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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 22

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 = 22
ObjectiveMaximize circle radius
Best known, unproven0.175500940767Source-precision interval: 0.175500940766–0.175500940768 (values inside match)exhibited site certificate 0.175500940Packomania (compiled by Eckard Specht), m=8, n=22. 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 = 8 · n = 22, 0.17550094
Current leader

0.17550094

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": "0.578883180",
      "y": "0.364393148"
    },
    {
      "turn": "0",
      "x": "0.903166634",
      "y": "0.230070543"
    },
    {
      "turn": "0",
      "x": "0.344873745",
      "y": "0.626007186"
    },
    {
      "turn": "0",
      "x": "1.248500722",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.783320450",
      "y": "0.649713535"
    },
    {
      "turn": "0",
      "x": "0.210551140",
      "y": "0.950290640"
    },
    {
      "turn": "0",
      "x": "1.572784176",
      "y": "0.427215824"
    },
    {
      "turn": "0",
      "x": "1.133414984",
      "y": "0.624491762"
    },
    {
      "turn": "0",
      "x": "0.560308881",
      "y": "0.920763643"
    },
    {
      "turn": "0",
      "x": "0.893422183",
      "y": "1.031388453"
    },
    {
      "turn": "0",
      "x": "1.536251060",
      "y": "0.776311301"
    },
    {
      "turn": "0",
      "x": "0.308932518",
      "y": "1.287223015"
    },
    {
      "turn": "0",
      "x": "1.237055256",
      "y": "0.959843838"
    },
    {
      "turn": "0",
      "x": "0.659879701",
      "y": "1.293419424"
    },
    {
      "turn": "0",
      "x": "1.808931653",
      "y": "0.997326265"
    },
    {
      "turn": "0",
      "x": "0.479039862",
      "y": "1.594250396"
    },
    {
      "turn": "0",
      "x": "0.984163155",
      "y": "1.427742029"
    },
    {
      "turn": "0",
      "x": "1.312144501",
      "y": "1.302719801"
    },
    {
      "turn": "0",
      "x": "1.676501947",
      "y": "1.322387327"
    },
    {
      "turn": "0",
      "x": "0.803323316",
      "y": "1.728573001"
    },
    {
      "turn": "0",
      "x": "1.153877213",
      "y": "1.746301122"
    },
    {
      "turn": "0",
      "x": "1.478160667",
      "y": "1.611978517"
    }
  ],
  "radius": "0.175500940"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 22,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.578883180",
      "y": "0.364393148"
    },
    {
      "turn": "0",
      "x": "0.903166634",
      "y": "0.230070543"
    },
    {
      "turn": "0",
      "x": "0.344873745",
      "y": "0.626007186"
    },
    {
      "turn": "0",
      "x": "1.248500722",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.783320450",
      "y": "0.649713535"
    },
    {
      "turn": "0",
      "x": "0.210551140",
      "y": "0.950290640"
    },
    {
      "turn": "0",
      "x": "1.572784176",
      "y": "0.427215824"
    },
    {
      "turn": "0",
      "x": "1.133414984",
      "y": "0.624491762"
    },
    {
      "turn": "0",
      "x": "0.560308881",
      "y": "0.920763643"
    },
    {
      "turn": "0",
      "x": "0.893422183",
      "y": "1.031388453"
    },
    {
      "turn": "0",
      "x": "1.536251060",
      "y": "0.776311301"
    },
    {
      "turn": "0",
      "x": "0.308932518",
      "y": "1.287223015"
    },
    {
      "turn": "0",
      "x": "1.237055256",
      "y": "0.959843838"
    },
    {
      "turn": "0",
      "x": "0.659879701",
      "y": "1.293419424"
    },
    {
      "turn": "0",
      "x": "1.808931653",
      "y": "0.997326265"
    },
    {
      "turn": "0",
      "x": "0.479039862",
      "y": "1.594250396"
    },
    {
      "turn": "0",
      "x": "0.984163155",
      "y": "1.427742029"
    },
    {
      "turn": "0",
      "x": "1.312144501",
      "y": "1.302719801"
    },
    {
      "turn": "0",
      "x": "1.676501947",
      "y": "1.322387327"
    },
    {
      "turn": "0",
      "x": "0.803323316",
      "y": "1.728573001"
    },
    {
      "turn": "0",
      "x": "1.153877213",
      "y": "1.746301122"
    },
    {
      "turn": "0",
      "x": "1.478160667",
      "y": "1.611978517"
    }
  ],
  "radius": "0.175500940"
}

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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