Uniform distribution of points on a sphere: only Platonic solids?

Question

I'm quite sure the only way to uniformly distribute $n$ points on the sphere $S^2$ is by inscribing one of the 5 Platonic solids, thus there only exists a solution for $n=4,6,8,12,20$.

But am I right? And how can this be proven?

Addendum

My intuitiv definition of a uniform distribution is: Any 2 adjacent points have the same (spherical) distance; the points adjacent to $p$ are those that have minimal distance from $p$.

This should describe the uniformity of the vertices of a Platonic solid distributed on its circumscribed sphere.

Answer 1

I can think of at least two more uniform distributions that fit your current description, one with 14 points, and one with 32 points.

You obtain them by combining the vertices of a pair of dual regular polytopes:

• six vertices of the octahedron + eight vertices of the cube. The adjacent points form a regular triangle or square.
• twelve vertices of the icosahedron + 20 vertices of the dodecahedron. The adjacent points form a regular triangle or regular pentagon.