From: ksbrown@seanet.com (Kevin Brown)
Newsgroups: rec.puzzles,sci.math
Subject: Re: 8 electrons on sphere (Re: What am I?)
Date: Thu, 09 Oct 1997 04:48:18 GMT

On 5 Oct 1997 elkies@ramanujan.harvard.edu (Noam Elkies) wrote:
> For maximizing the minimal distance between 8 points on the sphere,
> the best configuration is not the cube, as one might expect, but the
> Archimedean solid obtained from the cube by twisting one pair of
> opposite faces by 45 degrees.  Probably the least-energy problem is
> answered by a similar polyhedron, though with a slightly different 
> shape, there being no reason to expect the two kinds of edges to be 
> congruent. Figuring out the exact shape looks like an unpleasant 
> calculus problem which I do not want to work out without a very good 
> reason...

A description of at least one min-energy solution for each N from 
1 to 23 is given at http://www.seanet.com/~ksbrown/igeometr.htm
(I say "at least one" because there are actually multiple local
min-energy configurations for N=16 and many larger N.)  The
descriptions for N=7 and 8 are as follows (where "Q" is the sum 
of the inverses of the point-to-point distances assuming a sphere 
of radius 1).

N = 7:  Seven points give north/south poles and an equilateral 
pentagon at the equator.  Of the 21 separations between particles, 
one is a diameter of length 2, ten are of length sqrt(2), and five 
each of lengths a and b, where ab = sqrt(5) and a/b = (1+sqrt(5))/2.  
This gives
               _____________              _____________
              / 5 + sqrt(5)              /  2sqrt(5)
       a =   /  -----------       b =   /  -----------
           \/        2                \/   1 + sqrt(5)


N = 8:  Eight particles arrange themselves into two squares on 
parallel planes, with the squares rotated by 45 degrees relative to 
each other.  The 28 separations between particles come in the 
following four lengths:

  a = 1.2876935    b = 1.8968930    c = 1.6563945    d = 1.1712477

We observe that c = sqrt(2)d, so the two parallel square faces have 
edge lengths d.  This configuration has Q = 19.675..., as opposed 
to Q = 22.485... for the verticies of a cube inscribed in a sphere.  


By the way, it's interesting to note that some of the min-energy
configurations for larger N contain the min-energy configurations
for smaller N as subsets.  For example, N=22 contains N=6.  It's
also interesting to note the values of N for which the min-energy
configuration contains one or more diameters of the sphere.  These
properties are tabulated on the web page mentioned above.
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