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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 19

Pack n unit-side equilateral triangles into a disc. Contact is allowed; interiors cannot overlap. Minimise the container circle radius.

Instancen = 19
ObjectiveMinimize container circle radius
Best known, unproven1.825Source-precision interval: 1.825–1.826 (values inside match)exhibited site certificate 1.825741865Compiled by Erich Friedman; Found by Haym Hirsh in August 2019. Source expression: r = √(10/3) = 1.825+.

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • Objectivecontainer circle radius = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equilateral triangles in a disc n = 19, 1.825741865
Current leader

1.825741865

container circle radius

Matches the best known
Answer sourceHaym Hirsh
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267752141",
      "x": "1.042657193",
      "y": "1.790553783"
    },
    {
      "turn": "-0.268146263",
      "x": "1.316460312",
      "y": "1.632339206"
    },
    {
      "turn": "0.267752141",
      "x": "0.610608340",
      "y": "1.590232272"
    },
    {
      "turn": "-0.268146263",
      "x": "0.336688938",
      "y": "1.432219104"
    },
    {
      "turn": "-0.268146263",
      "x": "0.884411458",
      "y": "1.432017695"
    },
    {
      "turn": "-0.268146263",
      "x": "1.705968255",
      "y": "1.358334678"
    },
    {
      "turn": "0.267752141",
      "x": "1.316344028",
      "y": "1.316111461"
    },
    {
      "turn": "0.267752141",
      "x": "0.568067430",
      "y": "1.115906234"
    },
    {
      "turn": "0.267752141",
      "x": "1.705851972",
      "y": "1.042106934"
    },
    {
      "turn": "-0.577595442",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "-0.268146263",
      "x": "0.294148028",
      "y": "0.957893066"
    },
    {
      "turn": "-0.268146263",
      "x": "1.431932570",
      "y": "0.884093766"
    },
    {
      "turn": "-0.268146263",
      "x": "0.683655972",
      "y": "0.683888539"
    },
    {
      "turn": "0.267752141",
      "x": "0.294031745",
      "y": "0.641665322"
    },
    {
      "turn": "0.267752141",
      "x": "1.663311062",
      "y": "0.567780896"
    },
    {
      "turn": "0.267752141",
      "x": "1.115588542",
      "y": "0.567982305"
    },
    {
      "turn": "-0.268146263",
      "x": "1.389391660",
      "y": "0.409767728"
    },
    {
      "turn": "0.267752141",
      "x": "0.683539688",
      "y": "0.367660794"
    },
    {
      "turn": "-0.268146263",
      "x": "0.957342807",
      "y": "0.209446217"
    }
  ],
  "radius": "0.316227765"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 19,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267752141",
      "x": "1.042657193",
      "y": "1.790553783"
    },
    {
      "turn": "-0.268146263",
      "x": "1.316460312",
      "y": "1.632339206"
    },
    {
      "turn": "0.267752141",
      "x": "0.610608340",
      "y": "1.590232272"
    },
    {
      "turn": "-0.268146263",
      "x": "0.336688938",
      "y": "1.432219104"
    },
    {
      "turn": "-0.268146263",
      "x": "0.884411458",
      "y": "1.432017695"
    },
    {
      "turn": "-0.268146263",
      "x": "1.705968255",
      "y": "1.358334678"
    },
    {
      "turn": "0.267752141",
      "x": "1.316344028",
      "y": "1.316111461"
    },
    {
      "turn": "0.267752141",
      "x": "0.568067430",
      "y": "1.115906234"
    },
    {
      "turn": "0.267752141",
      "x": "1.705851972",
      "y": "1.042106934"
    },
    {
      "turn": "-0.577595442",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "-0.268146263",
      "x": "0.294148028",
      "y": "0.957893066"
    },
    {
      "turn": "-0.268146263",
      "x": "1.431932570",
      "y": "0.884093766"
    },
    {
      "turn": "-0.268146263",
      "x": "0.683655972",
      "y": "0.683888539"
    },
    {
      "turn": "0.267752141",
      "x": "0.294031745",
      "y": "0.641665322"
    },
    {
      "turn": "0.267752141",
      "x": "1.663311062",
      "y": "0.567780896"
    },
    {
      "turn": "0.267752141",
      "x": "1.115588542",
      "y": "0.567982305"
    },
    {
      "turn": "-0.268146263",
      "x": "1.389391660",
      "y": "0.409767728"
    },
    {
      "turn": "0.267752141",
      "x": "0.683539688",
      "y": "0.367660794"
    },
    {
      "turn": "-0.268146263",
      "x": "0.957342807",
      "y": "0.209446217"
    }
  ],
  "radius": "0.316227765"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

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