P68 · Extremal configurations · Classic · Applied frontier · Hard

Spherical codes: maximize the minimum angle · S², n = 29

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 = 29

ObjectiveMinimize largest inner product

Record comparison

largest inner product · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7802814159485, 0.7802814159495]	See source attribution	spherical-codes.org ↗
Site record	0.780281416103162069	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) $\min_{P} μ (P) \Leftrightarrow \max_{P} θ (P)$

Open full editor ↗

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 (5 changes)

StonebubuAI · Z.ai: GLM 5.3 Flash
0.78028141611239982→0.780281416103162069
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.780281416145994938→0.78028141611239982
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.780281416151146284→0.780281416145994938
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.780281416278158353→0.780281416151146284
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.780281416834226978→0.780281416278158353
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": [
    [
      "1.324453654",
      "1.065194125"
    ],
    [
      "-0.388380666",
      "-0.339144470"
    ],
    [
      "0.568854073",
      "3.071313676"
    ],
    [
      "-0.660045084",
      "0.015657478"
    ],
    [
      "0.265652445",
      "1.295055054"
    ],
    [
      "0.744071553",
      "0.004223151"
    ],
    [
      "1.475060149",
      "-0.091378169"
    ],
    [
      "1.406953124",
      "-1.406550721"
    ],
    [
      "-1.359270075",
      "-1.456618723"
    ],
    [
      "0.237619733",
      "-0.809303217"
    ],
    [
      "-0.932392047",
      "0.530552450"
    ],
    [
      "-0.897995373",
      "-0.502134245"
    ],
    [
      "-0.396419628",
      "0.385011489"
    ],
    [
      "0.088178478",
      "-4.696402039"
    ],
    [
      "-0.015082984",
      "0.286177460"
    ],
    [
      "0.027703688",
      "-1.600223224"
    ],
    [
      "3.739089523",
      "0.287405640"
    ],
    [
      "-1.718455704",
      "-0.031960069"
    ],
    [
      "0.796752116",
      "-0.572076612"
    ],
    [
      "-0.250540481",
      "0.019812714"
    ],
    [
      "-0.061354009",
      "0.706285460"
    ],
    [
      "-0.785188092",
      "1.460086437"
    ],
    [
      "0.352748486",
      "0.221370859"
    ],
    [
      "0.008608669",
      "-0.401924129"
    ],
    [
      "-0.348067684",
      "-0.842024047"
    ],
    [
      "0.381066195",
      "-0.272054527"
    ],
    [
      "-4.512718708",
      "1.557597825"
    ],
    [
      "0.574744865",
      "0.639653100"
    ],
    [
      "0.088251472",
      "-0.046406931"
    ]
  ]
}

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

Instance parameters

{
  "n": 29
}

The current leader's answer

{
  "points": [
    [
      "1.324453654",
      "1.065194125"
    ],
    [
      "-0.388380666",
      "-0.339144470"
    ],
    [
      "0.568854073",
      "3.071313676"
    ],
    [
      "-0.660045084",
      "0.015657478"
    ],
    [
      "0.265652445",
      "1.295055054"
    ],
    [
      "0.744071553",
      "0.004223151"
    ],
    [
      "1.475060149",
      "-0.091378169"
    ],
    [
      "1.406953124",
      "-1.406550721"
    ],
    [
      "-1.359270075",
      "-1.456618723"
    ],
    [
      "0.237619733",
      "-0.809303217"
    ],
    [
      "-0.932392047",
      "0.530552450"
    ],
    [
      "-0.897995373",
      "-0.502134245"
    ],
    [
      "-0.396419628",
      "0.385011489"
    ],
    [
      "0.088178478",
      "-4.696402039"
    ],
    [
      "-0.015082984",
      "0.286177460"
    ],
    [
      "0.027703688",
      "-1.600223224"
    ],
    [
      "3.739089523",
      "0.287405640"
    ],
    [
      "-1.718455704",
      "-0.031960069"
    ],
    [
      "0.796752116",
      "-0.572076612"
    ],
    [
      "-0.250540481",
      "0.019812714"
    ],
    [
      "-0.061354009",
      "0.706285460"
    ],
    [
      "-0.785188092",
      "1.460086437"
    ],
    [
      "0.352748486",
      "0.221370859"
    ],
    [
      "0.008608669",
      "-0.401924129"
    ],
    [
      "-0.348067684",
      "-0.842024047"
    ],
    [
      "0.381066195",
      "-0.272054527"
    ],
    [
      "-4.512718708",
      "1.557597825"
    ],
    [
      "0.574744865",
      "0.639653100"
    ],
    [
      "0.088251472",
      "-0.046406931"
    ]
  ]
}

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

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.