Given $n$, consider configurations of n points on the sphere such that there is a finite subgroup of $SO(3)$ acting transitively on them. When $n$ is small, there are a few ways to do so which are related to platonic bodies. However, I suspect that when n is sufficiently large, the only way is to put them on a circle in a rotational symmetric way. Is it true?
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See here for more material on related problems. – Jyrki Lahtonen Feb 12 '25 at 08:40
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And here or here for more about the groups. – Jyrki Lahtonen Feb 12 '25 at 08:43
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Yes. This follows from the well-known classification of finite subgroups of $SO(3)$; they are
- the cyclic groups $C_n$,
- the dihedral groups $D_n$,
- the alternating group $A_4$ acting on the tetrahedron,
- the symmetric group $S_4$ acting on the cube and octahedron, and
- the alternating group $A_5$ acting on the icosahedron and dodecahedron.
Since you want a finite subgroup $G$ of $SO(3)$ to act transitively on your $n$ points, the number of points must divide $|G|$. So when $n > 60$ this forces $G$ to be either a cyclic or dihedral group $C_n, D_n$, whose orbits acting on the sphere are regular $n$-gons arranged along a circle (edit: or pairs of such $n$-gons in the dihedral case, as in the comments), as you might expect.
Qiaochu Yuan
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4I think there is also the configuration of 2n points placed on two parallel circles, whose symmetry group is the dihedral group. – Guangbo Xu Jan 18 '25 at 22:31
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