The angle of two closest points among $N$ points evenly placed on the $d$-dimensional unit sphere

Question

Some previous questions (here and here) ask for algorithms to place $N$ points evenly on the $d$-dimensional unit sphere. In my case, what I am looking for is that, given these $N$ points that are evenly placed on the $d$-dimensional unit sphere (the center of the sphere is the origin in $\mathbb{R}^d$), I want to approximate/bound the angle of two closest points among such $N$ points. When $d=2$, this angle is $2 \pi / N$. Is there any similar result/approximation for $d > 2$? I would be grateful for hints or pointers to relevant results.

Thank you everyone for your time.

Answer 1

This is related to spherical codes. There is a large literature as well as open problems. Even in dimension $d=3,$ there are open questions. You can start with Wolfram Encyclopeadia of Mathematics, here where $d=3$ case is described:

How can $n$ points be distributed on a unit sphere such that they maximize the minimum distance between any pair of points? This maximum distance is called the covering radius, and the configuration is called a spherical code (or spherical packing). In 1943, Fejes Tóth proved that for $n$ points, there always exist two points whose distance $d$ is $$ d\leq \sqrt{4-\csc^2\left[\frac{\pi n}{6(n-2)}\right]},$$
and that the limit is exact for $n=3, 4, 6,$ and $12.$ The problem of spherical packing is therefore sometimes known as the Fejes Tóth's problem. The general problem has not been solved.

A large number of putatively optimal arrangements of points with their coordinates are given in Neil Sloane's homepage for some $d$ and $n,$ at http://neilsloane.com/packings/

You can compute the related angles from there.