P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 23

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 = 23

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7228469848685, 0.7228469848695]	See source attribution	spherical-codes.org ↗
Site record	0.722846985003268662	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 (6 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.722846985007606277→0.722846985003268662
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.722846985032651353→0.722846985007606277
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.7228469850332465→0.722846985032651353
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.722846985047700715→0.7228469850332465
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.7228469851226897→0.722846985047700715
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.72284698595527336→0.7228469851226897
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": [
    [
      "1.246509988",
      "-0.538334444"
    ],
    [
      "0.298837680",
      "0.267516804"
    ],
    [
      "0.368213672",
      "-0.159021672"
    ],
    [
      "0.010141752",
      "-0.400956786"
    ],
    [
      "1.579480810",
      "0.661013555"
    ],
    [
      "0.601708841",
      "-1.603011236"
    ],
    [
      "0.794931316",
      "0.129294662"
    ],
    [
      "3.545639363",
      "-1.531267146"
    ],
    [
      "-1.007362018",
      "-0.891836973"
    ],
    [
      "-2.129667572",
      "0.055400589"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.512271217",
      "1.849758663"
    ],
    [
      "0.450889685",
      "-0.667331551"
    ],
    [
      "-0.934136070",
      "-0.040652385"
    ],
    [
      "0.570446178",
      "0.746036818"
    ],
    [
      "-0.016321197",
      "0.618374711"
    ],
    [
      "-3.118396993",
      "5.827872923"
    ],
    [
      "-0.395560504",
      "0.066340689"
    ],
    [
      "-0.151989024",
      "-0.926757310"
    ],
    [
      "-0.460358002",
      "-0.405308864"
    ],
    [
      "-1.008347240",
      "1.375513530"
    ],
    [
      "-0.569421620",
      "0.546875243"
    ],
    [
      "-1.630878369",
      "-3.087157602"
    ]
  ]
}

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

Instance parameters

{
  "n": 23
}

The current leader's answer

{
  "points": [
    [
      "1.246509988",
      "-0.538334444"
    ],
    [
      "0.298837680",
      "0.267516804"
    ],
    [
      "0.368213672",
      "-0.159021672"
    ],
    [
      "0.010141752",
      "-0.400956786"
    ],
    [
      "1.579480810",
      "0.661013555"
    ],
    [
      "0.601708841",
      "-1.603011236"
    ],
    [
      "0.794931316",
      "0.129294662"
    ],
    [
      "3.545639363",
      "-1.531267146"
    ],
    [
      "-1.007362018",
      "-0.891836973"
    ],
    [
      "-2.129667572",
      "0.055400589"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.512271217",
      "1.849758663"
    ],
    [
      "0.450889685",
      "-0.667331551"
    ],
    [
      "-0.934136070",
      "-0.040652385"
    ],
    [
      "0.570446178",
      "0.746036818"
    ],
    [
      "-0.016321197",
      "0.618374711"
    ],
    [
      "-3.118396993",
      "5.827872923"
    ],
    [
      "-0.395560504",
      "0.066340689"
    ],
    [
      "-0.151989024",
      "-0.926757310"
    ],
    [
      "-0.460358002",
      "-0.405308864"
    ],
    [
      "-1.008347240",
      "1.375513530"
    ],
    [
      "-0.569421620",
      "0.546875243"
    ],
    [
      "-1.630878369",
      "-3.087157602"
    ]
  ]
}

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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