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

[Felix Fröhlich]

Thanks for the replies.

I think this is definitely something I need to look into more closely.

Felix

2018-05-31 19:22 GMT+02:00 Hugo Pfoertner <yae9911 at gmail.com>:

> 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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> >
>
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