Given a radius $R$ and points $N$, I want to distribute points in a sphere volumetrically so that they are as far apart as possible.

I know that for $N = 1$, I can place it anywhere. For $N = 2$, diametrically opposite. But what about $N > 2$ ?

I have seen a lot of questions which look like this but all of them deal with distribution of points on the surface of the sphere.

I will just state my use case, if that would be useful. If there are $N$ electrons contained in a sphere, what will be there positions ?

Also what exactly subfield do these kinds of problem lie in ? How do I go about approaching problems like these ?

Update: As per @Zang MingJie 's query, I want to minimize the sum total distance between point pairs. Because that would be the minimum energy configuration in my electron case (explained above)

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    $\begingroup$ define "as far apart as possible", for example biggest minimal distance, or total sum of all point pairs. $\endgroup$ – Zang MingJie Jan 19 '19 at 23:00
  • $\begingroup$ @ZangMingJie total sum of all point pairs. $\endgroup$ – May Rest in Peace Jan 19 '19 at 23:01
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    $\begingroup$ The equilibrium state of $N$ electrons contained in a sphere minimizes the sum of reciprocal of inter-distances, since it minimizes the electric potential. It may be possible that the configuration maximizing the sum of inter-distances is different from that of electrons (or, even if they coincide, one needs a justification for this). $\endgroup$ – Sangchul Lee Jan 19 '19 at 23:04
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    $\begingroup$ Clearly, for three points you want an equatorial equilateral tirangle, for $n=4$ a regular tetrahedron... $\endgroup$ – Hagen von Eitzen Jan 19 '19 at 23:07
  • $\begingroup$ @SangchulLee I was under the impression that if I maximize sum of inter-distances , I would minimize the potential. Now I think, i was wrong in my assumption $\endgroup$ – May Rest in Peace Jan 19 '19 at 23:08

To minimize the potential energy you want to minimize the sum of the reciprocal distances. We know that puts all the charges on the surface of a conductor, none in the interior, so you want them on the surface of the sphere.

For the number of vertices in the Platonic solids you can certainly use those. For other numbers, finding the right configuration is hard. You can look at the figures in packomania to see how strange things get in the plane. I am sure the surface of a sphere is no easier.


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