P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 28

Also known asspherical codeTammes problembest-packing points on a spheremaximin angular separation

Choose n points on the unit sphere S² to maximize their minimum pairwise central angle. Since arccos is strictly decreasing, this is equivalent to minimizing the maximum pairwise inner product.

InstanceS², n = 28

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7732302623375, 0.7732302623385]	See source attribution	spherical-codes.org ↗
Site record	0.773230262512644572	zzzcy	Record details ↓

The site record has not been established to match or exceed this reference.

Formal definition

ContainerThe unit sphere S² in three-dimensional Euclidean space
SubmissionExactly n stereographic pairs [u,v] in [-20,20]²
MapStereographic projection maps [u,v] exactly to the unit-sphere point below $(u, v) \mapsto \frac{(2 u, 2 v, 1 - u^{2} - v^{2})}{1 + u^{2} + v^{2}}$
Scorepᵢ is the mapped unit vector, and μ(P) is the largest inner product over all pairs; the verifier compares exact rationals by cross-multiplication and records ceil(μ·10¹⁸) $μ (P) = \max_{1 \leq i < j \leq n} p_{i} \cdot p_{j}$
Minimum angleBecause arccos is strictly decreasing on [-1, 1], the largest inner product corresponds to the smallest central angle $θ (P) = \arccos (μ (P))$
ObjectiveMinimize μ(P), equivalently maximize the minimum angle θ(P) $\min_{P} μ (P) \Leftrightarrow \max_{P} θ (P)$

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Layout shown: site record

VERIFIED CONSTRUCTIONOrthographic view of the spherical code

Getting a feel for it

Applications

Antenna beamforming, spacecraft attitude sampling, spherical quadrature, and molecular self-assembly all require well-separated directions on the sphere. Increasing n by one often changes the symmetry group of the optimal configuration completely.

Topological defects on the sphere

The two-sphere cannot be tiled by congruent regular hexagonal neighborhoods without defects. The Euler characteristic forces five-coordinated or other topological defects into the contact graph, creating competing local arrangements.

Open benchmark instances

spherical-codes.org lists n ≤ 14 and n = 24 as proved optimal. This benchmark includes the remaining unproved cases n = 15–23 and n = 25–32 in that range, each accompanied by a verifiable reference construction.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.773230262514504334→0.773230262512644572
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.773230262521368591→0.773230262514504334
2026-09-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.773230262529779711→0.773230262521368591
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.773230262658023704→0.773230262529779711
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.773230263147002607→0.773230262658023704
2026-09-01

ANSWER FORMAT

How to write your answer

[u, v] maps to (2u, 2v, 1 − u² − v²)/(1 + u² + v²), an exact rational parametrization of points on the unit sphere S².

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": [
    [
      "0.130233008",
      "0.634701177"
    ],
    [
      "-2.076982685",
      "0.266130674"
    ],
    [
      "-1.011002677",
      "-0.049766303"
    ],
    [
      "-0.339724537",
      "-1.836730312"
    ],
    [
      "-0.343000735",
      "0.557353010"
    ],
    [
      "0.317914789",
      "-0.456556773"
    ],
    [
      "2.495629529",
      "0.039058314"
    ],
    [
      "-0.050529910",
      "0.260958862"
    ],
    [
      "1.257993794",
      "-1.595852208"
    ],
    [
      "0.574278456",
      "-0.091894807"
    ],
    [
      "0.859469545",
      "-0.574070857"
    ],
    [
      "-0.949269369",
      "0.685381356"
    ],
    [
      "-1.243233176",
      "-0.894925495"
    ],
    [
      "-1.137080165",
      "1.952727096"
    ],
    [
      "1.190879606",
      "1.478139393"
    ],
    [
      "-0.502775516",
      "0.132837082"
    ],
    [
      "1.128631367",
      "0.106698855"
    ],
    [
      "2.593487344",
      "6.142882269"
    ],
    [
      "-0.104820755",
      "-0.440756823"
    ],
    [
      "-0.174786563",
      "-0.071123790"
    ],
    [
      "-0.515992049",
      "-0.309953940"
    ],
    [
      "0.325701489",
      "0.247745176"
    ],
    [
      "0.034530368",
      "1.279275517"
    ],
    [
      "0.677992785",
      "0.635702958"
    ],
    [
      "-3.560757651",
      "-3.326186546"
    ],
    [
      "-0.396783946",
      "-0.857066372"
    ],
    [
      "0.305578151",
      "-1.019029224"
    ],
    [
      "0.174092784",
      "-0.090881836"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 28
}

The current leader's answer

{
  "points": [
    [
      "0.130233008",
      "0.634701177"
    ],
    [
      "-2.076982685",
      "0.266130674"
    ],
    [
      "-1.011002677",
      "-0.049766303"
    ],
    [
      "-0.339724537",
      "-1.836730312"
    ],
    [
      "-0.343000735",
      "0.557353010"
    ],
    [
      "0.317914789",
      "-0.456556773"
    ],
    [
      "2.495629529",
      "0.039058314"
    ],
    [
      "-0.050529910",
      "0.260958862"
    ],
    [
      "1.257993794",
      "-1.595852208"
    ],
    [
      "0.574278456",
      "-0.091894807"
    ],
    [
      "0.859469545",
      "-0.574070857"
    ],
    [
      "-0.949269369",
      "0.685381356"
    ],
    [
      "-1.243233176",
      "-0.894925495"
    ],
    [
      "-1.137080165",
      "1.952727096"
    ],
    [
      "1.190879606",
      "1.478139393"
    ],
    [
      "-0.502775516",
      "0.132837082"
    ],
    [
      "1.128631367",
      "0.106698855"
    ],
    [
      "2.593487344",
      "6.142882269"
    ],
    [
      "-0.104820755",
      "-0.440756823"
    ],
    [
      "-0.174786563",
      "-0.071123790"
    ],
    [
      "-0.515992049",
      "-0.309953940"
    ],
    [
      "0.325701489",
      "0.247745176"
    ],
    [
      "0.034530368",
      "1.279275517"
    ],
    [
      "0.677992785",
      "0.635702958"
    ],
    [
      "-3.560757651",
      "-3.326186546"
    ],
    [
      "-0.396783946",
      "-0.857066372"
    ],
    [
      "0.305578151",
      "-1.019029224"
    ],
    [
      "0.174092784",
      "-0.090881836"
    ]
  ]
}

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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