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P63 · Extremal configurations · Formed here · Applied frontier · Hard

Optimal quadrature points on the torus · n = 8

Place n equal-weight quadrature points on the torus T^d. Fix the kernel K(x,y) = Π (1 + 6·B₂({x_r − y_r})) with B₂(t) = t² − t + 1/6; the score is the squared worst-case error of the equal-weight rule, E = (1/n²)Σ K(x_i,x_j) − 1. Make it as small as possible.

Instanced = 2, n = 8
ObjectiveMinimize the worst-case integration error

Formal definition

  • ContainerThe d-dimensional torus: coordinates modulo 1, written in [0, 1)
  • SubmissionExactly n points, each d decimal coordinates; coincidences are allowed
  • ObjectiveMinimize the squared worst-case error of the equal-weight rule, E = (1/n²)Σ K(x_i,x_j) − 1; the kernel integrates to one, so E is non-negative
  • KernelK = Π(1 + 6·B₂({x_r−y_r})); λ = 6 is this site's fixed kernel version, never to change — same family as the literature's periodic L2 discrepancy (λ = 3) and diaphony (λ = 2π²), deliberately its own parameter
  • ScoringThe record is the exact integer n²S^{2d}·E with the denominator cleared; the page shows the error √E, rounded up at the twelfth decimal
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1y0
0x1
VERIFIED CONSTRUCTIONthe first two coordinates of 8 higher-dimensional points

Getting a feel for it

What it optimizes

In numerical integration of periodic functions, a point set is only as good as the hardest function it faces. The tensor-product B₂ kernel is sensitive to holes and regularity in every coordinate direction: let one dimension clump, and the score decays at once.

Where the frontier is

Global optimality in this family has been proven only at tiny n: Fibonacci lattices were settled for a handful of n as recently as 2025, and minimizing tensor-product energies on the torus is active research. The site's λ = 6 version has no published per-instance optima at all; everything is open.

Source
Current leader

1.01550480058

the worst-case integration error

Record holderFounding benchmark
Solution methodHuman
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ANSWER FORMAT

How to write your answer

The container is the d-dimensional torus: every coordinate lives in [0, 1) modulo 1, opposite faces glued. Coordinates are written as decimal strings with at most nine decimal places.

Submit points: exactly n rows of d decimal-string coordinates in [0, 1). Coordinates are read modulo 1; write 1 as 0.

The current leader's answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.125",
      "0"
    ],
    [
      "0.25",
      "0"
    ],
    [
      "0.375",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "0.625",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "0.875",
      "0"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 8,
  "d": 2
}

The current leader's answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.125",
      "0"
    ],
    [
      "0.25",
      "0"
    ],
    [
      "0.375",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "0.625",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "0.875",
      "0"
    ]
  ]
}

Submit points: exactly n rows of d decimal-string coordinates in [0, 1). Coordinates are read modulo 1; write 1 as 0. · Verifier v1.0.0

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