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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 13

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 = 13
ObjectiveMaximize circle radius
Best known, unproven0.207999994722Source-precision interval: 0.207999994721–0.207999994723 (values inside match)exhibited site certificate 0.207999994Packomania (compiled by Eckard Specht), m=5, n=13. 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 = 13, 0.207999994
Current leader

0.207999994

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": "0.398983000",
      "y": "1.229041191"
    },
    {
      "turn": "0",
      "x": "0.398983000",
      "y": "0.813041201"
    },
    {
      "turn": "0",
      "x": "0.650430923",
      "y": "0.481635207"
    },
    {
      "turn": "0",
      "x": "0.686192317",
      "y": "1.529984374"
    },
    {
      "turn": "0",
      "x": "0.764119222",
      "y": "1.029713493"
    },
    {
      "turn": "0",
      "x": "1.008637881",
      "y": "0.693162432"
    },
    {
      "turn": "0",
      "x": "1.051328539",
      "y": "1.330656677"
    },
    {
      "turn": "0",
      "x": "1.174100813",
      "y": "0.311484546"
    },
    {
      "turn": "0",
      "x": "1.229568066",
      "y": "1.706537858"
    },
    {
      "turn": "0",
      "x": "1.295847197",
      "y": "0.994105616"
    },
    {
      "turn": "0",
      "x": "1.461310130",
      "y": "0.612427729"
    },
    {
      "turn": "0",
      "x": "1.474086725",
      "y": "1.369986796"
    },
    {
      "turn": "0",
      "x": "1.727598758",
      "y": "0.980727247"
    }
  ],
  "radius": "0.207999994"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 13,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.398983000",
      "y": "1.229041191"
    },
    {
      "turn": "0",
      "x": "0.398983000",
      "y": "0.813041201"
    },
    {
      "turn": "0",
      "x": "0.650430923",
      "y": "0.481635207"
    },
    {
      "turn": "0",
      "x": "0.686192317",
      "y": "1.529984374"
    },
    {
      "turn": "0",
      "x": "0.764119222",
      "y": "1.029713493"
    },
    {
      "turn": "0",
      "x": "1.008637881",
      "y": "0.693162432"
    },
    {
      "turn": "0",
      "x": "1.051328539",
      "y": "1.330656677"
    },
    {
      "turn": "0",
      "x": "1.174100813",
      "y": "0.311484546"
    },
    {
      "turn": "0",
      "x": "1.229568066",
      "y": "1.706537858"
    },
    {
      "turn": "0",
      "x": "1.295847197",
      "y": "0.994105616"
    },
    {
      "turn": "0",
      "x": "1.461310130",
      "y": "0.612427729"
    },
    {
      "turn": "0",
      "x": "1.474086725",
      "y": "1.369986796"
    },
    {
      "turn": "0",
      "x": "1.727598758",
      "y": "0.980727247"
    }
  ],
  "radius": "0.207999994"
}

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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