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

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.

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/