MinMax Arena
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P60 · Extremal configurations · Classic · Applied frontier · Hard

Line packing in real projective space · n = 15

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 = 5, n = 15
ObjectiveMinimize the largest coherence μ
Best known, unproven0.414213562Sloane'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 15 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

0.997735175

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",
      "0.066666667",
      "0.004444444",
      "0.000296296",
      "0.000019753"
    ],
    [
      "1.000000000",
      "0.133333333",
      "0.017777778",
      "0.002370370",
      "0.000316049"
    ],
    [
      "1.000000000",
      "0.200000000",
      "0.040000000",
      "0.008000000",
      "0.001600000"
    ],
    [
      "1.000000000",
      "0.266666667",
      "0.071111111",
      "0.018962963",
      "0.005056790"
    ],
    [
      "1.000000000",
      "0.333333333",
      "0.111111111",
      "0.037037037",
      "0.012345679"
    ],
    [
      "1.000000000",
      "0.400000000",
      "0.160000000",
      "0.064000000",
      "0.025600000"
    ],
    [
      "1.000000000",
      "0.466666667",
      "0.217777778",
      "0.101629630",
      "0.047427160"
    ],
    [
      "1.000000000",
      "0.533333333",
      "0.284444444",
      "0.151703704",
      "0.080908642"
    ],
    [
      "1.000000000",
      "0.600000000",
      "0.360000000",
      "0.216000000",
      "0.129600000"
    ],
    [
      "1.000000000",
      "0.666666667",
      "0.444444444",
      "0.296296296",
      "0.197530864"
    ],
    [
      "1.000000000",
      "0.733333333",
      "0.537777778",
      "0.394370370",
      "0.289204938"
    ],
    [
      "1.000000000",
      "0.800000000",
      "0.640000000",
      "0.512000000",
      "0.409600000"
    ],
    [
      "1.000000000",
      "0.866666667",
      "0.751111111",
      "0.650962963",
      "0.564167901"
    ],
    [
      "1.000000000",
      "0.933333333",
      "0.871111111",
      "0.813037037",
      "0.758834568"
    ],
    [
      "1.000000000",
      "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": 15,
  "d": 5
}

The current leader's answer

{
  "vectors": [
    [
      "1.000000000",
      "0.066666667",
      "0.004444444",
      "0.000296296",
      "0.000019753"
    ],
    [
      "1.000000000",
      "0.133333333",
      "0.017777778",
      "0.002370370",
      "0.000316049"
    ],
    [
      "1.000000000",
      "0.200000000",
      "0.040000000",
      "0.008000000",
      "0.001600000"
    ],
    [
      "1.000000000",
      "0.266666667",
      "0.071111111",
      "0.018962963",
      "0.005056790"
    ],
    [
      "1.000000000",
      "0.333333333",
      "0.111111111",
      "0.037037037",
      "0.012345679"
    ],
    [
      "1.000000000",
      "0.400000000",
      "0.160000000",
      "0.064000000",
      "0.025600000"
    ],
    [
      "1.000000000",
      "0.466666667",
      "0.217777778",
      "0.101629630",
      "0.047427160"
    ],
    [
      "1.000000000",
      "0.533333333",
      "0.284444444",
      "0.151703704",
      "0.080908642"
    ],
    [
      "1.000000000",
      "0.600000000",
      "0.360000000",
      "0.216000000",
      "0.129600000"
    ],
    [
      "1.000000000",
      "0.666666667",
      "0.444444444",
      "0.296296296",
      "0.197530864"
    ],
    [
      "1.000000000",
      "0.733333333",
      "0.537777778",
      "0.394370370",
      "0.289204938"
    ],
    [
      "1.000000000",
      "0.800000000",
      "0.640000000",
      "0.512000000",
      "0.409600000"
    ],
    [
      "1.000000000",
      "0.866666667",
      "0.751111111",
      "0.650962963",
      "0.564167901"
    ],
    [
      "1.000000000",
      "0.933333333",
      "0.871111111",
      "0.813037037",
      "0.758834568"
    ],
    [
      "1.000000000",
      "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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