solutions to the Thomson problem up to 12 points (excluding 11) have nice symmetry. (though I feel that the solution for 1 point also lacks pretty symmetry.)
2 . digon . vertex types: 1
3 . equilateral triangle . vertex types: 1
4 . regular tetrahedron . vertex types: 1
5 . triangular dipyramid . vertex types: 2
6 . regular octahedron . vertex types: 1
7 . pentagonal dipyramid . vertex types: 2
8 . square antiprism . vertex types: 1
9 . triaugmented triangular prism . vertex types: 2
10 . gyroelongated square dipyramid . vertex types: 2
12 . regular icosahedron . vertex types: 1
"vertex types" refers to vertex transitivity. we do not care about face or edge types of the convex hull polyhedron because the Thomson problem is about points (vertices).
https://www.mathpages.com/home/kmath005/kmath005.htm describes some (locally) minimum energy configurations.
maybe we care about the "balanced" entries in this Wikipedia table in which magnitude(sum(r[i])) = 0 (precisely), where r[i] are the positions of the charges as vectors. the table has no citation, but Laszlo Hars has replicated energy results for small solutions.
assuming we trust the Wikipedia table, the balanced sequence continues 2 3 4 5 6 7 8 9 10 12 14 15 16 17 18 20 22 23 24 27 28 29 30 32 which does not have an OEIS number.
there is also the sequence 2 3 4 6 8 12 24 in which the solution has only one vertex type. 24 is snub cube. there might be some more entries between 12 and 24, but probably not. the sequence probably ends at 24.
not all icosahedral geodesic spheres appear in the minimum energy table. GP(2,0) = 42, GP(3,0) = 92, GP(4,0) = 162 are the first few missing. clearly GP(n,0) seems to not be preferred. which geodesic spheres are more likely to be minimum energy solutions? this is likely similar to how cube is not minimum energy for 8 points, instead square antiprism. do icosahedral geodesic spheres continue to appear as minimum energy configurations, or is there a final geodesic sphere which is a solution?
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