P07 · Extremal configurations · Classic

Spreading points in a circle · n = 13

Place n points inside a circle of radius 1, maximizing the smallest distance between any two of them.

Instancen = 13

ObjectiveMaximize the smallest distance between two points

Formal definition

ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
SubmissionExactly n points, no two coinciding
ConstraintsEvery point lies inside the circle or on it
ObjectiveMake the smallest pairwise distance as large as possible; compared internally by its square, exactly

Open the full editor ↗

2y0

12345678910111213

0x2

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Where the room for improvement is

Spreading points IS packing equal circles: discs of half the minimum distance around each point must not overlap. Optima are jammed contact structures, and the container shape decides everything.

Where the frontier is

Spreading points in a disc is dual to packing equal circles in it: a point set with spacing d is a packing of radius d/2. Proofs in the cci table transfer; n = 4 is proven here, and the rest open or close as cci does.

Source

Current leader

0.239315663

the smallest distance between two points

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2. Coordinates are written as plain decimals such as "1.5", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "1.5".

The current leader's answer

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.442728013",
      "1.232361586"
    ],
    [
      "1.284032373",
      "1.411491933"
    ],
    [
      "1.06026834",
      "1.496354437"
    ],
    [
      "0.822697556",
      "1.467508121"
    ],
    [
      "0.625744626",
      "1.331561329"
    ],
    [
      "0.514529091",
      "1.119657832"
    ],
    [
      "0.514529091",
      "0.880342168"
    ],
    [
      "0.625744626",
      "0.668438671"
    ],
    [
      "0.822697556",
      "0.532491879"
    ],
    [
      "1.06026834",
      "0.503645563"
    ],
    [
      "1.284032373",
      "0.588508067"
    ],
    [
      "1.442728013",
      "0.767638414"
    ]
  ]
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.442728013",
      "1.232361586"
    ],
    [
      "1.284032373",
      "1.411491933"
    ],
    [
      "1.06026834",
      "1.496354437"
    ],
    [
      "0.822697556",
      "1.467508121"
    ],
    [
      "0.625744626",
      "1.331561329"
    ],
    [
      "0.514529091",
      "1.119657832"
    ],
    [
      "0.514529091",
      "0.880342168"
    ],
    [
      "0.625744626",
      "0.668438671"
    ],
    [
      "0.822697556",
      "0.532491879"
    ],
    [
      "1.06026834",
      "0.503645563"
    ],
    [
      "1.284032373",
      "0.588508067"
    ],
    [
      "1.442728013",
      "0.767638414"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "1.5". · Verifier v1.0.0