# $N$ points as far apart as possible in a sphere volumetrically?

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)

• define "as far apart as possible", for example biggest minimal distance, or total sum of all point pairs. – Zang MingJie Jan 19 '19 at 23:00
• @ZangMingJie total sum of all point pairs. – May Rest in Peace Jan 19 '19 at 23:01
• 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). – Sangchul Lee Jan 19 '19 at 23:04
• Clearly, for three points you want an equatorial equilateral tirangle, for $n=4$ a regular tetrahedron... – Hagen von Eitzen Jan 19 '19 at 23:07
• @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 – May Rest in Peace Jan 19 '19 at 23:08