P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · n = 12

Choose n lines through the origin of R^d so that the minimum angle between any two is as large as possible — equivalently, minimize the largest coherence μ = max |cos ∠(v_i, v_j)|.

Instanced = 4, n = 12

ObjectiveMinimize the largest coherence μ

Best known, unproven0.5Sloane's Grassmannian packing table (best packing found)

Formal definition

ContainerReal d-space R^d, every line through the origin; an answer is n points of real projective space RP^{d-1}
SubmissionExactly n nonzero vectors, d coordinates each; a vector stands for the line it spans
ObjectiveMinimize μ² = max (v_i·v_j)²/(|v_i|²|v_j|²), compared exactly by cross-multiplication — no normalization, no square roots
ScoringThe record is ceil(μ²·10¹⁸), rounded against the submitter; the page shows μ, rounded up at the ninth decimal

Challenge this record ↗

VERIFIED CONSTRUCTIONpairwise overlap heat of 12 directions; brighter is closer

Getting a feel for it

Another way to say it

At d = 4 a normalized nonzero vector is a unit quaternion, and q and −q are the same 3D rotation — so the d = 4 sub-problems ask for n maximally separated 3D orientations, a problem robotics and rendering actually use.

Where the frontier is

Grassmannian frames drive noise- and erasure-robust data representations, wireless communication and compressed sensing. Sloane's packing table maintains the best known values for these parameters and openly invites improvement; optimality proofs in d = 3 stop at n = 8.

Source

Current leader

0.996423143

the largest coherence μ

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

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",
      "0.083333333",
      "0.006944444",
      "0.000578704"
    ],
    [
      "1.000000000",
      "0.166666667",
      "0.027777778",
      "0.004629630"
    ],
    [
      "1.000000000",
      "0.250000000",
      "0.062500000",
      "0.015625000"
    ],
    [
      "1.000000000",
      "0.333333333",
      "0.111111111",
      "0.037037037"
    ],
    [
      "1.000000000",
      "0.416666667",
      "0.173611111",
      "0.072337963"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000"
    ],
    [
      "1.000000000",
      "0.583333333",
      "0.340277778",
      "0.198495370"
    ],
    [
      "1.000000000",
      "0.666666667",
      "0.444444444",
      "0.296296296"
    ],
    [
      "1.000000000",
      "0.750000000",
      "0.562500000",
      "0.421875000"
    ],
    [
      "1.000000000",
      "0.833333333",
      "0.694444444",
      "0.578703704"
    ],
    [
      "1.000000000",
      "0.916666667",
      "0.840277778",
      "0.770254630"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 12,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.083333333",
      "0.006944444",
      "0.000578704"
    ],
    [
      "1.000000000",
      "0.166666667",
      "0.027777778",
      "0.004629630"
    ],
    [
      "1.000000000",
      "0.250000000",
      "0.062500000",
      "0.015625000"
    ],
    [
      "1.000000000",
      "0.333333333",
      "0.111111111",
      "0.037037037"
    ],
    [
      "1.000000000",
      "0.416666667",
      "0.173611111",
      "0.072337963"
    ],
    [
      "1.000000000",
      "0.500000000",
      "0.250000000",
      "0.125000000"
    ],
    [
      "1.000000000",
      "0.583333333",
      "0.340277778",
      "0.198495370"
    ],
    [
      "1.000000000",
      "0.666666667",
      "0.444444444",
      "0.296296296"
    ],
    [
      "1.000000000",
      "0.750000000",
      "0.562500000",
      "0.421875000"
    ],
    [
      "1.000000000",
      "0.833333333",
      "0.694444444",
      "0.578703704"
    ],
    [
      "1.000000000",
      "0.916666667",
      "0.840277778",
      "0.770254630"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ]
  ]
}

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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