Spherical codes: maximize the minimum angle
Also known asspherical codeTammes problembest-packing points on a spheremaximin angular separation
Choose n points on the unit sphere S² to maximize their minimum central angle. Since arccos is decreasing, the verifier equivalently minimizes the largest pairwise inner product.
Formal definition
- ContainerThe unit sphere S² in three-dimensional Euclidean space
- SubmissionExactly n stereographic pairs [u,v] in [-20,20]²
- Map[u,v] maps exactly to (2u,2v,1-u²-v²)/(1+u²+v²)
- ObjectiveMaximize the minimum angle, equivalently minimize the largest pairwise inner product. Exact rationals are compared by cross-multiplication and scored as ceil(max dot · 10¹⁸)
Getting a feel for it
Where it is used
Antenna beams, spacecraft attitude sampling, spherical quadrature and molecular self-assembly all need directions that stay far apart. Adding one point often forces the entire symmetric structure to reorganize.
Why a latitude-longitude grid loses
The sphere has no boundary, but it cannot be tiled by identical small neighborhoods. Pentagonal and hexagonal defects and competing local contact graphs are unavoidable. 'Uniformly distributed' names the goal; realizing it is the problem.
Only open rows remain
spherical-codes.org marks n≤14 and n=24 as proved, so the arena does not offer them as challenges. Only the unproved rows n=15…23 and 25…32 remain, each with a complete exhibited construction.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Data and citation
Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.
GET https://minmaxarena.com/data/spherical-code-on-s2.json
The first frozen edition is published at the start of next month; until then, cite with the date you accessed it.
BibTeX (click to copy)
@misc{minmaxarena-spherical-code-on-s2,
title = {{Spherical codes: maximize the minimum angle} (P68)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, accessed 2026-08-29},
url = {https://minmaxarena.com/problems/spherical-code-on-s2},
license = {CC BY 4.0}
}Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
No threads yet. Be the first to talk about this problem?