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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 27

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

Instancen = 27
ObjectiveMinimize container circle radius
Best known, unproven2.138Source-precision interval: 2.138–2.139 (values inside match)exhibited site certificate 2.138465229Compiled by Erich Friedman; Found by David W. Cantrell in July 2005. Source expression: r = 2.138+.

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 = 27, 2.138465229
Current leader

2.138465229

container circle radius

Matches the best known
Answer sourceDavid W. Cantrell
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": "0.399178434",
      "x": "0.804231102",
      "y": "1.814081697"
    },
    {
      "turn": "0.144800318",
      "x": "1.195768898",
      "y": "1.814081697"
    },
    {
      "turn": "0.366295857",
      "x": "0.491652134",
      "y": "1.555777406"
    },
    {
      "turn": "0.174211903",
      "x": "1.508347866",
      "y": "1.555777406"
    },
    {
      "turn": "-0.174211903",
      "x": "0.745730445",
      "y": "1.464479605"
    },
    {
      "turn": "-0.366295857",
      "x": "1.254269555",
      "y": "1.464479605"
    },
    {
      "turn": "-0.238231847",
      "x": "0.252645603",
      "y": "1.377512751"
    },
    {
      "turn": "-0.298114758",
      "x": "1.747354397",
      "y": "1.377512751"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "1.196006996"
    },
    {
      "turn": "0.267949192",
      "x": "0.480994160",
      "y": "1.107012935"
    },
    {
      "turn": "0.267949192",
      "x": "1.519005840",
      "y": "1.107012935"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "0.247181612",
      "y": "0.972021198"
    },
    {
      "turn": "-0.267949192",
      "x": "1.752818388",
      "y": "0.972021198"
    },
    {
      "turn": "0.267949192",
      "x": "0.766187452",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "1.233812548",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "0.247181612",
      "y": "0.702037723"
    },
    {
      "turn": "0.267949192",
      "x": "1.752818388",
      "y": "0.702037723"
    },
    {
      "turn": "-0.267949192",
      "x": "1.000000000",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "0.532374905",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "1.467625095",
      "y": "0.656040046"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "0.532374905",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "1.467625095",
      "y": "0.386056572"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "0.251064834"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "0.251064834"
    }
  ],
  "radius": "0.269983473"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.399178434",
      "x": "0.804231102",
      "y": "1.814081697"
    },
    {
      "turn": "0.144800318",
      "x": "1.195768898",
      "y": "1.814081697"
    },
    {
      "turn": "0.366295857",
      "x": "0.491652134",
      "y": "1.555777406"
    },
    {
      "turn": "0.174211903",
      "x": "1.508347866",
      "y": "1.555777406"
    },
    {
      "turn": "-0.174211903",
      "x": "0.745730445",
      "y": "1.464479605"
    },
    {
      "turn": "-0.366295857",
      "x": "1.254269555",
      "y": "1.464479605"
    },
    {
      "turn": "-0.238231847",
      "x": "0.252645603",
      "y": "1.377512751"
    },
    {
      "turn": "-0.298114758",
      "x": "1.747354397",
      "y": "1.377512751"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "1.196006996"
    },
    {
      "turn": "0.267949192",
      "x": "0.480994160",
      "y": "1.107012935"
    },
    {
      "turn": "0.267949192",
      "x": "1.519005840",
      "y": "1.107012935"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "1.061015258"
    },
    {
      "turn": "-0.267949192",
      "x": "0.247181612",
      "y": "0.972021198"
    },
    {
      "turn": "-0.267949192",
      "x": "1.752818388",
      "y": "0.972021198"
    },
    {
      "turn": "0.267949192",
      "x": "0.766187452",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "1.233812548",
      "y": "0.791031784"
    },
    {
      "turn": "0.267949192",
      "x": "0.247181612",
      "y": "0.702037723"
    },
    {
      "turn": "0.267949192",
      "x": "1.752818388",
      "y": "0.702037723"
    },
    {
      "turn": "-0.267949192",
      "x": "1.000000000",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "0.532374905",
      "y": "0.656040046"
    },
    {
      "turn": "-0.267949192",
      "x": "1.467625095",
      "y": "0.656040046"
    },
    {
      "turn": "0.267949192",
      "x": "1.000000000",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "0.532374905",
      "y": "0.386056572"
    },
    {
      "turn": "0.267949192",
      "x": "1.467625095",
      "y": "0.386056572"
    },
    {
      "turn": "-0.267949192",
      "x": "0.766187452",
      "y": "0.251064834"
    },
    {
      "turn": "-0.267949192",
      "x": "1.233812548",
      "y": "0.251064834"
    }
  ],
  "radius": "0.269983473"
}

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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