[math-fun] Network on sphere puzzle

Let a network N on the sphere S^2 = {p in R^3 | ||p|| = 1} be defined as a finite set of K >= 2 points in S^2 with some pairs of these points connected by a shortest (or tied for shortest) geodesic. We say a network N is "minimal" if for some epsilon > 0, all networks obtained from N by varying the position of each point by less than epsilon (and connecting each pair of corresponding points by a geodesic) cannot decrease the total length of N (the sum of the lengths of all geodesics). Puzzle: ------- What are the minimal networks on S^2 ? Extra credit:* ------------- What are the minimal networks on the square torus T^2 = R^2 / Z^2 ? —Dan ————— * I don't know the answer to this one yet. But I suspect Neil Sloane does.