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P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 25

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

Instancen = 25
ObjectiveMinimize container circle radius
Best known, unproven2.055Source-precision interval: 2.055–2.056 (values inside match)exhibited site certificate 2.055249514Compiled by Erich Friedman; Found by David W. Cantrell in July 2005. Source expression: r = 2.055+.

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 = 25, 2.055249514
Current leader

2.055249514

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": "0.577352738",
      "x": "0.648853619",
      "y": "1.734272980"
    },
    {
      "turn": "0.000001852",
      "x": "0.929768549",
      "y": "1.734274021"
    },
    {
      "turn": "0.000001852",
      "x": "1.351140943",
      "y": "1.734275581"
    },
    {
      "turn": "0.000001852",
      "x": "0.640862641",
      "y": "1.414514607"
    },
    {
      "turn": "0.577352738",
      "x": "1.359134289",
      "y": "1.414517267"
    },
    {
      "turn": "0.577352738",
      "x": "0.359947803",
      "y": "1.389641325"
    },
    {
      "turn": "0.000001852",
      "x": "1.640049311",
      "y": "1.389646066"
    },
    {
      "turn": "0.577352738",
      "x": "0.859541734",
      "y": "1.216396807"
    },
    {
      "turn": "0.000001852",
      "x": "1.140456663",
      "y": "1.216397847"
    },
    {
      "turn": "0.000001852",
      "x": "0.219491239",
      "y": "1.146361340"
    },
    {
      "turn": "0.577352738",
      "x": "1.780507677",
      "y": "1.146367120"
    },
    {
      "turn": "0.000001852",
      "x": "0.719085170",
      "y": "0.973116821"
    },
    {
      "turn": "-0.577347800",
      "x": "1.280915029",
      "y": "0.973118902"
    },
    {
      "turn": "-0.577347800",
      "x": "0.438170332",
      "y": "0.948243539"
    },
    {
      "turn": "0.000001852",
      "x": "1.561830051",
      "y": "0.948247700"
    },
    {
      "turn": "0.577352738",
      "x": "0.859543536",
      "y": "0.729837876"
    },
    {
      "turn": "0.000001852",
      "x": "1.140458465",
      "y": "0.729838916"
    },
    {
      "turn": "0.000001852",
      "x": "0.297713769",
      "y": "0.704963554"
    },
    {
      "turn": "0.577352738",
      "x": "1.702288417",
      "y": "0.704968755"
    },
    {
      "turn": "0.000001852",
      "x": "0.719086972",
      "y": "0.486557890"
    },
    {
      "turn": "0.577352738",
      "x": "1.280916831",
      "y": "0.486559971"
    },
    {
      "turn": "-0.577347800",
      "x": "0.438172134",
      "y": "0.461684608"
    },
    {
      "turn": "0.000001852",
      "x": "1.561831853",
      "y": "0.461688769"
    },
    {
      "turn": "0.577352738",
      "x": "0.859545337",
      "y": "0.243278945"
    },
    {
      "turn": "0.000001852",
      "x": "1.140460267",
      "y": "0.243279985"
    }
  ],
  "radius": "0.280914928"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.577352738",
      "x": "0.648853619",
      "y": "1.734272980"
    },
    {
      "turn": "0.000001852",
      "x": "0.929768549",
      "y": "1.734274021"
    },
    {
      "turn": "0.000001852",
      "x": "1.351140943",
      "y": "1.734275581"
    },
    {
      "turn": "0.000001852",
      "x": "0.640862641",
      "y": "1.414514607"
    },
    {
      "turn": "0.577352738",
      "x": "1.359134289",
      "y": "1.414517267"
    },
    {
      "turn": "0.577352738",
      "x": "0.359947803",
      "y": "1.389641325"
    },
    {
      "turn": "0.000001852",
      "x": "1.640049311",
      "y": "1.389646066"
    },
    {
      "turn": "0.577352738",
      "x": "0.859541734",
      "y": "1.216396807"
    },
    {
      "turn": "0.000001852",
      "x": "1.140456663",
      "y": "1.216397847"
    },
    {
      "turn": "0.000001852",
      "x": "0.219491239",
      "y": "1.146361340"
    },
    {
      "turn": "0.577352738",
      "x": "1.780507677",
      "y": "1.146367120"
    },
    {
      "turn": "0.000001852",
      "x": "0.719085170",
      "y": "0.973116821"
    },
    {
      "turn": "-0.577347800",
      "x": "1.280915029",
      "y": "0.973118902"
    },
    {
      "turn": "-0.577347800",
      "x": "0.438170332",
      "y": "0.948243539"
    },
    {
      "turn": "0.000001852",
      "x": "1.561830051",
      "y": "0.948247700"
    },
    {
      "turn": "0.577352738",
      "x": "0.859543536",
      "y": "0.729837876"
    },
    {
      "turn": "0.000001852",
      "x": "1.140458465",
      "y": "0.729838916"
    },
    {
      "turn": "0.000001852",
      "x": "0.297713769",
      "y": "0.704963554"
    },
    {
      "turn": "0.577352738",
      "x": "1.702288417",
      "y": "0.704968755"
    },
    {
      "turn": "0.000001852",
      "x": "0.719086972",
      "y": "0.486557890"
    },
    {
      "turn": "0.577352738",
      "x": "1.280916831",
      "y": "0.486559971"
    },
    {
      "turn": "-0.577347800",
      "x": "0.438172134",
      "y": "0.461684608"
    },
    {
      "turn": "0.000001852",
      "x": "1.561831853",
      "y": "0.461688769"
    },
    {
      "turn": "0.577352738",
      "x": "0.859545337",
      "y": "0.243278945"
    },
    {
      "turn": "0.000001852",
      "x": "1.140460267",
      "y": "0.243279985"
    }
  ],
  "radius": "0.280914928"
}

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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