P22 · Extremal configurations · Classic

The smallest triangle in a disc · n = 10

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 = 10

ObjectiveMaximize the smallest triangle's area

Best known, unproven0.150383733180435296MathWorld's value for a disc of unit area, given in closed form, 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

12345678910

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

0.071844476057464545

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"
    ],
    [
      "1.647213595",
      "1.470228202"
    ],
    [
      "1.247213595",
      "1.760845213"
    ],
    [
      "0.752786405",
      "1.760845213"
    ],
    [
      "0.352786405",
      "1.470228202"
    ],
    [
      "0.2",
      "1"
    ],
    [
      "0.352786405",
      "0.529771798"
    ],
    [
      "0.752786405",
      "0.239154787"
    ],
    [
      "1.247213595",
      "0.239154787"
    ],
    [
      "1.647213595",
      "0.529771798"
    ]
  ]
}

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

Instance parameters

{
  "n": 10
}

The current leader's answer

{
  "points": [
    [
      "1.8",
      "1"
    ],
    [
      "1.647213595",
      "1.470228202"
    ],
    [
      "1.247213595",
      "1.760845213"
    ],
    [
      "0.752786405",
      "1.760845213"
    ],
    [
      "0.352786405",
      "1.470228202"
    ],
    [
      "0.2",
      "1"
    ],
    [
      "0.352786405",
      "0.529771798"
    ],
    [
      "0.752786405",
      "0.239154787"
    ],
    [
      "1.247213595",
      "0.239154787"
    ],
    [
      "1.647213595",
      "0.529771798"
    ]
  ]
}

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