Optimal distribution of points over the surface of a sphere How can one generate a distribution of N points over the surface of a sphere so that the all N voronoi cells have the same area? Which is the best algorithm for this?
 A: 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. 
A: 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.
A: A simple solution places the points evenly around the equator, i.e., point $i$ at longitude $2\pi i/N$ and latitude 0.  As required, the Voronoi cells will have the same area, $4\pi r^2/N$; they are sectors with two vertices at the poles, i.e., the arrangement looks like a beach ball.  This satisfies the question as you stated it, but I doubt it's what you want, because the title says you're looking for an optimal distribution.
