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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 36

Pack n equal circles into the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1), maximising their radius.

Instancem = 5 · n = 36
ObjectiveMaximize circle radius
Best known, unproven0.128215138245Source-precision interval: 0.128215138244–0.128215138246 (values inside match)exhibited site certificate 0.128215137Packomania (compiled by Eckard Specht), m=5, n=36. Paolo Amore, Circle packing in regular polygons , Phys. Fluids 35 , 027130 (2023). Coordinates imported from the public archive, normalized, decimalized and exactly verified by MinMax Arena. The web-table target and grid certificate are separate.

Formal definition

  • ContainerThe editor and certificate normalize to the true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1). 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.
  • ObjectiveCircle radius r, larger is better; the container circumradius is 1.
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 Equal circles in regular polygons m = 5 · n = 36, 0.128215137
Current leader

0.128215137

circle radius

Answer sourcePaolo Amore
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 true regular m-gon with centre (1,1), circumradius 1 and first vertex (2,1)

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",
      "x": "0.319198144",
      "y": "1.414318580"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "1.137584611"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.881154335"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.505368498"
    },
    {
      "turn": "0",
      "x": "0.493705273",
      "y": "0.693261417"
    },
    {
      "turn": "0",
      "x": "0.528404113",
      "y": "1.562606642"
    },
    {
      "turn": "0",
      "x": "0.535094101",
      "y": "1.275951596"
    },
    {
      "turn": "0",
      "x": "0.546975446",
      "y": "1.019796720"
    },
    {
      "turn": "0",
      "x": "0.565174206",
      "y": "0.432895997"
    },
    {
      "turn": "0",
      "x": "0.739681335",
      "y": "0.620788915"
    },
    {
      "turn": "0",
      "x": "0.759594195",
      "y": "0.876444865"
    },
    {
      "turn": "0",
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      "y": "1.385472959"
    },
    {
      "turn": "0",
      "x": "0.772283799",
      "y": "1.641847955"
    },
    {
      "turn": "0",
      "x": "0.797956380",
      "y": "1.130137250"
    },
    {
      "turn": "0",
      "x": "0.806771288",
      "y": "0.346946380"
    },
    {
      "turn": "0",
      "x": "0.981278417",
      "y": "0.534839299"
    },
    {
      "turn": "0",
      "x": "0.999370050",
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    },
    {
      "turn": "0",
      "x": "1.010839173",
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    },
    {
      "turn": "0",
      "x": "1.016163484",
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    {
      "turn": "0",
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.129670485",
      "y": "0.745795800"
    },
    {
      "turn": "0",
      "x": "1.236592575",
      "y": "0.510940190"
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    {
      "turn": "0",
      "x": "1.255721369",
      "y": "1.388627499"
    },
    {
      "turn": "0",
      "x": "1.255721369",
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    {
      "turn": "0",
      "x": "1.260043169",
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    },
    {
      "turn": "0",
      "x": "1.307059720",
      "y": "0.264382148"
    },
    {
      "turn": "0",
      "x": "1.388559163",
      "y": "0.754373293"
    },
    {
      "turn": "0",
      "x": "1.406447303",
      "y": "1.181171047"
    },
    {
      "turn": "0",
      "x": "1.410769104",
      "y": "1.592874130"
    },
    {
      "turn": "0",
      "x": "1.492858753",
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
      "x": "1.794310623",
      "y": "0.935025482"
    }
  ],
  "radius": "0.128215137"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 36,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "1.414318580"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "1.137584611"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.881154335"
    },
    {
      "turn": "0",
      "x": "0.319198144",
      "y": "0.505368498"
    },
    {
      "turn": "0",
      "x": "0.493705273",
      "y": "0.693261417"
    },
    {
      "turn": "0",
      "x": "0.528404113",
      "y": "1.562606642"
    },
    {
      "turn": "0",
      "x": "0.535094101",
      "y": "1.275951596"
    },
    {
      "turn": "0",
      "x": "0.546975446",
      "y": "1.019796720"
    },
    {
      "turn": "0",
      "x": "0.565174206",
      "y": "0.432895997"
    },
    {
      "turn": "0",
      "x": "0.739681335",
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
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    },
    {
      "turn": "0",
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      "y": "0.346946380"
    },
    {
      "turn": "0",
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      "y": "0.534839299"
    },
    {
      "turn": "0",
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      "y": "0.967351552"
    },
    {
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    {
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    {
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    {
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    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    {
      "turn": "0",
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    }
  ],
  "radius": "0.128215137"
}

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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