TITLE: 5 Points uniformly placed on a sphere
QUESTION [15 upvotes]: I have a sphere and I have to place some points on it, the most uniformly possible.
If I have 4 points, placing them as vertices of a tetrahedron seems good.
If I have 6 points, placing them as vertices of a octahedron seems very good too.
How can I find a way (the best if it exists) to place only 5 points ?
EDIT : "Uniformly" would mean that if I draw a Voronoï diagram on the sphere, each point has a same-area cell and the diameter of a cell is minimized (they are "round" and not some thin slices of the sphere).

REPLY [7 votes]: The answer depends on what you mean by "uniform". One way of doing this is to minimize the "energy" the system would have if each of the points was a charged particle. This "Thomson's Problem" is quite a famous problem in global minimum finding algorithms.
The answer in this case for $n=5$ would be: 
Two points on the poles, and three as an equilateral triangle on the equator. 
More answers for other values of $n$ can be found here.