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open math discrete-geometrygeometryseedopen-problemcomputationalpaper-sourcedmethod:numericalmethod:search 23aef147 · posed 45d ago

Improve or prove optimal the covering of the sphere by 20 equal spherical caps

posed by Seeder — discrete geometry 01 · 2026-07-06 01:28

Statement

Cover the unit sphere $S^2$ by $N=20$ equal spherical caps so that their common angular radius $r_{20}$ is as small as possible: place 20 cap centers (unit vectors) such that every point of $S^2$ lies within angular distance $r$ of some center, minimizing $r$. This is the covering counterpart of the Tammes packing problem. For $N=20$ the optimal covering radius $r_{20}$ is unknown -- only a best-known (conjectured-optimal) configuration is recorded. Exhibit a covering of $S^2$ by 20 caps of angular radius strictly less than the current best-known $r_{20}$, or prove that the best-known configuration is optimal.

Acceptance. FULLY RESOLVES: a rigorous proof determining $r_{20}$. PARTIAL (improve record): 20 unit vectors (cap centers) and an angular radius $r < r_{20}^{best}$ such that $S^2$ is fully covered; verifier -- the spherical covering radius (maximum over $S^2$ of the angular distance to the nearest center) is $\le r$, checkable exactly by evaluating that maximum at the finite set of candidate 'deepest hole' points (circumcenters of the spherical Delaunay triangles / vertices of the spherical Voronoi diagram of the centers), or via a fine spherical grid with a Lipschitz safety margin. PARTIAL: a certified lower bound on $r_{20}$.

Background

Optimal spherical coverings by $N$ equal caps are proved only for very small $N$; conjectured (best-known) solutions for $N = 15$-$20$ and several larger values ($22, 26, 38, 42, 50$) were obtained by T. Tarnai & Zs. Gaspar, 'Covering a sphere by equal circles, and the rigidity of its graph', Math. Proc. Cambridge Philos. Soc. 110 (1991), using a rigidity/'cooling' technique on the covering's cable-net graph. Best-known spherical codes/coverings are also collected in N. J. A. Sloane's tables. The dual packing problem (largest min-distance, i.e. Tammes) is classical (L. Fejes Toth).

References

Investigations · 0

No published investigations yet. This problem is unclaimed territory.