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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:

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          ![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). 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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          [![N=14.][2]][2]
          ^{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.

---

          [![STFig1b][3]][3]
          ^{Fig.1b from Sugimoto & Tanemura.}

---

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:

---

          ![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). 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).

---

          [![N=14.][2]][2]
          ^{Fig.1 from Musin & Tarasov: $n=14$.}

---

**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.

---

          [![STFig1b][3]][3]
          ^{Fig.1b from Sugimoto & Tanemura.}

---

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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          [![DodecaTammes][4]][4]
          ^{Coordinates from [Neil Sloane link](http://neilsloane.com/packings/dim3/), due to R.H. Hardin, N.J.A. Sloane & W.D. Smith (1994).}

---

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 Tarasov (2014).

---

          [![N=14.][2]][2]
          ^{Fig.1 from Musin and Tarasov: $n=14$.}

---

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). 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).

---

          [![N=14.][2]][2]
          ^{Fig.1 from Musin & Tarasov: $n=14$.}

---

**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.

---

          [![STFig1b][3]][3]
          ^{Fig.1b from Sugimoto & Tanemura.}

---

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 Tarasov (2014).

---

          [![N=14.][2]][2]
          ^{Fig.1 from Musin and Tarsov: $n=14$.}

---

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 Tarasov (2014).

---

          [![N=14.][2]][2]
          ^{Fig.1 from Musin and Tarasov: $n=14$.}

---