From: Kevin Brown <ksbrown@seanet.com>
To: "'Dave Rusin'" <rusin@math.niu.edu>
Subject: RE: Number of Distinct Min-Energy Configs
Date: Wed, 22 Apr 1998 21:33:44 -0700

Thanks for the message.  Coincidentally, I had just been
checking Neil's online encyclopedia for the integer sequence

0,1,1,1,3,2,4,4,6,7,20,3,28,9,21,12,14,18,52,26,69,17,48,...

and was surprised not to find it, given his long-time interest
in the Thomson problem, since this is the number of distinct
distances in the (absolute) min-energy configuration of n 
electrons on a sphere.  Also, I still haven't been able to
find a tabulation of the number of (local) minimal configs
for n electrons, although I did find a reference that says
the number eventually grows exponentially, even though it is
fairly small (often 1) for n less than 100.  Jim Buddenhagen
says he thinks there are at least two for n=22, but I've never 
been able to find more than 1.  After n=16, the next n with 
multiple minimums that I can find is n=32, and I think n=40 
has at least 3, and n=46 has at least 4.  Neil's info seems
to focus on the absolute global minimum, without trying to
count all the local minima, and even for the global min config
his tables list only the energy and a set of coordinates for
the particles, with no assessment of the distances or symmetries.
I'm particularly interested in the cases when a min configuration
for n points contains a min config for m points as a subset.
I'm eventually going to upload some pictures of these embeddings
to my web page on this subject, which currently just has
individual images, but doesn't show how any can be embedded
in others.  I've also just added a little Java applet to give
an interactively viewable image of the interesting n=24 case,
and will be adding Java's for the others when I get time.

Regards,
Kevin Brown

----------
From: 	Dave Rusin[SMTP:rusin@math.niu.edu]
Sent: 	Tuesday, April 21, 1998 8:40 AM
To: 	ksbrown@seanet.com
Subject: 	Re: Number of Distinct Min-Energy Configs

In article <352f2a6d.29657671@news.seanet.com> you write:
>Let k(n) denote the number of distinct (up to rotation and
>reflections) arrangements of particles on the surface of a
>sphere such that the total potential energy is a (local)
>minimum, assuming an inverse-square repulsive force between 
>the particles.
>
>(1) What are the values of k(n) for n=1 to 100?  (I believe
>    k(n)=1 for all n<16, and k(16)=2.)

You might consider sending these questions to Neil Sloane at bell labs;
he's been using coulomb potential for point-dispersal for some time.
He has a web page which a considerable amount of related information,
with which you may already be familiar.
dave