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P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · n = 32

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.

InstanceS², n = 32
ObjectiveMinimize largest inner product
Best known, unproven0.793616615466546398three-dimensional spherical code from spherical-codes.org, maintained by Cohn; the published largest inner product is 0.793616614872. The same directions are rewritten in stereographic coordinates, whose exact nine-decimal-grid certificate has largest inner product 0.793616615466546398; optimality is open

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)
Challenge this record ↗
VERIFIED CONSTRUCTIONOrthographic view of the spherical code

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.

Source
The record-holding arrangement for Spherical codes: maximize the minimum angle n = 32, 0.793616615466546398
Current leader

0.793616615466546398

largest inner product

Matches the best known
Answer sourceHenry Cohn's spherical-code archive
Solution methodPublished reference construction
Challenge this record ↗
ANSWER FORMAT

How to write your answer

[u,v] represents (2u, 2v, 1-u²-v²)/(1+u²+v²) on the sphere. This is an exact rational parametrization, not approximate normalization.

Submit points: exactly n stereographic pairs [u,v], each a decimal string in [-20,20] with at most nine places. The verifier maps them exactly to the unit sphere.

The current leader's answer

{
  "points": [
    [
      "-0.363146454",
      "-0.47871611"
    ],
    [
      "-1.317013042",
      "1.853843133"
    ],
    [
      "1.094924504",
      "1.296314458"
    ],
    [
      "0.157044246",
      "-1.299329666"
    ],
    [
      "-0.123502995",
      "0.736751725"
    ],
    [
      "0.332180875",
      "-0.350959105"
    ],
    [
      "4.082683793",
      "-5.762628966"
    ],
    [
      "1.248517504",
      "-0.321258419"
    ],
    [
      "-0.096949462",
      "0.052099345"
    ],
    [
      "-0.745199418",
      "0.232253064"
    ],
    [
      "0.451240317",
      "0.304629587"
    ],
    [
      "-1.564756828",
      "-0.761676369"
    ],
    [
      "2.532000491",
      "0.324838393"
    ],
    [
      "-0.309470306",
      "0.336944199"
    ],
    [
      "0.045409497",
      "-0.671420951"
    ],
    [
      "1.44272428",
      "4.091871413"
    ],
    [
      "0.665480153",
      "-0.080057647"
    ],
    [
      "-0.030657",
      "-0.276754227"
    ],
    [
      "-0.823336327",
      "-2.283145219"
    ],
    [
      "0.974825946",
      "0.387283017"
    ],
    [
      "-0.71040327",
      "0.83622507"
    ],
    [
      "-1.466098453",
      "0.408305896"
    ],
    [
      "0.065650101",
      "0.354749141"
    ],
    [
      "-0.602094197",
      "-1.003894114"
    ],
    [
      "0.411283712",
      "0.793566922"
    ],
    [
      "-0.833307032",
      "-0.305432235"
    ],
    [
      "0.001976071",
      "1.422630794"
    ],
    [
      "0.230733549",
      "0.000855862"
    ],
    [
      "1.318620968",
      "-1.484263578"
    ],
    [
      "-0.425111358",
      "-0.075248755"
    ],
    [
      "-4.333943832",
      "-0.016075929"
    ],
    [
      "0.645377439",
      "-0.740955517"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 32
}

The current leader's answer

{
  "points": [
    [
      "-0.363146454",
      "-0.47871611"
    ],
    [
      "-1.317013042",
      "1.853843133"
    ],
    [
      "1.094924504",
      "1.296314458"
    ],
    [
      "0.157044246",
      "-1.299329666"
    ],
    [
      "-0.123502995",
      "0.736751725"
    ],
    [
      "0.332180875",
      "-0.350959105"
    ],
    [
      "4.082683793",
      "-5.762628966"
    ],
    [
      "1.248517504",
      "-0.321258419"
    ],
    [
      "-0.096949462",
      "0.052099345"
    ],
    [
      "-0.745199418",
      "0.232253064"
    ],
    [
      "0.451240317",
      "0.304629587"
    ],
    [
      "-1.564756828",
      "-0.761676369"
    ],
    [
      "2.532000491",
      "0.324838393"
    ],
    [
      "-0.309470306",
      "0.336944199"
    ],
    [
      "0.045409497",
      "-0.671420951"
    ],
    [
      "1.44272428",
      "4.091871413"
    ],
    [
      "0.665480153",
      "-0.080057647"
    ],
    [
      "-0.030657",
      "-0.276754227"
    ],
    [
      "-0.823336327",
      "-2.283145219"
    ],
    [
      "0.974825946",
      "0.387283017"
    ],
    [
      "-0.71040327",
      "0.83622507"
    ],
    [
      "-1.466098453",
      "0.408305896"
    ],
    [
      "0.065650101",
      "0.354749141"
    ],
    [
      "-0.602094197",
      "-1.003894114"
    ],
    [
      "0.411283712",
      "0.793566922"
    ],
    [
      "-0.833307032",
      "-0.305432235"
    ],
    [
      "0.001976071",
      "1.422630794"
    ],
    [
      "0.230733549",
      "0.000855862"
    ],
    [
      "1.318620968",
      "-1.484263578"
    ],
    [
      "-0.425111358",
      "-0.075248755"
    ],
    [
      "-4.333943832",
      "-0.016075929"
    ],
    [
      "0.645377439",
      "-0.740955517"
    ]
  ]
}

Submit points: exactly n stereographic pairs [u,v], each a decimal string in [-20,20] with at most nine places. The verifier maps them exactly to the unit sphere. · Verifier v1.0.0

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