I'm interested in the subject of maximally separated points (e.g., the minimum of the distances between any two of the points is maximal) in various spaces, and I've been trying to think about how this works on a spherical surface.
For the simpler case of a circle, you can easily separate any number of points maximally simply by dividing the circle into the appropriate number of equal arcs and taking the points to be the vertices between the arcs.
On a spherical surface things are more complicated. For 0, 1, 2, or 3 points, the maximal spacing is on any great circle as above. For 4, 6, 8, 12, or 20 points the maximal spacing must be on the vertices of the relevant platonic solid inscribed in the sphere, otherwise there must be some points spaced more closely than this and you can separate them further. For the case of five points, the solution seems to be two points on opposite poles of a diameter plus three equally-separated points on a great circle lying in a plane perpendicular to that diameter. For seven or more points, other than the platonic cases, I have great difficulty determining if there's any solution better than the poles-plus-great-circle solution given for five points. Is there any better solution I'm missing?
