---
tag: 3C7V5X5FY8
kind: problem
name: Thomson problem
msc: [52C17, 78A30]
uses: [0NPPGHQNXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Thomson_problem
same_as:
  wikidata: Q7795912
  wikipedia: Thomson problem
history:
  posed_by: "J. J. Thomson"
  first_stated: "1904"
---

## Problem

Let $N \ge 2$ be an integer and let $S^2 \subset \R^3$ be the unit sphere.
Find the minimum of the energy
$$E(x_1, \dots, x_N) = \sum_{1 \le i < j \le N} \frac{1}{\abs{x_i - x_j}}$$
over all $N$-tuples of distinct points $x_1, \dots, x_N \in S^2$, where $\abs{x_i - x_j}$ is the [[Euclidean distance|0NPPGHQNXQ]].
Find also the configurations that attain it.

## Remarks

The energy is the electrostatic potential energy of $N$ equal point charges that repel by Coulomb's law and are confined to the sphere.
Thomson posed the question in 1904 in connection with his model of the atom.
By compactness a minimizer exists for each $N$.
Rigorous solutions are known for $N = 2, 3, 4, 5, 6$ and $12$.
They are two antipodal points, an equilateral triangle on a great circle, the regular tetrahedron, the triangular bipyramid, the regular octahedron and the regular icosahedron.
The case $N = 5$ was proved by Richard Schwartz with computer assistance.
For larger $N$ only numerical candidates are known.
The minimizers need not maximize the minimal distance, so the Thomson problem differs from the [[Tammes problem|9WD2TZ4ZB5]] in general.

## Explanations

### L1 human

Put $N$ electrons on the surface of a ball and let them push each other apart.
They settle in a pattern of lowest energy, and the problem asks which pattern this is.
For a few small numbers the answer is a regular shape, but in general it is unknown.