Explicit bound for symmetric point configurations on the sphere

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $S^2=\{x\in\mathbb R^3:\|x\|_2=1\}$. A nonempty finite subset $X\subset S^2$ is symmetric if a finite group acts transitively on $X$ by isometries of its induced Euclidean metric. A great circle is the intersection of $S^2$ with a plane through the origin; two parallel circles are intersections with two parallel affine planes.

Statement recorded in the supplied source.

Determine an explicit absolute constant $B$ such that $|X|\le B$ for every symmetric $X$ contained in neither a great circle nor a union of two parallel circles. Is the sharp bound $B=60$, attained by the vertex set of a truncated icosahedron centered at the origin and scaled onto $S^2$? [S2]

Short English statement

Give an explicit size bound for symmetric spherical point sets outside the exceptional circle configurations and decide whether the soccer-ball vertices are maximal.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000004 (AMR-099-0004). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Points in Equilibrium, May 2015; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Unnumbered sphere addendum following Question 0.1, p. 1. https://arquivo.pt/wayback/20201231041612id_/https://www.wisdom.weizmann.ac.il/~itai/equilibrium.pdf.