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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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The notion of 'best' seems ambiguous, but it seems that three points on the equator, and the other two points at each pole seems a good candidate for your problem. – Sangchul Lee Aug 22 '12 at 13:59
This might be useful – leonbloy Aug 22 '12 at 14:01
Searching for "distributing points on a sphere" turns up many things. It is a hard problem, with the usual definition being to maximize the minimum distance between the points. I believe sos440's solution has been proven optimal in the 5 point case, but higher cases are difficult to prove. – Ross Millikan Aug 22 '12 at 14:05
A triangular dipyramid has 5 vertices. You might have to "squash" it a bit to get the points onto a sphere, but similar to the octahedron for 6 points you'll get a point at each "pole" and the rest distributed around the "equator". – KeithS Aug 22 '12 at 14:46
I'd go with @Keith's and sos440's take. The VSEPR theory in chemistry maintains that molecules that consist of one central atom with five pendant atoms (e.g. phosphorus pentachloride, $\require{mhchem}\cf{PCl5}$) take on a trigonal bipyramidal arrangement, like this. – J. M. Aug 22 '12 at 15:10
up vote 3 down vote accepted

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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The OP has clarified that s/he had in mind sphere packing, not minimization of a potential energy function. – Ben Crowell Aug 22 '12 at 15:21
@Ben: could you please clarify where did OP clarify that thing? – Ilya Aug 22 '12 at 15:28
@BenCrowell - "have a sphere and I have to place some points on it, the most uniformly possible." Sphere packing is something else entirely. – nbubis Aug 22 '12 at 15:28
This answer is not really what I was asking for, but it's nice to know. – Xoff Aug 22 '12 at 16:07

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