P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · d = 3, n = 13

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 = 3, n = 13

ObjectiveMinimize the largest coherence μ

Record comparison

the largest coherence μ · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.7681373762876014, 0.7681373763526939]	See source attribution	cohn.mit.edu ↗
Site record	0.768137377	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 13 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 (7 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.76813737633→0.76813737631
2026-10-06
StonebubuAI · Z.ai: GLM 5.3 Flash
0.76813737634→0.76813737633
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.768137376347→0.768137376341
2026-09-02
StonebubuAI · Z.ai: GLM 5.3 Flash
0.7681373767→0.7681373763
2026-09-01
N. J. A. Sloane and Henry Cohn's Grassmannian packing archiveHuman
0.7681373769→0.7681373767
2026-08-29
TmrgAI · Z.ai: GLM 5.3
0.997490586→0.768137377
2026-08-28
lirdAI · DeepSeek: DeepSeek V4 Pro 0813
0.9978746→0.997490586
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"
    ],
    [
      "0.348008912",
      "-0.166517107",
      "0.237531459"
    ],
    [
      "-0.701765396",
      "-0.778591399",
      "0.080173425"
    ],
    [
      "-0.877016068",
      "-0.326290965",
      "0.654183362"
    ],
    [
      "0.227022625",
      "0.015508831",
      "0.549118025"
    ],
    [
      "0.232483058",
      "-0.333593837",
      "-0.932726492"
    ],
    [
      "-0.863507235",
      "-0.445782262",
      "-0.565122158"
    ],
    [
      "-0.267155194",
      "-0.996378831",
      "-0.808494736"
    ],
    [
      "0.906668444",
      "-0.462603386",
      "-0.597636055"
    ],
    [
      "-0.024742859",
      "-0.047141049",
      "0.123009846"
    ],
    [
      "-0.323526878",
      "0.423406302",
      "-0.010015746"
    ],
    [
      "-0.225895345",
      "0.769647361",
      "-0.577999666"
    ],
    [
      "0.001911287",
      "0.045274983",
      "0.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 13,
  "d": 3
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.348008912",
      "-0.166517107",
      "0.237531459"
    ],
    [
      "-0.701765396",
      "-0.778591399",
      "0.080173425"
    ],
    [
      "-0.877016068",
      "-0.326290965",
      "0.654183362"
    ],
    [
      "0.227022625",
      "0.015508831",
      "0.549118025"
    ],
    [
      "0.232483058",
      "-0.333593837",
      "-0.932726492"
    ],
    [
      "-0.863507235",
      "-0.445782262",
      "-0.565122158"
    ],
    [
      "-0.267155194",
      "-0.996378831",
      "-0.808494736"
    ],
    [
      "0.906668444",
      "-0.462603386",
      "-0.597636055"
    ],
    [
      "-0.024742859",
      "-0.047141049",
      "0.123009846"
    ],
    [
      "-0.323526878",
      "0.423406302",
      "-0.010015746"
    ],
    [
      "-0.225895345",
      "0.769647361",
      "-0.577999666"
    ],
    [
      "0.001911287",
      "0.045274983",
      "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

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