From: skaufman5@juno.com
Date: Sat, 5 Aug 2000 14:42:57 -0500
Subject: points on sphere

Professor Davis

I stumbled into a problem which I think might be related to one of your
interest areas.  Here's how it went:

It starts with a physical problem:  What is the minimum-energy
configuration of n>1 identical particles of equal charge, constrained to
move on the surface of a sphere?  The particles repel each other, and the
total energy is well defined in terms of the distances between each of
the charges (for a given pair of particles at distance r, it is 1/r).

Clearly (I think) the minimum energy will occur when each particle is the
same distance from its nearest neighbors.  This would be equally true
whether the distance is measured on the sphere's surface or directly
through 3D space.

Now we begin to get closer to mathematics:  The positions are:

n     positions
2     the ends of a diameter
3     the vertices of an inscribed equilaterial triangle
4     vertices of an inscribed tetrahedron
5     ???!
6     tangent points to a superscribed cube
7    ???!!
8     vertices of an inscribed cube

etc.

It is disturbing that n=5 and 7 (and certainly many others, at least all
prime n) do not have obvious solutions, and even more disturbing to think
that they might not have ANY solutions.  I'm sure this is related to a
problem, probably a "well-known" one in "real" mathematics, but my
background is physics and I don't have a clue as to where to start
looking.  However, I came across the topic "algebraic topology" and it
sounded like it might be related.  You were listed on "algebraic
topologists' web sites," hence this e-mail.

I would appreciate it if you could point me in the right direction here.

Stan Kaufman
New Brighton, MN