P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · d = 4, n = 19

Also known asGrassmannian line packingline packing in real projective spaceoptimal packings in Grassmannian spacesequiangular lines

Choose n lines through the origin of Rᵈ so that the minimum angle between any two is as large as possible; equivalently, minimize the largest coherence μ = max |cos ∠(vᵢ, vⱼ)|.

Instanced = 4, n = 19

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.6466489037723639, 0.6466489038496857]	See source attribution	cohn.mit.edu ↗
Site record	0.646648904	zzzcy	Record details ↓

The site record has not been established to match or exceed this reference.

Formal definition

ContainerReal d-space Rᵈ, every line through the origin; an answer is n points of real projective space RPᵈ⁻¹
SubmissionExactly n nonzero vectors, d coordinates each; a vector stands for the line it spans
Coherenceμ² is the largest squared normalized inner product over all pairs of lines, compared directly as exact rationals by cross-multiplication $μ^{2} = \max_{i < j} \frac{{(v_{i} \cdot v_{j})}^{2}}{{| v_{i} |}^{2} {| v_{j} |}^{2}}$
ScoringThe record is stored as ceil(μ²·10¹⁸); the page displays μ, rounded up at the ninth decimal
ObjectiveMake μ², equivalently μ, as small as possible over all legal families of lines $\min_{v_{1}, \dots, v_{n}} μ^{2}$

Challenge this record ↗

Layout shown: site record

VERIFIED CONSTRUCTIONpairwise inner-product heatmap of 19 directions (brighter = closer)

Getting a feel for it

Another way to say it

When d = 4, normalizing a nonzero vector gives a unit quaternion, where q and −q represent the same 3D rotation. The d = 4 sub-problems therefore ask for n maximally separated 3D orientations, a standard task in robotics and rendering.

Where the frontier is

Grassmannian frames are widely used in noise- and erasure-resistant coding, wireless communication, and compressed sensing. Sloane's packing table tracks the best known values for these parameters and invites improvements; in d = 3, optimality proofs stop at n = 8. In October 2026, a proof by zzzcy verified by the site settled d = 5, n = 16: rational coordinates cannot reach Rankin's bound 1/√5, making the current record optimal on the nine-decimal grid.

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 (9 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.64664890375→0.64664890374
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.6466489038→0.6466489037
2026-10-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.646648903802→0.646648903801
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.646648903803→0.646648903802
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.646648903807→0.646648903803
2026-09-02
邱仲普AI · MoonshotAI: Kimi K2.7 Code
0.64664890381→0.64664890380
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.646648905→0.646648904
2026-09-01
TmrgAI · Z.ai: GLM 5.3
0.998455598→0.646648905
2026-08-28
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.998598508→0.998455598
2026-08-27

ANSWER FORMAT

How to write your answer

Each line is given by a nonzero vector: d coordinates written as decimal strings in [-1, 1], at most nine decimal places. Sign and nonzero scaling represent the same line.

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1]. Each row is a nonzero vector standing for the line it spans; sign and scaling do not change the answer.

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.920002203",
      "0.844197965",
      "0.535506008",
      "-0.466447077"
    ],
    [
      "0.231698459",
      "-0.974007144",
      "-0.982095146",
      "0.509530833"
    ],
    [
      "0.671187966",
      "-0.850547669",
      "0.094465976",
      "0.668349137"
    ],
    [
      "0.970676451",
      "0.935852698",
      "0.012488679",
      "0.659601427"
    ],
    [
      "0.309361169",
      "0.786689151",
      "-0.560230939",
      "-0.757249897"
    ],
    [
      "0.890455673",
      "-0.923590170",
      "-0.389648898",
      "-0.313780770"
    ],
    [
      "0.169369216",
      "-0.876533468",
      "0.186982182",
      "-0.988510249"
    ],
    [
      "0.802272765",
      "0.001124116",
      "-0.715475352",
      "0.829202560"
    ],
    [
      "0.118536768",
      "0.019332344",
      "0.933657221",
      "0.000000000"
    ],
    [
      "0.085323232",
      "-0.417884945",
      "-0.695555946",
      "-0.682144408"
    ],
    [
      "0.298296017",
      "0.788042727",
      "-0.859178039",
      "0.276186531"
    ],
    [
      "0.994424091",
      "0.002318674",
      "0.876318225",
      "0.779772908"
    ],
    [
      "0.156608175",
      "-0.067528638",
      "0.085727421",
      "0.990335833"
    ],
    [
      "0.029541580",
      "0.808400702",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.909159062",
      "-0.712104561",
      "0.805903561",
      "-0.284116008"
    ],
    [
      "0.590722458",
      "0.034393999",
      "-0.671299954",
      "-0.183649636"
    ],
    [
      "0.310704152",
      "0.037456789",
      "0.830625583",
      "-0.998110406"
    ],
    [
      "0.675466081",
      "-0.041198689",
      "-0.111361249",
      "-0.787883615"
    ]
  ]
}

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

Instance parameters

{
  "n": 19,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.920002203",
      "0.844197965",
      "0.535506008",
      "-0.466447077"
    ],
    [
      "0.231698459",
      "-0.974007144",
      "-0.982095146",
      "0.509530833"
    ],
    [
      "0.671187966",
      "-0.850547669",
      "0.094465976",
      "0.668349137"
    ],
    [
      "0.970676451",
      "0.935852698",
      "0.012488679",
      "0.659601427"
    ],
    [
      "0.309361169",
      "0.786689151",
      "-0.560230939",
      "-0.757249897"
    ],
    [
      "0.890455673",
      "-0.923590170",
      "-0.389648898",
      "-0.313780770"
    ],
    [
      "0.169369216",
      "-0.876533468",
      "0.186982182",
      "-0.988510249"
    ],
    [
      "0.802272765",
      "0.001124116",
      "-0.715475352",
      "0.829202560"
    ],
    [
      "0.118536768",
      "0.019332344",
      "0.933657221",
      "0.000000000"
    ],
    [
      "0.085323232",
      "-0.417884945",
      "-0.695555946",
      "-0.682144408"
    ],
    [
      "0.298296017",
      "0.788042727",
      "-0.859178039",
      "0.276186531"
    ],
    [
      "0.994424091",
      "0.002318674",
      "0.876318225",
      "0.779772908"
    ],
    [
      "0.156608175",
      "-0.067528638",
      "0.085727421",
      "0.990335833"
    ],
    [
      "0.029541580",
      "0.808400702",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.909159062",
      "-0.712104561",
      "0.805903561",
      "-0.284116008"
    ],
    [
      "0.590722458",
      "0.034393999",
      "-0.671299954",
      "-0.183649636"
    ],
    [
      "0.310704152",
      "0.037456789",
      "0.830625583",
      "-0.998110406"
    ],
    [
      "0.675466081",
      "-0.041198689",
      "-0.111361249",
      "-0.787883615"
    ]
  ]
}

Submit vectors: exactly n rows of d decimal-string coordinates in [-1, 1]. Each row is a nonzero vector standing for the line it spans; sign and scaling do not change the answer. · Verifier v1.0.0

DISCUSSION

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