Date: Fri, 20 Jan 95 23:34:29 CST
From: rusin (Dave Rusin)
To: jcarney@mit.edu
Subject: Re: HOW? Equally spaced points on a sphere??

There is no easy answer to your question. At least you have expressed 
the question in an unambiguous way with the electrons; keep in mind that
the distribution that results will not necessarily be "equally spaced"
in the sense of, say, having each point be the same distance from its nearest
neighbor.

If you need to do this approximately, you can spread the points as follows:
Put  N+1  points on the meridian from north to south poles, equally spaced.
(You can write out their positions using spherical coordinates if you like).
If you swing this meridian around the sphere, you'll sweep out the entire 
surface; in the process, each of the points will sweep out a circle. You
can show that the  ith  point will sweep out a circle of radius sin(pi i/N).
If you space points equally far apart on this circle, keeping the
displacement roughly the same as on that original meridian, you'll be
able to fit about  2N sin(pi i/N) points here. This process will put points
pretty evenly spaced on the sphere; the number of such points is about
2+ 2N*Sum(i=1 to N-1) sin(pi i/N). [There is a closed form for this sum which
I am forgetting, but you should just try a few N's to see if you get a
good number of points. N=6, 7, or 8 will give you 40-80 points I think.]

If you want a better approximation to the best position, just imagine what
would happen if electrons _were_ placed at these locations. For each of
them it's not hard to sum the electromagnetic forces applied by each of the
other electrons. This will give an acceleration which you can then allow the
electron to take on. Do so for a very short time to compute new positions
for each of the particles. Repeat until the particles sort of stabilize.
This is a purely computational matter which I would hope would not strain
the resources of _your_ institution.

dave

==============================================================================

From: jcarney@MIT.EDU
To: rusin@math.niu.edu (Dave Rusin)
Subject: Re: HOW? Equally spaced points on a sphere?? 
Date: Sat, 21 Jan 1995 02:07:42 EST

Hi there,

Thank you for your reply.  I like the approximation technique you
told me about.  I think it will be fine for what we want to do.
If not, I do think we have enough computation power to handle 
a simulation of 40-80 electrons on a sphere.  I think however
that the electrons will never really become stable, but if
the program runs long enough, they will spread themselves out
well enough.

Thanks again.

John Carney