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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · 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 = 8 · n = 36
ObjectiveMaximize circle radius
Best known, unproven0.139934447382Source-precision interval: 0.139934447381–0.139934447383 (values inside match)exhibited site certificate 0.139934446Packomania (compiled by Eckard Specht), m=8, n=36. Eckard Specht, program coc , 2020. 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 = 8 · n = 36, 0.139934446
Current leader

0.139934446

circle radius

Answer sourceEckard Specht
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": "0.485703658",
      "y": "0.364492474"
    },
    {
      "turn": "0",
      "x": "0.744268802",
      "y": "0.257391285"
    },
    {
      "turn": "0",
      "x": "1.003986078",
      "y": "0.153115042"
    },
    {
      "turn": "0",
      "x": "0.320980483",
      "y": "0.590750897"
    },
    {
      "turn": "0",
      "x": "0.856283749",
      "y": "0.513865942"
    },
    {
      "turn": "0",
      "x": "0.598895338",
      "y": "0.623765008"
    },
    {
      "turn": "0",
      "x": "1.116001025",
      "y": "0.409589699"
    },
    {
      "turn": "0",
      "x": "1.376562523",
      "y": "0.307441258"
    },
    {
      "turn": "0",
      "x": "0.213879294",
      "y": "0.849316041"
    },
    {
      "turn": "0",
      "x": "0.491926270",
      "y": "0.885323100"
    },
    {
      "turn": "0",
      "x": "1.619340476",
      "y": "0.446672948"
    },
    {
      "turn": "0",
      "x": "0.821721712",
      "y": "0.793101966"
    },
    {
      "turn": "0",
      "x": "1.080794796",
      "y": "0.687235372"
    },
    {
      "turn": "0",
      "x": "1.341356294",
      "y": "0.585086931"
    },
    {
      "turn": "0",
      "x": "0.204914620",
      "y": "1.129041322"
    },
    {
      "turn": "0",
      "x": "0.482389198",
      "y": "1.165571543"
    },
    {
      "turn": "0",
      "x": "0.742157700",
      "y": "1.061422979"
    },
    {
      "turn": "0",
      "x": "1.584134248",
      "y": "0.724318621"
    },
    {
      "turn": "0",
      "x": "1.042941413",
      "y": "0.964532541"
    },
    {
      "turn": "0",
      "x": "1.303502911",
      "y": "0.862384100"
    },
    {
      "turn": "0",
      "x": "0.312015809",
      "y": "1.387606465"
    },
    {
      "turn": "0",
      "x": "1.546280865",
      "y": "1.001615790"
    },
    {
      "turn": "0",
      "x": "1.805353949",
      "y": "0.895749196"
    },
    {
      "turn": "0",
      "x": "0.642171457",
      "y": "1.397434802"
    },
    {
      "turn": "0",
      "x": "0.902369456",
      "y": "1.290898013"
    },
    {
      "turn": "0",
      "x": "1.171252613",
      "y": "1.213254945"
    },
    {
      "turn": "0",
      "x": "0.460279390",
      "y": "1.624976449"
    },
    {
      "turn": "0",
      "x": "1.446638678",
      "y": "1.263145965"
    },
    {
      "turn": "0",
      "x": "1.776414089",
      "y": "1.174117806"
    },
    {
      "turn": "0",
      "x": "0.737889415",
      "y": "1.660462673"
    },
    {
      "turn": "0",
      "x": "0.996855597",
      "y": "1.554334851"
    },
    {
      "turn": "0",
      "x": "1.265738754",
      "y": "1.476691782"
    },
    {
      "turn": "0",
      "x": "1.669312900",
      "y": "1.432682950"
    },
    {
      "turn": "0",
      "x": "0.959281896",
      "y": "1.831670055"
    },
    {
      "turn": "0",
      "x": "1.228165053",
      "y": "1.754026986"
    },
    {
      "turn": "0",
      "x": "1.488412976",
      "y": "1.646228767"
    }
  ],
  "radius": "0.139934446"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 36,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.485703658",
      "y": "0.364492474"
    },
    {
      "turn": "0",
      "x": "0.744268802",
      "y": "0.257391285"
    },
    {
      "turn": "0",
      "x": "1.003986078",
      "y": "0.153115042"
    },
    {
      "turn": "0",
      "x": "0.320980483",
      "y": "0.590750897"
    },
    {
      "turn": "0",
      "x": "0.856283749",
      "y": "0.513865942"
    },
    {
      "turn": "0",
      "x": "0.598895338",
      "y": "0.623765008"
    },
    {
      "turn": "0",
      "x": "1.116001025",
      "y": "0.409589699"
    },
    {
      "turn": "0",
      "x": "1.376562523",
      "y": "0.307441258"
    },
    {
      "turn": "0",
      "x": "0.213879294",
      "y": "0.849316041"
    },
    {
      "turn": "0",
      "x": "0.491926270",
      "y": "0.885323100"
    },
    {
      "turn": "0",
      "x": "1.619340476",
      "y": "0.446672948"
    },
    {
      "turn": "0",
      "x": "0.821721712",
      "y": "0.793101966"
    },
    {
      "turn": "0",
      "x": "1.080794796",
      "y": "0.687235372"
    },
    {
      "turn": "0",
      "x": "1.341356294",
      "y": "0.585086931"
    },
    {
      "turn": "0",
      "x": "0.204914620",
      "y": "1.129041322"
    },
    {
      "turn": "0",
      "x": "0.482389198",
      "y": "1.165571543"
    },
    {
      "turn": "0",
      "x": "0.742157700",
      "y": "1.061422979"
    },
    {
      "turn": "0",
      "x": "1.584134248",
      "y": "0.724318621"
    },
    {
      "turn": "0",
      "x": "1.042941413",
      "y": "0.964532541"
    },
    {
      "turn": "0",
      "x": "1.303502911",
      "y": "0.862384100"
    },
    {
      "turn": "0",
      "x": "0.312015809",
      "y": "1.387606465"
    },
    {
      "turn": "0",
      "x": "1.546280865",
      "y": "1.001615790"
    },
    {
      "turn": "0",
      "x": "1.805353949",
      "y": "0.895749196"
    },
    {
      "turn": "0",
      "x": "0.642171457",
      "y": "1.397434802"
    },
    {
      "turn": "0",
      "x": "0.902369456",
      "y": "1.290898013"
    },
    {
      "turn": "0",
      "x": "1.171252613",
      "y": "1.213254945"
    },
    {
      "turn": "0",
      "x": "0.460279390",
      "y": "1.624976449"
    },
    {
      "turn": "0",
      "x": "1.446638678",
      "y": "1.263145965"
    },
    {
      "turn": "0",
      "x": "1.776414089",
      "y": "1.174117806"
    },
    {
      "turn": "0",
      "x": "0.737889415",
      "y": "1.660462673"
    },
    {
      "turn": "0",
      "x": "0.996855597",
      "y": "1.554334851"
    },
    {
      "turn": "0",
      "x": "1.265738754",
      "y": "1.476691782"
    },
    {
      "turn": "0",
      "x": "1.669312900",
      "y": "1.432682950"
    },
    {
      "turn": "0",
      "x": "0.959281896",
      "y": "1.831670055"
    },
    {
      "turn": "0",
      "x": "1.228165053",
      "y": "1.754026986"
    },
    {
      "turn": "0",
      "x": "1.488412976",
      "y": "1.646228767"
    }
  ],
  "radius": "0.139934446"
}

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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