What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?

The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points. All the points are on a single sphere of radius $R$.

Here's a visualization: enter image description here

Acknowledgements: Based on this Physics S.E. question. Image from StackOverflow.

  • $\begingroup$ Mean distance between 2 points within a sphere math.stackexchange.com/questions/167932/… might help $\endgroup$
    – raindrop
    Sep 5, 2012 at 4:17
  • $\begingroup$ I seem to recall hearing that the problem of maximizing the minimum is unsolved. But maximizing the sum is another matter. $\endgroup$ Sep 5, 2012 at 4:24
  • 7
    $\begingroup$ A bunch of previous questions are closely related. None is an exact duplicate of this one, but you may find the answers and references there of interest. $\endgroup$
    – user856
    Sep 5, 2012 at 4:56
  • $\begingroup$ "Straight line"? Through the sphere or on the surface? $\endgroup$
    – Henry
    Sep 5, 2012 at 6:37
  • $\begingroup$ Straight line through the sphere. $\endgroup$
    – raindrop
    Sep 6, 2012 at 16:43

1 Answer 1


As far as I know the answer to the general question is unknown. For the computer approach you can look at this article by Berman and Hanes. Here it is shown that the result for 5 points on the sphere can be found in finite time by computer. Also you can find some interesting references in the introduction part.

Hope, this will help


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