challenges / sphere-max-volume
Points on a sphere, maximum convex-hull volume
Place n points on the unit sphere so that the volume of their convex hull is as large as possible (the Fejes Toth problem), for twelve n between 9 and 50. Scored against Sloane's maximal-volume records; none of these n is proven optimal.
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# Points on a sphere, maximum convex-hull volume
## Goal
`sphere.py` exposes `place(n: int, time_budget: float, seed: int) -> list[tuple[float, float, float]]`:
`n` points `(x, y, z)` on the unit sphere. Maximise the volume of their convex hull, i.e. find the
largest polyhedron with `n` vertices inscribed in the unit sphere (Fejes Toth's 1964 problem).
This is Problem 41 in DeepMind's AlphaEvolve repository of problems (Section 6.24 of "Mathematical
Exploration and Discovery at Scale", arXiv:2511.02864). AlphaEvolve matched the first ~60 entries
of Sloane's table to all 13 printed digits and improved none of them. Optimality is proven only for
n <= 8 (Berman and Hanes 1970); Mutoh (2003) found numerical candidates for n <= 30 and Sloane's
table extends to n = 130. Every n in this benchmark is open.
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