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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 20

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

Instancen = 20
ObjectiveMinimize container circle radius
Best known, unproven1.874Source-precision interval: 1.874–1.875 (values inside match)exhibited site certificate 1.874369584Compiled by Erich Friedman; Found by Erich Friedman in 1997. Source expression: r = 1.874+.

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 = 20, 1.874369584
Current leader

1.874369584

container circle radius

Matches the best known
Answer sourceErich Friedman
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": "0.267460348",
      "x": "1.263642146",
      "y": "1.693892164"
    },
    {
      "turn": "0.267460348",
      "x": "0.732123471",
      "y": "1.690918209"
    },
    {
      "turn": "-0.268438157",
      "x": "0.998739212",
      "y": "1.536663054"
    },
    {
      "turn": "-0.268438157",
      "x": "0.465226718",
      "y": "1.537149782"
    },
    {
      "turn": "-0.268438157",
      "x": "1.530257886",
      "y": "1.539637009"
    },
    {
      "turn": "0.267460348",
      "x": "1.535721211",
      "y": "1.231608187"
    },
    {
      "turn": "0.267460348",
      "x": "0.466696095",
      "y": "1.229124603"
    },
    {
      "turn": "0.267460348",
      "x": "1.000208588",
      "y": "1.228637876"
    },
    {
      "turn": "-0.268438157",
      "x": "0.733311835",
      "y": "1.074869449"
    },
    {
      "turn": "-0.268438157",
      "x": "1.268824459",
      "y": "1.077839760"
    },
    {
      "turn": "-0.268438157",
      "x": "0.298646005",
      "y": "0.903787448"
    },
    {
      "turn": "-0.268438157",
      "x": "1.701177277",
      "y": "0.902507908"
    },
    {
      "turn": "0.267460348",
      "x": "0.733030823",
      "y": "0.766845867"
    },
    {
      "turn": "0.267460348",
      "x": "1.268543446",
      "y": "0.769816178"
    },
    {
      "turn": "-0.268438157",
      "x": "1.001646693",
      "y": "0.616047751"
    },
    {
      "turn": "0.267460348",
      "x": "0.298364993",
      "y": "0.595763867"
    },
    {
      "turn": "0.267460348",
      "x": "1.700896264",
      "y": "0.594484326"
    },
    {
      "turn": "-0.268438157",
      "x": "0.564980733",
      "y": "0.441508712"
    },
    {
      "turn": "-0.268438157",
      "x": "1.433999511",
      "y": "0.440715899"
    },
    {
      "turn": "0.267460348",
      "x": "1.001365681",
      "y": "0.308024170"
    }
  ],
  "radius": "0.308023708"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267460348",
      "x": "1.263642146",
      "y": "1.693892164"
    },
    {
      "turn": "0.267460348",
      "x": "0.732123471",
      "y": "1.690918209"
    },
    {
      "turn": "-0.268438157",
      "x": "0.998739212",
      "y": "1.536663054"
    },
    {
      "turn": "-0.268438157",
      "x": "0.465226718",
      "y": "1.537149782"
    },
    {
      "turn": "-0.268438157",
      "x": "1.530257886",
      "y": "1.539637009"
    },
    {
      "turn": "0.267460348",
      "x": "1.535721211",
      "y": "1.231608187"
    },
    {
      "turn": "0.267460348",
      "x": "0.466696095",
      "y": "1.229124603"
    },
    {
      "turn": "0.267460348",
      "x": "1.000208588",
      "y": "1.228637876"
    },
    {
      "turn": "-0.268438157",
      "x": "0.733311835",
      "y": "1.074869449"
    },
    {
      "turn": "-0.268438157",
      "x": "1.268824459",
      "y": "1.077839760"
    },
    {
      "turn": "-0.268438157",
      "x": "0.298646005",
      "y": "0.903787448"
    },
    {
      "turn": "-0.268438157",
      "x": "1.701177277",
      "y": "0.902507908"
    },
    {
      "turn": "0.267460348",
      "x": "0.733030823",
      "y": "0.766845867"
    },
    {
      "turn": "0.267460348",
      "x": "1.268543446",
      "y": "0.769816178"
    },
    {
      "turn": "-0.268438157",
      "x": "1.001646693",
      "y": "0.616047751"
    },
    {
      "turn": "0.267460348",
      "x": "0.298364993",
      "y": "0.595763867"
    },
    {
      "turn": "0.267460348",
      "x": "1.700896264",
      "y": "0.594484326"
    },
    {
      "turn": "-0.268438157",
      "x": "0.564980733",
      "y": "0.441508712"
    },
    {
      "turn": "-0.268438157",
      "x": "1.433999511",
      "y": "0.440715899"
    },
    {
      "turn": "0.267460348",
      "x": "1.001365681",
      "y": "0.308024170"
    }
  ],
  "radius": "0.308023708"
}

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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