P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 30

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 = 30

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7815518750945, 0.7815518750955]	See source attribution	spherical-codes.org ↗
Site record	0.781551875216503283	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
0.781551875266463232→0.781551875216503283
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.781551875305512436→0.781551875266463232
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.781551875307850813→0.781551875305512436
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.781551875412752461→0.781551875307850813
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.781551875763660801→0.781551875412752461
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": [
    [
      "-0.155194097",
      "-0.705524828"
    ],
    [
      "0.155194097",
      "0.705524828"
    ],
    [
      "-0.747331334",
      "-0.750284307"
    ],
    [
      "0.747331334",
      "0.750284307"
    ],
    [
      "0.381363497",
      "-0.891845045"
    ],
    [
      "-0.381363497",
      "0.891845045"
    ],
    [
      "0.192464983",
      "-0.420971363"
    ],
    [
      "-0.192464983",
      "0.420971363"
    ],
    [
      "-0.230650973",
      "-1.413767500"
    ],
    [
      "0.230650973",
      "1.413767500"
    ],
    [
      "1.014617832",
      "-1.687265579"
    ],
    [
      "-1.014617832",
      "1.687265579"
    ],
    [
      "-0.411407327",
      "-0.323311564"
    ],
    [
      "0.411407327",
      "0.323311564"
    ],
    [
      "-4.635493352",
      "1.484938549"
    ],
    [
      "4.635493352",
      "-1.484938549"
    ],
    [
      "0.863672987",
      "0.122210104"
    ],
    [
      "-0.863672987",
      "-0.122210104"
    ],
    [
      "-0.342446573",
      "0.065645491"
    ],
    [
      "0.342446573",
      "-0.065645491"
    ],
    [
      "-0.645855391",
      "0.380777542"
    ],
    [
      "0.645855391",
      "-0.380777542"
    ],
    [
      "-1.073046096",
      "-3.349699586"
    ],
    [
      "1.073046096",
      "3.349699586"
    ],
    [
      "-0.065744317",
      "-0.156796740"
    ],
    [
      "0.065744317",
      "0.156796740"
    ],
    [
      "-1.381448654",
      "0.452876029"
    ],
    [
      "1.381448654",
      "-0.452876029"
    ],
    [
      "1.805539129",
      "0.756951051"
    ],
    [
      "-1.805539129",
      "-0.756951051"
    ]
  ]
}

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

Instance parameters

{
  "n": 30
}

The current leader's answer

{
  "points": [
    [
      "-0.155194097",
      "-0.705524828"
    ],
    [
      "0.155194097",
      "0.705524828"
    ],
    [
      "-0.747331334",
      "-0.750284307"
    ],
    [
      "0.747331334",
      "0.750284307"
    ],
    [
      "0.381363497",
      "-0.891845045"
    ],
    [
      "-0.381363497",
      "0.891845045"
    ],
    [
      "0.192464983",
      "-0.420971363"
    ],
    [
      "-0.192464983",
      "0.420971363"
    ],
    [
      "-0.230650973",
      "-1.413767500"
    ],
    [
      "0.230650973",
      "1.413767500"
    ],
    [
      "1.014617832",
      "-1.687265579"
    ],
    [
      "-1.014617832",
      "1.687265579"
    ],
    [
      "-0.411407327",
      "-0.323311564"
    ],
    [
      "0.411407327",
      "0.323311564"
    ],
    [
      "-4.635493352",
      "1.484938549"
    ],
    [
      "4.635493352",
      "-1.484938549"
    ],
    [
      "0.863672987",
      "0.122210104"
    ],
    [
      "-0.863672987",
      "-0.122210104"
    ],
    [
      "-0.342446573",
      "0.065645491"
    ],
    [
      "0.342446573",
      "-0.065645491"
    ],
    [
      "-0.645855391",
      "0.380777542"
    ],
    [
      "0.645855391",
      "-0.380777542"
    ],
    [
      "-1.073046096",
      "-3.349699586"
    ],
    [
      "1.073046096",
      "3.349699586"
    ],
    [
      "-0.065744317",
      "-0.156796740"
    ],
    [
      "0.065744317",
      "0.156796740"
    ],
    [
      "-1.381448654",
      "0.452876029"
    ],
    [
      "1.381448654",
      "-0.452876029"
    ],
    [
      "1.805539129",
      "0.756951051"
    ],
    [
      "-1.805539129",
      "-0.756951051"
    ]
  ]
}

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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