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

[Hugo Pfoertner]

There are related sequences in the OEIS, e.g., http://oeis.org/A033177 ,
http://oeis.org/A133491 or http://oeis.org/A242617

This special case is know as Thomson problem. See links provided in the
cited sequences.

Hugo Pfoertner

On Thu, May 31, 2018 at 5:18 PM, Fred Lunnon <fred.lunnon at gmail.com> wrote:

>   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/a726df877d03e8373a7b54c8d927db
> aab097.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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