6
$\begingroup$

My goal is to put $n$ points on a sphere in $\mathbb{R}^3$ to divide it in $n$ parts, so that their disposition would be as "equivalent" as possible. I don't exactly know what "equivalent" mathematically means, perhaps that the min distance between two points is maximal.

Anyway in $2$ dimensions it is simple to divide a circle in $n$ parts. In $3$ dimensions I can figure out some good re-partitions for particular values of $n$ but I lack a more general approach.

  • 0
    The ones cited by Rahul Narain may be better, but you could also look at http://math.stackexchange.com/questions/11499/possible-to-imitate-a-sphere-with-1000-congruent-polygons/11512#115122012-04-14

2 Answers 2

1

A nontrivial problem, I think. You might find some links into the research literature on Ed Saff's homepage.

  • 0
    That link is broken but https://my.vanderbilt.edu/edsaff/spheres-manifolds/#equal-area-points and http://eqsp.sourceforge.net/ may help2018-07-22
0

You can try the physical method: treat each point as an electron constrained in a sphere, and randomly distribute these point particles in the sphere, then you can solve the equations of motion to reach a stable (minimum energy) state, where each particle maximally separated from its closest neighbours (electric repulsive forces). Here is how to apply this method on a sphere surface.