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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 5 · n = 26

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 = 26
ObjectiveMaximize circle radius
Best known, unproven0.150683336700Source-precision interval: 0.150683336699–0.150683336701 (values inside match)exhibited site certificate 0.150683336Packomania (compiled by Eckard Specht), m=5, n=26. 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 = 26, 0.150683336
Current leader

0.150683336

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": "0.341666342",
      "y": "1.478307400"
    },
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "1.176940727"
    },
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "0.875574053"
    },
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "0.574207380"
    },
    {
      "turn": "0",
      "x": "0.608885955",
      "y": "0.736234358"
    },
    {
      "turn": "0",
      "x": "0.608885955",
      "y": "0.434867685"
    },
    {
      "turn": "0",
      "x": "0.628283081",
      "y": "1.571434824"
    },
    {
      "turn": "0",
      "x": "0.628283081",
      "y": "1.270068150"
    },
    {
      "turn": "0",
      "x": "0.758074629",
      "y": "0.998082809"
    },
    {
      "turn": "0",
      "x": "0.895502693",
      "y": "0.341740261"
    },
    {
      "turn": "0",
      "x": "0.910247702",
      "y": "0.737957401"
    },
    {
      "turn": "0",
      "x": "0.914899819",
      "y": "1.664562247"
    },
    {
      "turn": "0",
      "x": "0.928725070",
      "y": "1.246478258"
    },
    {
      "turn": "0",
      "x": "1.080898143",
      "y": "0.986352850"
    },
    {
      "turn": "0",
      "x": "1.129699490",
      "y": "0.531407686"
    },
    {
      "turn": "0",
      "x": "1.138662879",
      "y": "1.462691088"
    },
    {
      "turn": "0",
      "x": "1.218666337",
      "y": "0.237841012"
    },
    {
      "turn": "0",
      "x": "1.251279600",
      "y": "1.772382729"
    },
    {
      "turn": "0",
      "x": "1.291878833",
      "y": "1.202303636"
    },
    {
      "turn": "0",
      "x": "1.300349931",
      "y": "0.779803135"
    },
    {
      "turn": "0",
      "x": "1.423012530",
      "y": "0.462202682"
    },
    {
      "turn": "0",
      "x": "1.433999680",
      "y": "1.522674803"
    },
    {
      "turn": "0",
      "x": "1.512108788",
      "y": "0.994232747"
    },
    {
      "turn": "0",
      "x": "1.600151416",
      "y": "0.706013442"
    },
    {
      "turn": "0",
      "x": "1.611138566",
      "y": "1.278864043"
    },
    {
      "turn": "0",
      "x": "1.813571160",
      "y": "1.000001673"
    }
  ],
  "radius": "0.150683336"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "1.478307400"
    },
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "1.176940727"
    },
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "0.875574053"
    },
    {
      "turn": "0",
      "x": "0.341666342",
      "y": "0.574207380"
    },
    {
      "turn": "0",
      "x": "0.608885955",
      "y": "0.736234358"
    },
    {
      "turn": "0",
      "x": "0.608885955",
      "y": "0.434867685"
    },
    {
      "turn": "0",
      "x": "0.628283081",
      "y": "1.571434824"
    },
    {
      "turn": "0",
      "x": "0.628283081",
      "y": "1.270068150"
    },
    {
      "turn": "0",
      "x": "0.758074629",
      "y": "0.998082809"
    },
    {
      "turn": "0",
      "x": "0.895502693",
      "y": "0.341740261"
    },
    {
      "turn": "0",
      "x": "0.910247702",
      "y": "0.737957401"
    },
    {
      "turn": "0",
      "x": "0.914899819",
      "y": "1.664562247"
    },
    {
      "turn": "0",
      "x": "0.928725070",
      "y": "1.246478258"
    },
    {
      "turn": "0",
      "x": "1.080898143",
      "y": "0.986352850"
    },
    {
      "turn": "0",
      "x": "1.129699490",
      "y": "0.531407686"
    },
    {
      "turn": "0",
      "x": "1.138662879",
      "y": "1.462691088"
    },
    {
      "turn": "0",
      "x": "1.218666337",
      "y": "0.237841012"
    },
    {
      "turn": "0",
      "x": "1.251279600",
      "y": "1.772382729"
    },
    {
      "turn": "0",
      "x": "1.291878833",
      "y": "1.202303636"
    },
    {
      "turn": "0",
      "x": "1.300349931",
      "y": "0.779803135"
    },
    {
      "turn": "0",
      "x": "1.423012530",
      "y": "0.462202682"
    },
    {
      "turn": "0",
      "x": "1.433999680",
      "y": "1.522674803"
    },
    {
      "turn": "0",
      "x": "1.512108788",
      "y": "0.994232747"
    },
    {
      "turn": "0",
      "x": "1.600151416",
      "y": "0.706013442"
    },
    {
      "turn": "0",
      "x": "1.611138566",
      "y": "1.278864043"
    },
    {
      "turn": "0",
      "x": "1.813571160",
      "y": "1.000001673"
    }
  ],
  "radius": "0.150683336"
}

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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