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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 16

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

Instancen = 16
ObjectiveMinimize container circle radius
Best known, unproven1.687Source-precision interval: 1.687–1.688 (values inside match)exhibited site certificate 1.687452419Compiled by Erich Friedman; Found by Maurizio Morandi in April 2008. Source expression: r = 1.687+.

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 = 16, 1.687452419
Current leader

1.687452419

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": "0.268220009",
      "x": "0.894658075",
      "y": "1.744835028"
    },
    {
      "turn": "-0.267678413",
      "x": "0.598439905",
      "y": "1.573613752"
    },
    {
      "turn": "-0.267678413",
      "x": "1.400980178",
      "y": "1.574019288"
    },
    {
      "turn": "0.268220009",
      "x": "0.598612795",
      "y": "1.231470653"
    },
    {
      "turn": "0.268220009",
      "x": "1.401153068",
      "y": "1.231876189"
    },
    {
      "turn": "-0.267678413",
      "x": "0.999877508",
      "y": "1.242407488"
    },
    {
      "turn": "-0.267678413",
      "x": "0.302394625",
      "y": "1.060249377"
    },
    {
      "turn": "-0.267678413",
      "x": "1.697544128",
      "y": "1.060954367"
    },
    {
      "turn": "0.268220009",
      "x": "1.000050398",
      "y": "0.900264390"
    },
    {
      "turn": "-0.267678413",
      "x": "1.296441458",
      "y": "0.729342568"
    },
    {
      "turn": "0.268220009",
      "x": "0.302567516",
      "y": "0.718106278"
    },
    {
      "turn": "0.268220009",
      "x": "1.697717019",
      "y": "0.718811269"
    },
    {
      "turn": "-0.267678413",
      "x": "0.703832228",
      "y": "0.729043113"
    },
    {
      "turn": "0.268220009",
      "x": "1.296614348",
      "y": "0.387199469"
    },
    {
      "turn": "0.268220009",
      "x": "0.704005118",
      "y": "0.386900015"
    },
    {
      "turn": "-0.267678413",
      "x": "1.000396179",
      "y": "0.215978193"
    }
  ],
  "radius": "0.342143140"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.268220009",
      "x": "0.894658075",
      "y": "1.744835028"
    },
    {
      "turn": "-0.267678413",
      "x": "0.598439905",
      "y": "1.573613752"
    },
    {
      "turn": "-0.267678413",
      "x": "1.400980178",
      "y": "1.574019288"
    },
    {
      "turn": "0.268220009",
      "x": "0.598612795",
      "y": "1.231470653"
    },
    {
      "turn": "0.268220009",
      "x": "1.401153068",
      "y": "1.231876189"
    },
    {
      "turn": "-0.267678413",
      "x": "0.999877508",
      "y": "1.242407488"
    },
    {
      "turn": "-0.267678413",
      "x": "0.302394625",
      "y": "1.060249377"
    },
    {
      "turn": "-0.267678413",
      "x": "1.697544128",
      "y": "1.060954367"
    },
    {
      "turn": "0.268220009",
      "x": "1.000050398",
      "y": "0.900264390"
    },
    {
      "turn": "-0.267678413",
      "x": "1.296441458",
      "y": "0.729342568"
    },
    {
      "turn": "0.268220009",
      "x": "0.302567516",
      "y": "0.718106278"
    },
    {
      "turn": "0.268220009",
      "x": "1.697717019",
      "y": "0.718811269"
    },
    {
      "turn": "-0.267678413",
      "x": "0.703832228",
      "y": "0.729043113"
    },
    {
      "turn": "0.268220009",
      "x": "1.296614348",
      "y": "0.387199469"
    },
    {
      "turn": "0.268220009",
      "x": "0.704005118",
      "y": "0.386900015"
    },
    {
      "turn": "-0.267678413",
      "x": "1.000396179",
      "y": "0.215978193"
    }
  ],
  "radius": "0.342143140"
}

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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