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I was wondering about the following problem.

We want to place $n$ points $x_1,\ldots,x_n\in \mathbb R^3$ on the sphere of $\mathbb R^3$, $\mathbb S^2$, such that they are as far as possible.

We note that the problem is easy on $\mathbb R^2$.

Here what it would look like on $\mathbb R^3$:

$\qquad\qquad\quad$enter image description here

In other words, let $d$ be the usual distance on $\mathbb R^3$ and let's define the functions $f_n$ for $n\geqslant 2$:

$$\begin{matrix} f_n\colon & (\mathbb S^3)^n &\to &[0,2] \\ &(x_1,\ldots,x_n)&\mapsto &\min\big\{d(x_i,x_j),\ (i,j)\in \{1,\ldots,n\}^2, i\ne j\big\}. \end{matrix}$$

Since $(\mathbb S^3)^n$ is compact, and $f$ is continuous, $f$ attain a maximum $M_n:= \max f$.

Here is are two examples, the first one shows a configuration that has no chance to attain the maximum, the second one is a more serious candidate.

$\qquad\qquad\qquad\qquad\qquad\quad$enter image description here

  • If $n=2$, we can place the two points opposite two each other, $(-1,0)$ and $(1,0)$ for instance, and we get $M_2=2$.

  • If $n=3$, I would tend think that and equilateral triangle would be the solution to our problem, thus $M_3$ would be equal to $\frac 32$.

  • If $n=4$, I think the solution is to put a regular tetrahedron in the sphere, and some geometry show that $M_4$ then equal to $\frac{2\sqrt 6}3$.

Some questions.

Is there a generic way to calculate $M_n$?

If we can not find a formula for $M_n$, can we at least conduct an asymptotic analysis of the sequence $(M_n)$?

Do you know any references where this problem is treated?

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    related: http://mathoverflow.net/questions/24688/efficiently-sampling-points-uniformly-from-the-surface-of-an-n-sphere – 2017-01-08
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    @Max Related indeed, but it is not answering the same problem unfortunately. – 2017-01-08
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    You might be interested in [**Thomson Problem**](https://en.wikipedia.org/wiki/Thomson_problem). – 2017-01-08
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    @NgChungTak It look a lot like my problem indeed ! Thank you for the link. – 2017-01-08
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    I think that this issue has a kind of intrinsic instability that, IMHO, gives few hope for a nice analytic treatment. For example, case n=4 is "nice", whereas case n=5 (which would be interesting to study for its own sake) is not obtained by a "perturbation" of the former case. – 2017-01-08
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    You might also be interested in "spherical codes". – 2017-01-08

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The isoperimetric inequality gives reasonable bounds for large values of $n$.
If $n$ points on a unit sphere are $\geq d$ apart from each other (with respect to the geodetic distance on the sphere) by considering a circle centered at each point with radius $\frac{d}{2}$ we get almost disjoint circles, so the sum of their areas has to be less than $4\pi$, i.e. the surface of the unit sphere. It follows that, roughly, $$ \pi n\left(\frac{d}{2}\right)^2 \approx 4\pi $$ so it is reasonable to expect that $d(n)$ behaves like $\color{red}{\large\frac{C}{\sqrt{n}}}$ with $C\approx 4$.

Such problem is approximately solved up to $n\approx 500$: if we consider $n$ electrons on a unit sphere, they repel due to the electric force, and tend to assume a configuration in which the potential energy is minimized, i.e. the electrons are as far as possible from each other.