P07 · Extremal configurations · Classic

Spreading points in a circle · n = 19

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

Instancen = 19

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

12345678910111213141516171819

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

0.164594589

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"
    ],
    [
      "1.472908621",
      "1.162349735"
    ],
    [
      "1.394570255",
      "1.307106356"
    ],
    [
      "1.273474079",
      "1.418583239"
    ],
    [
      "1.122742744",
      "1.484700133"
    ],
    [
      "0.958710327",
      "1.498292247"
    ],
    [
      "0.799152288",
      "1.457886663"
    ],
    [
      "0.661359214",
      "1.367861955"
    ],
    [
      "0.560263124",
      "1.237973697"
    ],
    [
      "0.506819348",
      "1.082297295"
    ],
    [
      "0.506819348",
      "0.917702705"
    ],
    [
      "0.560263124",
      "0.762026303"
    ],
    [
      "0.661359214",
      "0.632138045"
    ],
    [
      "0.799152288",
      "0.542113337"
    ],
    [
      "0.958710327",
      "0.501707753"
    ],
    [
      "1.122742744",
      "0.515299867"
    ],
    [
      "1.273474079",
      "0.581416761"
    ],
    [
      "1.394570255",
      "0.692893644"
    ],
    [
      "1.472908621",
      "0.837650265"
    ]
  ]
}

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

Instance parameters

{
  "n": 19
}

The current leader's answer

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.472908621",
      "1.162349735"
    ],
    [
      "1.394570255",
      "1.307106356"
    ],
    [
      "1.273474079",
      "1.418583239"
    ],
    [
      "1.122742744",
      "1.484700133"
    ],
    [
      "0.958710327",
      "1.498292247"
    ],
    [
      "0.799152288",
      "1.457886663"
    ],
    [
      "0.661359214",
      "1.367861955"
    ],
    [
      "0.560263124",
      "1.237973697"
    ],
    [
      "0.506819348",
      "1.082297295"
    ],
    [
      "0.506819348",
      "0.917702705"
    ],
    [
      "0.560263124",
      "0.762026303"
    ],
    [
      "0.661359214",
      "0.632138045"
    ],
    [
      "0.799152288",
      "0.542113337"
    ],
    [
      "0.958710327",
      "0.501707753"
    ],
    [
      "1.122742744",
      "0.515299867"
    ],
    [
      "1.273474079",
      "0.581416761"
    ],
    [
      "1.394570255",
      "0.692893644"
    ],
    [
      "1.472908621",
      "0.837650265"
    ]
  ]
}

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