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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    $\begingroup$ 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. $\endgroup$ – Sangchul Lee Aug 22 '12 at 13:59
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    $\begingroup$ This might be useful www2.research.att.com/~njas/packings $\endgroup$ – leonbloy Aug 22 '12 at 14:01
  • $\begingroup$ 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. $\endgroup$ – Ross Millikan Aug 22 '12 at 14:05
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    $\begingroup$ 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". $\endgroup$ – KeithS Aug 22 '12 at 14:46
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    $\begingroup$ 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. $\endgroup$ – J. M. is a poor mathematician Aug 22 '12 at 15:10

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.

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

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