P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · d = 4, n = 15

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 = 15

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.6087394116943637, 0.6087394117765007]	See source attribution	cohn.mit.edu ↗
Site record	0.608739412	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 15 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 (11 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.60873941169→0.60873941168
2026-10-06
zzzcyAI · OpenAI: GPT-6 Astra
0.6087394117→0.6087394116
2026-10-04
StonebubuAI · Z.ai: GLM 5.3 Flash
0.608739411742→0.608739411741
2026-09-08
StonebubuAI · Z.ai: GLM 5.3 Flash
0.608739411745→0.608739411742
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.608739411746→0.608739411745
2026-09-07
StonebubuAI · Z.ai: GLM 5.3 Flash
0.60873941175→0.60873941174
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.60873941176→0.60873941175
2026-09-01
StonebubuAI · Z.ai: GLM 5.3 Flash
0.608739413→0.608739412
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archiveHuman
0.6087394122→0.6087394121
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.997445718→0.608739413
2026-08-28
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.997735218→0.997445718
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"
    ],
    [
      "0.327847850",
      "0.277092037",
      "0.010598484",
      "0.325083763"
    ],
    [
      "-0.400472708",
      "0.499062470",
      "-0.497077678",
      "0.013808129"
    ],
    [
      "0.162454534",
      "0.007052417",
      "-0.387375148",
      "0.506220802"
    ],
    [
      "0.871899693",
      "-0.123219039",
      "-0.630557720",
      "-0.937282645"
    ],
    [
      "0.317115817",
      "0.741259358",
      "-0.030051907",
      "-0.921772115"
    ],
    [
      "-0.963682108",
      "-0.108751860",
      "-0.879141949",
      "0.890357193"
    ],
    [
      "-0.659860283",
      "0.822306979",
      "0.818779294",
      "-0.030644892"
    ],
    [
      "-0.408804775",
      "-0.650273630",
      "-0.749822261",
      "-0.095843071"
    ],
    [
      "-0.283267536",
      "-0.012297107",
      "-0.731515034",
      "-0.836812894"
    ],
    [
      "0.155639120",
      "-0.127076252",
      "0.005151886",
      "0.158022080"
    ],
    [
      "0.004208451",
      "0.295618801",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.715501512",
      "-0.705388080",
      "0.598411291",
      "0.117951906"
    ],
    [
      "-0.247238530",
      "0.627750396",
      "-0.167190603",
      "0.753777584"
    ],
    [
      "-0.083817934",
      "-0.067122557",
      "0.876507462",
      "0.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 15,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.327847850",
      "0.277092037",
      "0.010598484",
      "0.325083763"
    ],
    [
      "-0.400472708",
      "0.499062470",
      "-0.497077678",
      "0.013808129"
    ],
    [
      "0.162454534",
      "0.007052417",
      "-0.387375148",
      "0.506220802"
    ],
    [
      "0.871899693",
      "-0.123219039",
      "-0.630557720",
      "-0.937282645"
    ],
    [
      "0.317115817",
      "0.741259358",
      "-0.030051907",
      "-0.921772115"
    ],
    [
      "-0.963682108",
      "-0.108751860",
      "-0.879141949",
      "0.890357193"
    ],
    [
      "-0.659860283",
      "0.822306979",
      "0.818779294",
      "-0.030644892"
    ],
    [
      "-0.408804775",
      "-0.650273630",
      "-0.749822261",
      "-0.095843071"
    ],
    [
      "-0.283267536",
      "-0.012297107",
      "-0.731515034",
      "-0.836812894"
    ],
    [
      "0.155639120",
      "-0.127076252",
      "0.005151886",
      "0.158022080"
    ],
    [
      "0.004208451",
      "0.295618801",
      "0.000000000",
      "0.000000000"
    ],
    [
      "-0.715501512",
      "-0.705388080",
      "0.598411291",
      "0.117951906"
    ],
    [
      "-0.247238530",
      "0.627750396",
      "-0.167190603",
      "0.753777584"
    ],
    [
      "-0.083817934",
      "-0.067122557",
      "0.876507462",
      "0.000000000"
    ]
  ]
}

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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