, pp 5-11.