Convex and discrete geometry Packing and covering Problem

Thomson problem

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. 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 in general.

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Euclidean distance concept
Thomson problem problem

Tag	`3C7V5X5FY8`
URI	`m2://3C7V5X5FY8`
Kind	problem
Writing	W0 stub
MSC	52C17, 78A30
Status	This copy of the site has no build data. Run `m2 status 3C7V5X5FY8`.
Uses	Euclidean distance
Same as	Wikidata Q7795912 Wikipedia Thomson problem

Thomson problem

Problem

Remarks

Reading path

Further reading

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