Thomson Problem
Place N identical charges on the surface of a sphere and let them repel each other until they find equilibrium. The resulting configurations are beautiful, symmetric, and — for most N — still not proven optimal.
The Problem
The Thomson problem asks: how should N identical point charges be arranged on the surface of a unit sphere to minimize the total electrostatic potential energy? The energy is the sum of all 1/rij terms for every pair of particles, where rij is the Euclidean distance between particles i and j.
Why It Matters
J.J. Thomson posed this problem in 1904 as part of his "plum pudding" model of the atom — he imagined electrons embedded in a uniform positive charge, and wanted to know how they'd arrange themselves. The atomic model was wrong, but the mathematical problem turned out to be deep and beautiful.
It connects to coding theory (distributing points on a sphere to maximize minimum distance), viral capsid geometry (many viruses have icosahedral symmetry), fullerene chemistry (C60 buckminsterfullerene), and computational geometry.
Known Optimal Solutions
For small N the optimal configurations are known and often correspond to familiar polyhedra: N=4 gives a tetrahedron, N=6 an octahedron, N=8 a square antiprism (not a cube!), and N=12 an icosahedron. For larger N, the problem is computationally hard and optimal solutions are known only through extensive numerical search.
The Simulation
This simulation starts with random particle placements and uses gradient descent with damping: each particle feels the repulsive Coulomb force from every other particle. The tangential component of the force (projected onto the sphere surface) moves the particle, and damping gradually removes kinetic energy until the system settles. The result is usually a local minimum — not always the global one. Hit Reset to try a different random start.