From: needham@symcom.math.uiuc.edu (James Needham)
Subject: Re: Points on Sphere
Organization: University of Illinois at Urbana
Date: Tue, 12 Jan 1993 01:44:09 GMT

Since there seems to be interest in placing n points on a 
sphere, I thought that I would post what little I know.  I 
know that I've been working on the problem for a year, 
and it has stubbornly resisted every means of attack. 

The generalized problem is this:  place n points on the 
unit sphere in order to extremize Sum(i<j, d(p(i), p(j)) ** 
alpha), where d is the Euclidean distance.  For alpha > 0, 
we want to maximize the sum, for alpha < 0, we want to 
minimize the sum.  alpha = -.5 corresponds to the 
potential energy determined by n point charges.  I've 
been concentrating on alpha = 1.

The corresponding problem in the plane, putting points 
on the circle, has been solved for many values of alpha.  
See (1) and (2).  Suffice it to say that the regular n-gon 
usually works; but for 1 < alpha < 2, I think the problem 
is still open.

For the sphere, tough, most of the interesting cases are 
still unknown.  There is an algorithm (see (3)) that takes 
n given points and moves them around (basically 
LaGrange multipliers) to what *seems* to be the best 
possible configurations.  Unfortunately, for alpha = 1, the 
optimal configuration can only be *proved* for n = 3 
(equilateral triangle), n = 4 (regular tetrahedron), n = 12 
(regular whatever), and n = 24 (regular whatever).  The 
algorithm indicates that the following are optimal:

n = 5:  bipyramid
n = 6:  bipyramid
n = 7:  one north pole and three pairs symmetric through 
the z axis
n = 8:  skewed cube
n = 9:  equilateral triangle on the equator with skewed 
equilateral triangles        
above and below it

I have "optimal" solutions for n through about 50.  Again, 
hardly any of these can be proven.  The hope that there 
is one method that will take in n and spit out the best 
configuration is fading.  The placement really seems to 
depend on the particular properties of n.  At one point I 
even had a conjecture relating it to twin primes.

The problem with this algorithm is that it behaves 
somewhat chaotically.  It is possible that an initial 
configuration gets sucked into a *local* extremal 
configuration (for example, with 5 points one can start 
with configurations that converge to the pyramid rather 
than the optimal bipyramid).  This seems to happen, 
though, with probability 0.  Does anybody out there know 
anything about chaotic dynamical systems on higher 
dimensional manifolds? If so, you might find this 
problem interesting.

One might also suspect that the optimal configurations 
remain the same as one varies alpha.  This is not the case.  
With seven points , the bipyramid is minimal for alpha = 
-.5, while for alpha = 1 the configuration described above 
is better.  (I think that the strange configuration 
described above for 7 points gives way to the bipyramid 
at about alpha = .6, if I remember correctly.)

There are lots of interesting problems in here, but don't 
get obsessed with it.  Any questions or comments will be 
cheerfully entertained!

1.  R. Alexander and K. Stolarsky, Extremal problems of 
distance geometry related to energy integrals, Trans. of 
AMS 193 (1974).

2.  G. Bjorck, Distributions of positive mass which 
maximize a certain generalized energy integral, Ark. Mat. 
3 (1956), 255-269.

3.  J. Berman and K. Hanes, Optimizing the arrangement of 
points on the unit sphere, Math. Comp. 31 (1977), 1006-
1008.


Jim Needham
needham@symcom.math.uiuc.edu