P22 · Extremal configurations · Classic

The smallest triangle in a disc · n = 14

Place n points inside a circle of radius 1 so that the smallest triangle formed by any three of them is as large as possible.

Instancen = 14

ObjectiveMaximize the smallest triangle's area

Best known, unproven0.075857251060901MathWorld's numerical value for a disc of unit area, converted to radius 1 and doubled; optimality open

Formal definition

ContainerThe container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2.
SubmissionExactly n points, no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the smallest triangle over all triples as large as possible; compared internally by twice the area, exactly

Open the full editor ↗

2y0

1234567891011121314

0x2

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Where the room for improvement is

Even spreading is not enough: no three points may come close to collinear, and near-collinearity is exactly what tidy arrangements love to do. Optima are often asymmetric and hard even to guess.

Where the frontier is

Heilbronn in a disc: MathWorld collects the known values, some in closed form, none proven; the proof techniques from the square version have not been carried over.

Source

Current leader

0.027499518584008526

the smallest triangle's area

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 "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": [
    [
      "1.8",
      "1"
    ],
    [
      "1.720775094",
      "1.347106991"
    ],
    [
      "1.498791841",
      "1.625465186"
    ],
    [
      "1.178016747",
      "1.77994233"
    ],
    [
      "0.821983253",
      "1.77994233"
    ],
    [
      "0.501208159",
      "1.625465186"
    ],
    [
      "0.279224906",
      "1.347106991"
    ],
    [
      "0.2",
      "1"
    ],
    [
      "0.279224906",
      "0.652893009"
    ],
    [
      "0.501208159",
      "0.374534814"
    ],
    [
      "0.821983253",
      "0.22005767"
    ],
    [
      "1.178016747",
      "0.22005767"
    ],
    [
      "1.498791841",
      "0.374534814"
    ],
    [
      "1.720775094",
      "0.652893009"
    ]
  ]
}

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

Instance parameters

{
  "n": 14
}

The current leader's answer

{
  "points": [
    [
      "1.8",
      "1"
    ],
    [
      "1.720775094",
      "1.347106991"
    ],
    [
      "1.498791841",
      "1.625465186"
    ],
    [
      "1.178016747",
      "1.77994233"
    ],
    [
      "0.821983253",
      "1.77994233"
    ],
    [
      "0.501208159",
      "1.625465186"
    ],
    [
      "0.279224906",
      "1.347106991"
    ],
    [
      "0.2",
      "1"
    ],
    [
      "0.279224906",
      "0.652893009"
    ],
    [
      "0.501208159",
      "0.374534814"
    ],
    [
      "0.821983253",
      "0.22005767"
    ],
    [
      "1.178016747",
      "0.22005767"
    ],
    [
      "1.498791841",
      "0.374534814"
    ],
    [
      "1.720775094",
      "0.652893009"
    ]
  ]
}

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