P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 15

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

Instancen = 15

ObjectiveMinimize container circle radius

Best known, unproven1.636Source-precision interval: 1.636–1.637 (values inside match)exhibited site certificate 1.636720719Compiled by Erich Friedman; Found by David W. Cantrell in July 2002. Source expression: r = 1.636+.

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

Current leader

1.636720719

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": "0.267840825",
      "x": "1.305613309",
      "y": "1.615211854"
    },
    {
      "turn": "0.267840825",
      "x": "0.694635533",
      "y": "1.615335407"
    },
    {
      "turn": "-0.311118512",
      "x": "0.361114628",
      "y": "1.440141510"
    },
    {
      "turn": "-0.225904512",
      "x": "1.639063332",
      "y": "1.439883082"
    },
    {
      "turn": "-0.268057565",
      "x": "1.000088755",
      "y": "1.438899539"
    },
    {
      "turn": "0.225691981",
      "x": "0.333046944",
      "y": "1.088511746"
    },
    {
      "turn": "0.310896731",
      "x": "1.666988799",
      "y": "1.088241995"
    },
    {
      "turn": "0.267840825",
      "x": "1.000017422",
      "y": "1.086151355"
    },
    {
      "turn": "-0.268057565",
      "x": "0.679536143",
      "y": "0.935756947"
    },
    {
      "turn": "-0.268057565",
      "x": "1.320437848",
      "y": "0.935627344"
    },
    {
      "turn": "0.302613075",
      "x": "0.362006075",
      "y": "0.563805630"
    },
    {
      "turn": "0.233662595",
      "x": "1.637817457",
      "y": "0.563547634"
    },
    {
      "turn": "0.267840825",
      "x": "0.999910422",
      "y": "0.557029080"
    },
    {
      "turn": "-0.268057565",
      "x": "0.690128417",
      "y": "0.388094329"
    },
    {
      "turn": "-0.268057565",
      "x": "1.309624077",
      "y": "0.387969054"
    }
  ],
  "radius": "0.352748189"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267840825",
      "x": "1.305613309",
      "y": "1.615211854"
    },
    {
      "turn": "0.267840825",
      "x": "0.694635533",
      "y": "1.615335407"
    },
    {
      "turn": "-0.311118512",
      "x": "0.361114628",
      "y": "1.440141510"
    },
    {
      "turn": "-0.225904512",
      "x": "1.639063332",
      "y": "1.439883082"
    },
    {
      "turn": "-0.268057565",
      "x": "1.000088755",
      "y": "1.438899539"
    },
    {
      "turn": "0.225691981",
      "x": "0.333046944",
      "y": "1.088511746"
    },
    {
      "turn": "0.310896731",
      "x": "1.666988799",
      "y": "1.088241995"
    },
    {
      "turn": "0.267840825",
      "x": "1.000017422",
      "y": "1.086151355"
    },
    {
      "turn": "-0.268057565",
      "x": "0.679536143",
      "y": "0.935756947"
    },
    {
      "turn": "-0.268057565",
      "x": "1.320437848",
      "y": "0.935627344"
    },
    {
      "turn": "0.302613075",
      "x": "0.362006075",
      "y": "0.563805630"
    },
    {
      "turn": "0.233662595",
      "x": "1.637817457",
      "y": "0.563547634"
    },
    {
      "turn": "0.267840825",
      "x": "0.999910422",
      "y": "0.557029080"
    },
    {
      "turn": "-0.268057565",
      "x": "0.690128417",
      "y": "0.388094329"
    },
    {
      "turn": "-0.268057565",
      "x": "1.309624077",
      "y": "0.387969054"
    }
  ],
  "radius": "0.352748189"
}

DISCUSSION

Discussion

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