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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 22

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

Instancen = 22
ObjectiveMinimize container circle radius
Best known, unproven1.943Source-precision interval: 1.943–1.944 (values inside match)exhibited site certificate 1.94398837Compiled by Erich Friedman; Found by Maurizio Morandi in June 2008. Source expression: r = 1.943+.

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 = 22, 1.94398837
Current leader

1.94398837

container circle radius

Matches the best known
Answer sourceMaurizio Morandi
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": "0.268200414",
      "x": "0.867571454",
      "y": "1.772565558"
    },
    {
      "turn": "0.268200414",
      "x": "1.321102191",
      "y": "1.667252860"
    },
    {
      "turn": "-0.267698002",
      "x": "0.610437905",
      "y": "1.623948680"
    },
    {
      "turn": "-0.267698002",
      "x": "1.063968642",
      "y": "1.518635982"
    },
    {
      "turn": "-0.267698002",
      "x": "1.578374957",
      "y": "1.518877113"
    },
    {
      "turn": "0.268200414",
      "x": "0.514707568",
      "y": "1.326911116"
    },
    {
      "turn": "0.268200414",
      "x": "1.482644620",
      "y": "1.221839549"
    },
    {
      "turn": "0.268200414",
      "x": "0.968238305",
      "y": "1.221598418"
    },
    {
      "turn": "-0.267698002",
      "x": "0.257574019",
      "y": "1.178294239"
    },
    {
      "turn": "-0.267698002",
      "x": "1.739917387",
      "y": "1.073463802"
    },
    {
      "turn": "-0.267698002",
      "x": "1.225511071",
      "y": "1.073222671"
    },
    {
      "turn": "-0.267698002",
      "x": "0.711104756",
      "y": "1.072981540"
    },
    {
      "turn": "0.268200414",
      "x": "0.257713236",
      "y": "0.881301614"
    },
    {
      "turn": "0.268200414",
      "x": "1.740056604",
      "y": "0.776471178"
    },
    {
      "turn": "0.268200414",
      "x": "0.711243973",
      "y": "0.775988916"
    },
    {
      "turn": "0.268200414",
      "x": "1.225650289",
      "y": "0.776230047"
    },
    {
      "turn": "-0.267698002",
      "x": "1.482923055",
      "y": "0.627854300"
    },
    {
      "turn": "-0.267698002",
      "x": "0.968516740",
      "y": "0.627613169"
    },
    {
      "turn": "-0.267698002",
      "x": "0.454110424",
      "y": "0.627372038"
    },
    {
      "turn": "0.268200414",
      "x": "0.743110611",
      "y": "0.330514819"
    },
    {
      "turn": "0.268200414",
      "x": "1.257516926",
      "y": "0.330755950"
    },
    {
      "turn": "-0.267698002",
      "x": "1.000383377",
      "y": "0.182139072"
    }
  ],
  "radius": "0.296992656"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 22,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.268200414",
      "x": "0.867571454",
      "y": "1.772565558"
    },
    {
      "turn": "0.268200414",
      "x": "1.321102191",
      "y": "1.667252860"
    },
    {
      "turn": "-0.267698002",
      "x": "0.610437905",
      "y": "1.623948680"
    },
    {
      "turn": "-0.267698002",
      "x": "1.063968642",
      "y": "1.518635982"
    },
    {
      "turn": "-0.267698002",
      "x": "1.578374957",
      "y": "1.518877113"
    },
    {
      "turn": "0.268200414",
      "x": "0.514707568",
      "y": "1.326911116"
    },
    {
      "turn": "0.268200414",
      "x": "1.482644620",
      "y": "1.221839549"
    },
    {
      "turn": "0.268200414",
      "x": "0.968238305",
      "y": "1.221598418"
    },
    {
      "turn": "-0.267698002",
      "x": "0.257574019",
      "y": "1.178294239"
    },
    {
      "turn": "-0.267698002",
      "x": "1.739917387",
      "y": "1.073463802"
    },
    {
      "turn": "-0.267698002",
      "x": "1.225511071",
      "y": "1.073222671"
    },
    {
      "turn": "-0.267698002",
      "x": "0.711104756",
      "y": "1.072981540"
    },
    {
      "turn": "0.268200414",
      "x": "0.257713236",
      "y": "0.881301614"
    },
    {
      "turn": "0.268200414",
      "x": "1.740056604",
      "y": "0.776471178"
    },
    {
      "turn": "0.268200414",
      "x": "0.711243973",
      "y": "0.775988916"
    },
    {
      "turn": "0.268200414",
      "x": "1.225650289",
      "y": "0.776230047"
    },
    {
      "turn": "-0.267698002",
      "x": "1.482923055",
      "y": "0.627854300"
    },
    {
      "turn": "-0.267698002",
      "x": "0.968516740",
      "y": "0.627613169"
    },
    {
      "turn": "-0.267698002",
      "x": "0.454110424",
      "y": "0.627372038"
    },
    {
      "turn": "0.268200414",
      "x": "0.743110611",
      "y": "0.330514819"
    },
    {
      "turn": "0.268200414",
      "x": "1.257516926",
      "y": "0.330755950"
    },
    {
      "turn": "-0.267698002",
      "x": "1.000383377",
      "y": "0.182139072"
    }
  ],
  "radius": "0.296992656"
}

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

DISCUSSION

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