# [seqfan] Re: Configurations of n repelling points on the sphere that are in "equilibrium"

Fred Lunnon fred.lunnon at gmail.com
Thu May 31 17:18:17 CEST 2018

```  This is a classical problem with many references in the literature,
both historical and current; although your attempt to define it more
precisely leaves something to be desired!

Searching with Google for "stable configuration on sphere" finds
numerous references, including

Harvey Cohn (1956)
Stability Configurations of Electrons on a Sphere
https://www.ams.org/journals/mcom/1956-10-055/S0025-5718-1956-0081133-0/S0025-5718-1956-0081133-0.pdf

Michael Goldberg (1912)
Stability Configurations of Electrons on a Sphere
https://pdfs.semanticscholar.org/c83c/a726df877d03e8373a7b54c8d927dbaab097.pdf

Fred Lunnon

On 5/31/18, Felix Fröhlich <felix.froe at gmail.com> wrote:
> Dear SeqFans,
>
> I am no expert in this subject area, so the following explanations are
> probably very imprecise or possibly wrong, but here we go:
>
> Suppose we have a sphere and want to place n charged points on the sphere
> that repel each other with equal charge. I am interested in configurations
> where the n points are in "equilibrium" or as close to equilibrium as
> possible. What I mean by this is the following:
>
> If we draw straight spherical lines from each point in all directions to
> the nearest neighboring points, then a configuration is close to
> equilibrium if the sum of the absolute differences between the lengths of
> any two lines is small.
>
> Consider the following example: If we take two points and place them close
> to each other, then some of the lines are very short, while some of the
> lines are very long. This means the absolute difference between the lenght
> of a line in the direction where the second point is and a line in the
> opposite direction will be large, since their lenghts are very different,
> so the configuration is "not in equilibrium" or "far from equilibrium".
>
> Consider another example: If we take the two diametral points
> (Diametralpunkte) of the sphere, then all the lines have the same length
> and thus the sum of the absolute differences is 0, so that configuration of
> 2 points is in equilibrium.
>
> Consider the following example for, say, n = 10: Suppose we place the 10
> points on the circumference of the sphere. This configuration could
> possibly be rigid, but by the above definition it is far from equilibrium,
> because the difference between the length of a line from a point to a
> neighboring point and the length of the line from the point to a point on
> the "opposite" side of the sphere is large.
>
> -------------
> With these above definitions, is there a unique configuration of n points
> for each n that is as close to equilibrium as possible? I assume this is
> not the case. Under this assumption, how many configurations exist for each
> n that are as close to equilibrium as possible?
>
> Regards
> Felix
>
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>

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