How do you optimally arrange N repulsive points on a sphere?
Just over 100 years ago, Thomson considered this very problem in an attempt
to explain the periodic table in terms of rigid electron shells, the
"plum pudding" model of the atom.
The problem has resurfaced in many fields including multielectron bubbles in superfluid helium,
virus morphology, protein s-layers, coding theory, is equivalent to many other problems in biology, math,
physics, and computer science, and can be applied to such problems as structural chemsitry, the design of
multibeam laser implosion devices, and the optimum placement of communication satellites.
Spherical packing on a flat surface is most efficient in a simple lattice of triangles (Imagine billiard
balls at the start of a game). Each sphere will own a single six-coordinate point on the lattice. However,
when this packing is attempted on a sphere, defects are forced on the lattice (Soccer Balls, C60 fullerenes,
etc). From Euler's Invariant F - E + V = 2 of polyhedra, we can show that if we characterize each point by
their topological or disclination charge, qi, which is the departure of their coordination number, ci, for
some particle i from the preferred flat space value of 6 (q = 6 - c), that the total disclination charge of
any triangulation of the sphere must be 12. In consideration of repulsive particles, the configurations that
minimize the number of nonzero qi are not guaranteed to be minimal in energy.