Equally spaced points on a sphere If you have an image on a plane,  you can represent it discreetly by sampling equally spaced points and putting the values in a matrix of some sort. (I'm not worried about the potential lost of information this may imply)
If you were to do the same for an image on a sphere, how could you pick the points to sample from. The Platonic Solids provide five examples of equally spaced points on a sphere. But are these the only five configurations possible? This would limit the maximum resolution of the image to 20 points of the vertices of the dodecaedro. (without resorting to imperfect configurations). This is a disappointingly low number of points. And my expectations say that this problem has less restrictions than the platonic solids do. So maybe it could have more solutions.
So, in one sentence, my question is:
For an arbitrarily large N, can you always find a configuration of n > N points that are even spaced within the surface of a sphere?
Maybe group-theory and the study of symmetries have something to say about this problem?
EDIT: After consulting some professors of mine, It turns out that this problem is closely related to the sphere packing problem.
One of my professors pointed to Sphere Codes:
http://neilsloane.com/packings/
The formal formulation of this problem is:  Place n points on a sphere in d dimensions so as to maximize the minimal distance (or equivalently the minimal angle) between them.
 A: Since regular polygons include the equilateral triangle, square, and regular pentagon, which form the five Platonic solids, then since these solids can all be inscribed in a sphere, and Euclid shows that no other regular solids are possible (Elements, XIII, 18), then wouldn't the twenty vertices of the regular dodecahedron be the maximum number of points that can be equally spaced on the surface of a sphere? Each of the twenty points is separated from its three nearest neighbors by an arc whose chord is the edge of the regular pentagon which makes up the inscribed solid. If this is a good argument, then there would be no solution for an arbitrary number of points.
A: The best method I can think of is unfortunately complex.  If you construct a standing spherical wave super imposed on the earth you can adjust the parameters of the wave to produce the desired number of nodes required.  I believe this will allow you to pick an arbitrary number of points and evenly distribute them.  A visual example of this can be seen on page 440 of this pdf http://www.mr-fusion.hellblazer.com/pdfs/spherical_nonlinear_waves.pdf
Perhaps someone here with a better grasp of differential equations and linear algebra can comment on plausibility of this approach.
