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 $\mathbb R^3$:
$\qquad\qquad\quad$
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.
$\qquad\qquad\qquad\qquad\qquad\quad$
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?
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.