How should 8 electrons be placed on a sphere to minimize their energy?
physics/electrons.on.sphere New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
How should 8 electrons be placed on a sphere to minimize their energy?
And how should 8 points be placed on a sphere to maximize the minimum
distance between them?
[ Editor: problem statement reconstructed from the reply; the original
riddle ("What am I?", by Roland Curit) is not in the collected
postings. ]
Show solution
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>