P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · d = 4, n = 17

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 = 17

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.6308517112998903, 0.6308517113791481]	See source attribution	cohn.mit.edu ↗
Site record	0.630851712	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 17 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 (8 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.63085171127→0.63085171126
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.6308517113→0.6308517112
2026-10-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.63085171132→0.63085171131
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.63085171134→0.63085171132
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.6308517117→0.6308517113
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archiveHuman
0.63085171179→0.63085171178
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.998044965→0.630851712
2026-08-28
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.998244174→0.998044965
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"
    ],
    [
      "0.819926128",
      "0.130810039",
      "0.362988711",
      "-0.975779958"
    ],
    [
      "-0.126019231",
      "0.754489093",
      "0.866707933",
      "0.331775479"
    ],
    [
      "-0.552083815",
      "0.557327278",
      "0.136060957",
      "-0.363246323"
    ],
    [
      "-0.133472161",
      "-0.109298689",
      "0.742991244",
      "0.000000000"
    ],
    [
      "0.515514998",
      "0.109750873",
      "-0.409210178",
      "0.471717932"
    ],
    [
      "0.145764788",
      "-0.187647013",
      "-0.001200687",
      "0.889664487"
    ],
    [
      "0.000943018",
      "0.099759573",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.023729045",
      "0.492592962",
      "0.803192568",
      "-0.632172759"
    ],
    [
      "0.897167836",
      "0.888727075",
      "-0.629128126",
      "-0.178806707"
    ],
    [
      "0.798398407",
      "0.208549625",
      "0.949115874",
      "0.939515638"
    ],
    [
      "-0.419320933",
      "-0.022110116",
      "0.535753309",
      "0.660648763"
    ],
    [
      "0.613644306",
      "-0.619472445",
      "-0.059556016",
      "-0.427011370"
    ],
    [
      "-0.975284233",
      "-0.966108535",
      "-0.700560499",
      "0.121345057"
    ],
    [
      "0.131167103",
      "0.461715817",
      "0.004464412",
      "0.555061766"
    ],
    [
      "-0.154971054",
      "0.747444306",
      "-0.683675095",
      "-0.589974810"
    ],
    [
      "-0.604504743",
      "0.307141219",
      "-0.698406097",
      "0.117541331"
    ]
  ]
}

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

Instance parameters

{
  "n": 17,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.819926128",
      "0.130810039",
      "0.362988711",
      "-0.975779958"
    ],
    [
      "-0.126019231",
      "0.754489093",
      "0.866707933",
      "0.331775479"
    ],
    [
      "-0.552083815",
      "0.557327278",
      "0.136060957",
      "-0.363246323"
    ],
    [
      "-0.133472161",
      "-0.109298689",
      "0.742991244",
      "0.000000000"
    ],
    [
      "0.515514998",
      "0.109750873",
      "-0.409210178",
      "0.471717932"
    ],
    [
      "0.145764788",
      "-0.187647013",
      "-0.001200687",
      "0.889664487"
    ],
    [
      "0.000943018",
      "0.099759573",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.023729045",
      "0.492592962",
      "0.803192568",
      "-0.632172759"
    ],
    [
      "0.897167836",
      "0.888727075",
      "-0.629128126",
      "-0.178806707"
    ],
    [
      "0.798398407",
      "0.208549625",
      "0.949115874",
      "0.939515638"
    ],
    [
      "-0.419320933",
      "-0.022110116",
      "0.535753309",
      "0.660648763"
    ],
    [
      "0.613644306",
      "-0.619472445",
      "-0.059556016",
      "-0.427011370"
    ],
    [
      "-0.975284233",
      "-0.966108535",
      "-0.700560499",
      "0.121345057"
    ],
    [
      "0.131167103",
      "0.461715817",
      "0.004464412",
      "0.555061766"
    ],
    [
      "-0.154971054",
      "0.747444306",
      "-0.683675095",
      "-0.589974810"
    ],
    [
      "-0.604504743",
      "0.307141219",
      "-0.698406097",
      "0.117541331"
    ]
  ]
}

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

Discussion

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