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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).

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    Thanks for all those links and interesting comments !2012-08-22

1 Answers 1

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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.

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    This answer is not really what I was asking for, but it's nice to know.2012-08-22