The Thomson problem (minimum-energy charges on a sphere)
Find configurations of n unit point charges on the sphere that minimise Coulomb energy. Global optimality is proven only for a few n, so the tasks are to lower best known energies and to prove new cases optimal.
Cite
@misc{cairn-thomson-problem,
title = {The Thomson problem (minimum-energy charges on a sphere)},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/thomson-problem}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
Place n points on the unit sphere S^2 so that the Coulomb energy Σ_{i<j} 1/|x_i − x_j| is minimal. The problem goes back to J. J. Thomson (1904). It is related to Smale's 7th problem, which uses the logarithmic potential.
Known status. Rigorous global minimisers are known only for n = 2, 3, 4, 5, 6 and 12 (per Wikipedia). The n = 5 case, the triangular bipyramid, has a computer-assisted proof by R. E. Schwartz. The octahedron (n = 6) and icosahedron (n = 12) cases are older results by Yudin and Andreev. For larger n, putative global minima from numerical searches are tabulated, e.g. in the Cambridge Cluster Database (Wales et al.).
What counts as progress
- A configuration for some n with energy strictly below the best tabulated value, given as coordinates.
- A rigorous (e.g. interval-arithmetic) proof that a known configuration is globally optimal for a new n, with code, starting from n = 7.
- Independent reproduction of tabulated minima for a range of n with open-source code, flagging any entries that cannot be reproduced.
- Studies of the energy landscape (number of local minima, defect structure at large n) with reproducible data.
How it is checked. Configurations are verified by recomputing the energy in high precision after normalising to the sphere, and comparing against the reference tables. Optimality proofs are checked by re-running the published verification code, and reviewers examine the case-splitting argument.