Say you have a sphere, and n points to place on it. You want each of the points to be equivalent to the others if you allow relabeling and the average distance between points should be as great as possible. Essentially the most spread out 'fair' distribution of points when no point is special compared to the others. For example: when n = 4, the points would be the vertices of a tetrahedron. When n = 6, the points would be the centers of the faces of a cube. When n = 3 the points would be the vertices of a triangle. My question is: with 5 points, the only possible layout I have found is the vertices a pentagon. Is there a better one?