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

Felix Fröhlich felix.froe at gmail.com
Thu May 31 16:31:00 CEST 2018

```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.

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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

```