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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.

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    Mean distance between 2 points within a sphere http://math.stackexchange.com/questions/167932/mean-distance-between-2-points-within-a-sphere might help2012-09-05
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    I seem to recall hearing that the problem of maximizing the minimum is unsolved. But maximizing the sum is another matter.2012-09-05
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    A [bunch](http://math.stackexchange.com/q/66365/856) of [previous](http://math.stackexchange.com/q/165819/856) questions [are](http://math.stackexchange.com/q/31619/856) closely [related](http://math.stackexchange.com/q/9846/856). None is an exact duplicate of this one, but you may find the answers and references there of interest.2012-09-05
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    "Straight line"? Through the sphere or on the surface?2012-09-05
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    Straight line through the sphere.2012-09-06
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    Are you looking for an analytical expression of the configuration or a numerical solution? This is a nonconvex optimization problem and I suspect it has several local solutions that are not global maximizers.2013-03-06
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    analytical, I just want to understand the steps involved in solving the problem2013-03-07
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    You may be able to find an analytical expression if you measure geodesic distance (on the surface of the sphere) and use spherical coordinates. This is just a suggestion that could put you on track for the harder question you ask.2013-08-18

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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