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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 17

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

Instancen = 17
ObjectiveMinimize container circle radius
Best known, unproven1.730Source-precision interval: 1.73–1.731 (values inside match)exhibited site certificate 1.730310721Compiled by Erich Friedman; Found by David W. Cantrell in July 2005. Source expression: r = 1.730+.

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 = 17, 1.730310721
Current leader

1.730310721

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": "0.119639667",
      "x": "1.222598852",
      "y": "1.758517176"
    },
    {
      "turn": "0.267731693",
      "x": "0.742727348",
      "y": "1.670722579"
    },
    {
      "turn": "-0.268166715",
      "x": "0.453694215",
      "y": "1.504005603"
    },
    {
      "turn": "-0.268166715",
      "x": "1.546714738",
      "y": "1.503561966"
    },
    {
      "turn": "-0.268166715",
      "x": "1.000085438",
      "y": "1.210500383"
    },
    {
      "turn": "0.267731693",
      "x": "0.591465461",
      "y": "1.170281105"
    },
    {
      "turn": "0.267731693",
      "x": "1.408672632",
      "y": "1.169949416"
    },
    {
      "turn": "-0.268166715",
      "x": "0.302432328",
      "y": "1.003564129"
    },
    {
      "turn": "-0.268166715",
      "x": "1.697570335",
      "y": "1.002997869"
    },
    {
      "turn": "0.267731693",
      "x": "0.999950008",
      "y": "0.876831860"
    },
    {
      "turn": "-0.268166715",
      "x": "0.710916875",
      "y": "0.710114883"
    },
    {
      "turn": "-0.268166715",
      "x": "1.288847712",
      "y": "0.709880312"
    },
    {
      "turn": "0.267731693",
      "x": "0.302296898",
      "y": "0.669895605"
    },
    {
      "turn": "0.267731693",
      "x": "1.697434906",
      "y": "0.669329345"
    },
    {
      "turn": "0.267731693",
      "x": "0.710781446",
      "y": "0.376446360"
    },
    {
      "turn": "0.267731693",
      "x": "1.288712282",
      "y": "0.376211788"
    },
    {
      "turn": "-0.268166715",
      "x": "0.999679149",
      "y": "0.209494812"
    }
  ],
  "radius": "0.333668550"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.119639667",
      "x": "1.222598852",
      "y": "1.758517176"
    },
    {
      "turn": "0.267731693",
      "x": "0.742727348",
      "y": "1.670722579"
    },
    {
      "turn": "-0.268166715",
      "x": "0.453694215",
      "y": "1.504005603"
    },
    {
      "turn": "-0.268166715",
      "x": "1.546714738",
      "y": "1.503561966"
    },
    {
      "turn": "-0.268166715",
      "x": "1.000085438",
      "y": "1.210500383"
    },
    {
      "turn": "0.267731693",
      "x": "0.591465461",
      "y": "1.170281105"
    },
    {
      "turn": "0.267731693",
      "x": "1.408672632",
      "y": "1.169949416"
    },
    {
      "turn": "-0.268166715",
      "x": "0.302432328",
      "y": "1.003564129"
    },
    {
      "turn": "-0.268166715",
      "x": "1.697570335",
      "y": "1.002997869"
    },
    {
      "turn": "0.267731693",
      "x": "0.999950008",
      "y": "0.876831860"
    },
    {
      "turn": "-0.268166715",
      "x": "0.710916875",
      "y": "0.710114883"
    },
    {
      "turn": "-0.268166715",
      "x": "1.288847712",
      "y": "0.709880312"
    },
    {
      "turn": "0.267731693",
      "x": "0.302296898",
      "y": "0.669895605"
    },
    {
      "turn": "0.267731693",
      "x": "1.697434906",
      "y": "0.669329345"
    },
    {
      "turn": "0.267731693",
      "x": "0.710781446",
      "y": "0.376446360"
    },
    {
      "turn": "0.267731693",
      "x": "1.288712282",
      "y": "0.376211788"
    },
    {
      "turn": "-0.268166715",
      "x": "0.999679149",
      "y": "0.209494812"
    }
  ],
  "radius": "0.333668550"
}

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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