Optimal distribution of points over the surface of a sphere

Solution 1:

The following is a classical paper on distributing points on a sphere:

Edward Saff, Arno Kuijlaars, Distributing Many Points on a Sphere, The Mathematical Intelligencer, Volume 19, Number 1, 1997, pages 5-11.

Solution 2:

This is a classical problem, and let me just explain what is and isn't possible.

Think of the five platonic solids. Tetrahedron, cube, octahedron, dodecahedron, and the icosahedrone. They have respectively: 4, 6, 8, 12 and 20 vertices. If you were to find a way of getting say, 7 points perfectly distributed on a sphere, guess what? You would have discovered a new platonic solid!

All you would have to do is just connect all the vertices with edges, and voilà, you would have found a sixth platonic solid. It's already been proven that this impossible, so..

You are going to have to come to grips with the sad reality that you won't find a way of getting an arbitrary number of points evenly distributed on a sphere, other than those mentioned. There are some pretty good estimates of even distributions, but other than the cases already mentioned it is not possible.