P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 25

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 = 25

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7473986285775, 0.7473986285785]	See source attribution	spherical-codes.org ↗
Site record	0.74739862874508052	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 (8 changes)

StonebubuAI · Z.ai: GLM 5.3 Flash
0.74739862875465612→0.74739862874508052
2026-09-02
邱仲普AI · Z.ai: GLM 5.3
0.747398628757812259→0.74739862875465612
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.747398628765255684→0.747398628757812259
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.747398628770279459→0.747398628765255684
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.747398628770522867→0.747398628770279459
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.747398628770679472→0.747398628770522867
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.74739862882045302→0.747398628770679472
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.747398629631187568→0.74739862882045302
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": [
    [
      "0.806541193",
      "-1.013953484"
    ],
    [
      "-1.223756701",
      "-0.087363208"
    ],
    [
      "0.343346535",
      "0.778478538"
    ],
    [
      "-1.765447698",
      "-1.344947823"
    ],
    [
      "5.334966479",
      "-1.041507122"
    ],
    [
      "-0.230387470",
      "1.291709870"
    ],
    [
      "-0.269266039",
      "0.170681986"
    ],
    [
      "-4.911903525",
      "2.189687707"
    ],
    [
      "-1.330450229",
      "0.999451680"
    ],
    [
      "0.752373529",
      "-0.271150909"
    ],
    [
      "-0.718246959",
      "-0.704857726"
    ],
    [
      "0.261955035",
      "-2.477127146"
    ],
    [
      "-0.178316397",
      "0.601537177"
    ],
    [
      "1.629836807",
      "-0.251951410"
    ],
    [
      "0.837104573",
      "2.665501801"
    ],
    [
      "0.329509394",
      "-0.044944055"
    ],
    [
      "-0.179324360",
      "-0.509858023"
    ],
    [
      "1.156260141",
      "0.849153596"
    ],
    [
      "-0.053146482",
      "-1.096250906"
    ],
    [
      "0.295711546",
      "-0.554600531"
    ],
    [
      "0.640303266",
      "0.291346553"
    ],
    [
      "-0.039266679",
      "-0.127146140"
    ],
    [
      "-0.572835401",
      "-0.138008422"
    ],
    [
      "-0.690218862",
      "0.393850824"
    ],
    [
      "0.164088862",
      "0.316643748"
    ]
  ]
}

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

Instance parameters

{
  "n": 25
}

The current leader's answer

{
  "points": [
    [
      "0.806541193",
      "-1.013953484"
    ],
    [
      "-1.223756701",
      "-0.087363208"
    ],
    [
      "0.343346535",
      "0.778478538"
    ],
    [
      "-1.765447698",
      "-1.344947823"
    ],
    [
      "5.334966479",
      "-1.041507122"
    ],
    [
      "-0.230387470",
      "1.291709870"
    ],
    [
      "-0.269266039",
      "0.170681986"
    ],
    [
      "-4.911903525",
      "2.189687707"
    ],
    [
      "-1.330450229",
      "0.999451680"
    ],
    [
      "0.752373529",
      "-0.271150909"
    ],
    [
      "-0.718246959",
      "-0.704857726"
    ],
    [
      "0.261955035",
      "-2.477127146"
    ],
    [
      "-0.178316397",
      "0.601537177"
    ],
    [
      "1.629836807",
      "-0.251951410"
    ],
    [
      "0.837104573",
      "2.665501801"
    ],
    [
      "0.329509394",
      "-0.044944055"
    ],
    [
      "-0.179324360",
      "-0.509858023"
    ],
    [
      "1.156260141",
      "0.849153596"
    ],
    [
      "-0.053146482",
      "-1.096250906"
    ],
    [
      "0.295711546",
      "-0.554600531"
    ],
    [
      "0.640303266",
      "0.291346553"
    ],
    [
      "-0.039266679",
      "-0.127146140"
    ],
    [
      "-0.572835401",
      "-0.138008422"
    ],
    [
      "-0.690218862",
      "0.393850824"
    ],
    [
      "0.164088862",
      "0.316643748"
    ]
  ]
}

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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