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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 11

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

Instancen = 11
ObjectiveMinimize container circle radius
Best known, unproven1.466Source-precision interval: 1.466–1.467 (values inside match)exhibited site certificate 1.466988273Compiled by Erich Friedman; Found by David W. Cantrell in August 2012. Source expression: r = 1.466+.

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 = 11, 1.466988273
Current leader

1.466988273

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": "0.468342575",
      "x": "1.088097982",
      "y": "1.638072529"
    },
    {
      "turn": "-0.468083702",
      "x": "0.523850926",
      "y": "1.570824158"
    },
    {
      "turn": "-0.085805959",
      "x": "1.475906629",
      "y": "1.571026304"
    },
    {
      "turn": "-0.510470890",
      "x": "1.703100014",
      "y": "1.144684267"
    },
    {
      "turn": "-0.051442567",
      "x": "0.296838609",
      "y": "1.144385681"
    },
    {
      "turn": "0.577491828",
      "x": "0.803231265",
      "y": "0.943125158"
    },
    {
      "turn": "0.000106163",
      "x": "1.196792869",
      "y": "0.943208721"
    },
    {
      "turn": "0.088231316",
      "x": "0.493371083",
      "y": "0.602225004"
    },
    {
      "turn": "0.465669099",
      "x": "1.506797787",
      "y": "0.602440181"
    },
    {
      "turn": "-0.088017341",
      "x": "1.253505057",
      "y": "0.301220102"
    },
    {
      "turn": "-0.465410756",
      "x": "0.746791705",
      "y": "0.301112513"
    }
  ],
  "radius": "0.393561612"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 11,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.468342575",
      "x": "1.088097982",
      "y": "1.638072529"
    },
    {
      "turn": "-0.468083702",
      "x": "0.523850926",
      "y": "1.570824158"
    },
    {
      "turn": "-0.085805959",
      "x": "1.475906629",
      "y": "1.571026304"
    },
    {
      "turn": "-0.510470890",
      "x": "1.703100014",
      "y": "1.144684267"
    },
    {
      "turn": "-0.051442567",
      "x": "0.296838609",
      "y": "1.144385681"
    },
    {
      "turn": "0.577491828",
      "x": "0.803231265",
      "y": "0.943125158"
    },
    {
      "turn": "0.000106163",
      "x": "1.196792869",
      "y": "0.943208721"
    },
    {
      "turn": "0.088231316",
      "x": "0.493371083",
      "y": "0.602225004"
    },
    {
      "turn": "0.465669099",
      "x": "1.506797787",
      "y": "0.602440181"
    },
    {
      "turn": "-0.088017341",
      "x": "1.253505057",
      "y": "0.301220102"
    },
    {
      "turn": "-0.465410756",
      "x": "0.746791705",
      "y": "0.301112513"
    }
  ],
  "radius": "0.393561612"
}

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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