Eight equally charged particles repel each other inside a sphere and come to rest; what figure do they form?

Since everybody seems to want to note that the puzzle looks like a
homework problem and not to solve it, I thought that I might note that
the figure is, in fact, an icosahedron.

-- Richard Barnes <bifurcation@mindspring.com>

An icosahedron has 20 faces, 30 edges, and 12 (not 8) vertices.

The answer is really much simpler, it's a cube!

-- Mike Naylor

Granting that the "homework" flames were out of line, this is probably
too hard a problem for rec.puzzles -- unless you have a neat *proof*
that your configuration is optimal, but even in that case sci.math or
sci.math.research is a more appropriate venue.  For known computational
results, see http://www.research.att.com/~njas/electrons/

Meanwhile what I suspect to be the intended answer to the riddle
follows.

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...

-- Noam D. Elkies <elkies@math.harvard.edu>

Thanks for the reply.  Maybe it was too hard but I certainly did not
want someone to back it up with a mathematical proof.  In fact, that
would be beyond me as well.  It was meant more as a conversational
riddle.  I had written a computer program to compute such objects and
was pleasantly surprised to find that 8 points did not create a sphere
[Editor: presumably "a cube"].  As I recall, the other 4 regular
polyhedrons were the results of 4, 6, 12 and 20 particles.

The answer you gave is the one I was looking for although I did not
know the name of such an object.  In fact, the real reason I posed the
question is because I figured everybody would jump on the 'Sphere'
[Editor: presumably "Cube"] bandwagon.

-- Roland Curit <rolandc@norfolk.infi.net>

I've seen it referred to as a square antiprism (the generalized
antiprism being where you take a prism and rotate one face so that each
vertex moves halfway towards the next one; since the vertices in a
square are 90deg apart, this gives a 45 deg rotation).

-- Martin DeMello <mdemello@rice.edu>

Mike Naylor wrote:

    Is "maximizing the minimal distance between points" the same as
    finding the rest position of particles that all repel each other
    according to the inverse-square law?  Does each particle seek a
    position where its average distance from the other particles is at a
    maximum, or does it seek a position where its distance to the
    closest particle is at a maximum, or is it some complicated function
    like maximizing the average of the square roots of the distances?

No, it is none of these (though the third is close), and I have no idea
how you might have got most of them from "maximizing the minimal
distance".  To each configuration is associated a minimal distance d,
i.e. the smallest distance occurring between different particles in the
configuration; and maximizing the minimal distance means maximizing d,
i.e. finding the configuration which makes d as large as possible.
Equivalently, find the radius d/2 of the largest spherical cap 8 of
which can be accommodated on the surface of the sphere.  It is not a
minimum energy problem, though it can be approximated by one using the
energy function exp(-Mr) for very large M.

    it might be possible to approximate the answer with a computer
    program.

Indeed; that is what was done in the computations reported in
N.J.A. Sloane's Web site.

    In any case, if the answer to the problem is not a square after all,
    and I believe you when you say that it isn't, then it's not only
    beyond a puzzle, it's beyond what most people would call homework.

No -- it's easy enough to verify that the cube (not "square") is not the
optimal solution: that only requires finding one configuration that does
better.  Checking that the other configuration *is* optimal is much
harder.

It wouldn't be much of a riddle if the answer was the obvious
symmetrical configuration!

-- Noam D. Elkies <elkies@math.harvard.edu>

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

     a = sqrt( (5 + sqrt(5)) / 2 )      b = sqrt( 2sqrt(5) / (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 vertices of a cube inscribed in a sphere.

[ Editor: a cube inscribed in the unit sphere has 12 edges of length
2/sqrt(3), 12 face diagonals of length 2sqrt(2)/sqrt(3) and 4 diameters
of length 2, so Q = 10.392 + 7.348 + 2 = 19.740, not 22.485. ]

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.

-- Kevin Brown <ksbrown@seanet.com>