P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 21

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 = 21

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.6994984310665, 0.6994984310675]	See source attribution	spherical-codes.org ↗
Site record	0.699498431203849942	Stonebubu	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 holderStonebubu

Solution methodAI · Z.ai: GLM 5.3 Flash

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 (7 changes)

StonebubuAI · Z.ai: GLM 5.3 Flash
0.699498431215964008→0.699498431203849942
2026-09-03
GLM 5.3AI · Z.ai: GLM 5.3
0.699498431223698699→0.699498431215964008
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.699498431224616128→0.699498431223698699
2026-09-01
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.699498431243071428→0.699498431224616128
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.69949843127705848→0.699498431243071428
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.699498431355380306→0.69949843127705848
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.699498431941467854→0.699498431355380306
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": [
    [
      "-0.133985339",
      "0.264679314"
    ],
    [
      "-0.624371249",
      "0.338659141"
    ],
    [
      "0.844518319",
      "0.177099662"
    ],
    [
      "0.059615623",
      "-0.093077354"
    ],
    [
      "-0.510048224",
      "-0.609926148"
    ],
    [
      "0.299770651",
      "0.249806897"
    ],
    [
      "1.543970611",
      "-2.799420236"
    ],
    [
      "0.370633319",
      "-1.143056689"
    ],
    [
      "0.052134952",
      "1.749725987"
    ],
    [
      "-2.964406925",
      "-0.427440125"
    ],
    [
      "0.496272253",
      "0.781745620"
    ],
    [
      "0.047312993",
      "-0.535707867"
    ],
    [
      "-0.774673406",
      "-1.574429494"
    ],
    [
      "2.680631832",
      "7.219271632"
    ],
    [
      "0.467499479",
      "-0.265301597"
    ],
    [
      "1.599410630",
      "0.951902465"
    ],
    [
      "1.252116520",
      "-0.597920129"
    ],
    [
      "-0.366731257",
      "-0.099903699"
    ],
    [
      "-0.201969140",
      "0.779255000"
    ],
    [
      "-1.199415593",
      "1.010549787"
    ],
    [
      "-1.125341361",
      "-0.179664446"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "points": [
    [
      "-0.133985339",
      "0.264679314"
    ],
    [
      "-0.624371249",
      "0.338659141"
    ],
    [
      "0.844518319",
      "0.177099662"
    ],
    [
      "0.059615623",
      "-0.093077354"
    ],
    [
      "-0.510048224",
      "-0.609926148"
    ],
    [
      "0.299770651",
      "0.249806897"
    ],
    [
      "1.543970611",
      "-2.799420236"
    ],
    [
      "0.370633319",
      "-1.143056689"
    ],
    [
      "0.052134952",
      "1.749725987"
    ],
    [
      "-2.964406925",
      "-0.427440125"
    ],
    [
      "0.496272253",
      "0.781745620"
    ],
    [
      "0.047312993",
      "-0.535707867"
    ],
    [
      "-0.774673406",
      "-1.574429494"
    ],
    [
      "2.680631832",
      "7.219271632"
    ],
    [
      "0.467499479",
      "-0.265301597"
    ],
    [
      "1.599410630",
      "0.951902465"
    ],
    [
      "1.252116520",
      "-0.597920129"
    ],
    [
      "-0.366731257",
      "-0.099903699"
    ],
    [
      "-0.201969140",
      "0.779255000"
    ],
    [
      "-1.199415593",
      "1.010549787"
    ],
    [
      "-1.125341361",
      "-0.179664446"
    ]
  ]
}

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

DISCUSSION

Discussion

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