P27 · Extremal configurations · Formed here

Spreading points in an annulus · n = 16

Place n points inside an annulus with outer radius 1 and inner radius 0.5, maximizing the smallest distance between any two of them.

Instancen = 16

ObjectiveMaximize the smallest distance between two points

Formal definition

ContainerThe container is an annulus of outer radius 1 and inner radius 0.5 centred at (1, 1), so coordinates run from 0 to 2 and the disc of radius 0.5 in the middle is not part of it.
SubmissionExactly n points, no two coinciding
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the smallest pairwise distance as large as possible; compared internally by its square, exactly

Open the full editor ↗

2y0

12345678910111213141516

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's shape decides everything.

Where the frontier is

Our own variant: point spreading in an annulus with outer radius 1 and inner radius 0.5 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.285714285

the smallest distance between two points

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is an annulus of outer radius 1 and inner radius 0.5 centred at (1, 1), so coordinates run from 0 to 2 and the disc of radius 0.5 in the middle is not part of it. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "0.57142857",
      "0.285714285"
    ],
    [
      "0.857142855",
      "0.285714285"
    ],
    [
      "1.14285714",
      "0.285714285"
    ],
    [
      "1.428571425",
      "0.285714285"
    ],
    [
      "0.285714285",
      "0.57142857"
    ],
    [
      "0.57142857",
      "0.57142857"
    ],
    [
      "1.428571425",
      "0.57142857"
    ],
    [
      "1.71428571",
      "0.57142857"
    ],
    [
      "0.285714285",
      "0.857142855"
    ],
    [
      "1.71428571",
      "0.857142855"
    ],
    [
      "0.285714285",
      "1.14285714"
    ],
    [
      "1.71428571",
      "1.14285714"
    ],
    [
      "0.285714285",
      "1.428571425"
    ],
    [
      "0.57142857",
      "1.428571425"
    ],
    [
      "1.428571425",
      "1.428571425"
    ],
    [
      "1.71428571",
      "1.428571425"
    ]
  ]
}

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

Instance parameters

{
  "n": 16
}

The current leader's answer

{
  "points": [
    [
      "0.57142857",
      "0.285714285"
    ],
    [
      "0.857142855",
      "0.285714285"
    ],
    [
      "1.14285714",
      "0.285714285"
    ],
    [
      "1.428571425",
      "0.285714285"
    ],
    [
      "0.285714285",
      "0.57142857"
    ],
    [
      "0.57142857",
      "0.57142857"
    ],
    [
      "1.428571425",
      "0.57142857"
    ],
    [
      "1.71428571",
      "0.57142857"
    ],
    [
      "0.285714285",
      "0.857142855"
    ],
    [
      "1.71428571",
      "0.857142855"
    ],
    [
      "0.285714285",
      "1.14285714"
    ],
    [
      "1.71428571",
      "1.14285714"
    ],
    [
      "0.285714285",
      "1.428571425"
    ],
    [
      "0.57142857",
      "1.428571425"
    ],
    [
      "1.428571425",
      "1.428571425"
    ],
    [
      "1.71428571",
      "1.428571425"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.5". · Verifier v1.0.0