[math-fun] Tammes problem for the square and cubical torus

The Tammes problem is the well-known problem to find the arrangement of N points on the unit sphere S^2 that maximizes the minimum distance between any pair of them. What about the same question for N points on the square torus T^2 = R^2/Z^2 or the cubical torus T^3 = R^3/Z^3 ??? (Note that the maximum distance between any two points on T^2 is sqrt(2)/2 and between any two points on T^3 is sqrt(3)/2.) This gets very difficult on S^2 even for slightly large N, so I don't expect it will be easy to solve on T^2 or T^3. But what would seem to be the best arrangements for 1 <= N <= 10 on T^2 and T^3 ??? —Dan