# Maximum total distance between points on a sphere

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:

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

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Mean distance between 2 points within a sphere math.stackexchange.com/questions/167932/… might help –  raindrop Sep 5 '12 at 4:17
I seem to recall hearing that the problem of maximizing the minimum is unsolved. But maximizing the sum is another matter. –  Michael Hardy Sep 5 '12 at 4:24
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. –  Rahul Sep 5 '12 at 4:56
"Straight line"? Through the sphere or on the surface? –  Henry Sep 5 '12 at 6:37
Straight line through the sphere. –  raindrop Sep 6 '12 at 16:43

An easy counterexample is a $n=6$, where an octahedron has larger sum of distances than a hexagon. –  Rahul May 30 at 0:56