P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · d = 4, n = 14

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 = 14

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.590076515250353, 0.5900765153350878]	See source attribution	cohn.mit.edu ↗
Site record	0.590076516	zzzcy	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

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 14 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
0.5900765153→0.5900765152
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.59007651532→0.59007651531
2026-09-23
StonebubuAI · Z.ai: GLM 5.3 Flash
0.590076515322→0.590076515320
2026-09-05
StonebubuAI · Z.ai: GLM 5.3 Flash
0.590076515326→0.590076515322
2026-09-03
StonebubuAI · Z.ai: GLM 5.3 Flash
0.59007651533→0.59007651532
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.5900765158→0.5900765153
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archiveHuman
0.590076517→0.590076516
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.997037032→0.590076517
2026-08-28
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.997392788→0.997037032
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"
    ],
    [
      "0.857491701",
      "-0.935834792",
      "0.328034051",
      "-0.626951537"
    ],
    [
      "0.107938540",
      "0.366913371",
      "-0.771350774",
      "0.889251994"
    ],
    [
      "0.109302738",
      "0.618812705",
      "-0.386843285",
      "-0.762164308"
    ],
    [
      "0.357247629",
      "-0.093890123",
      "-0.313317130",
      "-0.363224065"
    ],
    [
      "0.725822167",
      "-0.195671461",
      "-0.034482336",
      "0.972996806"
    ],
    [
      "0.714727044",
      "0.256103764",
      "0.365722805",
      "-0.870021055"
    ],
    [
      "-0.796082841",
      "-0.729439037",
      "-0.802933067",
      "-0.097921585"
    ],
    [
      "0.941625331",
      "0.862797487",
      "-0.956706970",
      "0.010493466"
    ],
    [
      "-0.087099629",
      "-0.365768076",
      "-0.147321561",
      "-0.484043250"
    ],
    [
      "0.784637762",
      "-0.460930864",
      "0.957492261",
      "0.381601590"
    ],
    [
      "0.893355451",
      "-0.974975165",
      "-0.696250856",
      "0.242209753"
    ],
    [
      "-0.087020501",
      "0.098345312",
      "0.834321996",
      "0.000000000"
    ],
    [
      "0.068422239",
      "0.781621187",
      "0.000000000",
      "0.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 14,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.857491701",
      "-0.935834792",
      "0.328034051",
      "-0.626951537"
    ],
    [
      "0.107938540",
      "0.366913371",
      "-0.771350774",
      "0.889251994"
    ],
    [
      "0.109302738",
      "0.618812705",
      "-0.386843285",
      "-0.762164308"
    ],
    [
      "0.357247629",
      "-0.093890123",
      "-0.313317130",
      "-0.363224065"
    ],
    [
      "0.725822167",
      "-0.195671461",
      "-0.034482336",
      "0.972996806"
    ],
    [
      "0.714727044",
      "0.256103764",
      "0.365722805",
      "-0.870021055"
    ],
    [
      "-0.796082841",
      "-0.729439037",
      "-0.802933067",
      "-0.097921585"
    ],
    [
      "0.941625331",
      "0.862797487",
      "-0.956706970",
      "0.010493466"
    ],
    [
      "-0.087099629",
      "-0.365768076",
      "-0.147321561",
      "-0.484043250"
    ],
    [
      "0.784637762",
      "-0.460930864",
      "0.957492261",
      "0.381601590"
    ],
    [
      "0.893355451",
      "-0.974975165",
      "-0.696250856",
      "0.242209753"
    ],
    [
      "-0.087020501",
      "0.098345312",
      "0.834321996",
      "0.000000000"
    ],
    [
      "0.068422239",
      "0.781621187",
      "0.000000000",
      "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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