P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 27

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 = 27

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7583892107765, 0.7583892107775]	See source attribution	spherical-codes.org ↗
Site record	0.758389210892928813	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
0.758389210955193209→0.758389210892928813
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.758389210968574519→0.758389210955193209
2026-09-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.758389210971543708→0.758389210968574519
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.758389211108826776→0.758389210971543708
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.758389211722986307→0.758389211108826776
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": [
    [
      "-0.914895487",
      "0.400295451"
    ],
    [
      "0.914895487",
      "-0.400295451"
    ],
    [
      "-0.055392491",
      "0.366519147"
    ],
    [
      "0.055392491",
      "-0.366519147"
    ],
    [
      "0.306786470",
      "0.208054500"
    ],
    [
      "-0.306786470",
      "-0.208054500"
    ],
    [
      "5.085477824",
      "-2.225055948"
    ],
    [
      "-5.085477824",
      "2.225055948"
    ],
    [
      "0.291439049",
      "0.666098678"
    ],
    [
      "-0.291439049",
      "-0.666098678"
    ],
    [
      "-1.234556583",
      "-2.821641465"
    ],
    [
      "1.234556583",
      "2.821641465"
    ],
    [
      "-0.952997384",
      "-0.953558096"
    ],
    [
      "0.952997384",
      "0.953558096"
    ],
    [
      "0.053600565",
      "1.347072395"
    ],
    [
      "-0.053600565",
      "-1.347072395"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.784856073",
      "0.219921716"
    ],
    [
      "-0.784856073",
      "-0.219921716"
    ],
    [
      "-1.793055259",
      "-0.097857961"
    ],
    [
      "1.793055259",
      "0.097857961"
    ],
    [
      "-0.419871201",
      "0.183706810"
    ],
    [
      "0.419871201",
      "-0.183706810"
    ],
    [
      "-1.144983792",
      "1.383342133"
    ],
    [
      "1.144983792",
      "-1.383342133"
    ],
    [
      "0.371118352",
      "-0.725696759"
    ],
    [
      "-0.371118352",
      "0.725696759"
    ]
  ]
}

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

Instance parameters

{
  "n": 27
}

The current leader's answer

{
  "points": [
    [
      "-0.914895487",
      "0.400295451"
    ],
    [
      "0.914895487",
      "-0.400295451"
    ],
    [
      "-0.055392491",
      "0.366519147"
    ],
    [
      "0.055392491",
      "-0.366519147"
    ],
    [
      "0.306786470",
      "0.208054500"
    ],
    [
      "-0.306786470",
      "-0.208054500"
    ],
    [
      "5.085477824",
      "-2.225055948"
    ],
    [
      "-5.085477824",
      "2.225055948"
    ],
    [
      "0.291439049",
      "0.666098678"
    ],
    [
      "-0.291439049",
      "-0.666098678"
    ],
    [
      "-1.234556583",
      "-2.821641465"
    ],
    [
      "1.234556583",
      "2.821641465"
    ],
    [
      "-0.952997384",
      "-0.953558096"
    ],
    [
      "0.952997384",
      "0.953558096"
    ],
    [
      "0.053600565",
      "1.347072395"
    ],
    [
      "-0.053600565",
      "-1.347072395"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.784856073",
      "0.219921716"
    ],
    [
      "-0.784856073",
      "-0.219921716"
    ],
    [
      "-1.793055259",
      "-0.097857961"
    ],
    [
      "1.793055259",
      "0.097857961"
    ],
    [
      "-0.419871201",
      "0.183706810"
    ],
    [
      "0.419871201",
      "-0.183706810"
    ],
    [
      "-1.144983792",
      "1.383342133"
    ],
    [
      "1.144983792",
      "-1.383342133"
    ],
    [
      "0.371118352",
      "-0.725696759"
    ],
    [
      "-0.371118352",
      "0.725696759"
    ]
  ]
}

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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