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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 24

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 = 24
ObjectiveMaximize circle radius
Best known, unproven0.168652864098Source-precision interval: 0.168652864097–0.168652864099 (values inside match)exhibited site certificate 0.168652863Packomania (compiled by Eckard Specht), m=8, n=24. 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 = 24, 0.168652863
Current leader

0.168652863

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": "1.010852103",
      "y": "0.187043633"
    },
    {
      "turn": "0",
      "x": "0.415677701",
      "y": "0.437176430"
    },
    {
      "turn": "0",
      "x": "0.751177710",
      "y": "0.402321045"
    },
    {
      "turn": "0",
      "x": "0.280523993",
      "y": "0.763466344"
    },
    {
      "turn": "0",
      "x": "1.342601305",
      "y": "0.324458652"
    },
    {
      "turn": "0",
      "x": "1.067450614",
      "y": "0.519566959"
    },
    {
      "turn": "0",
      "x": "0.613613354",
      "y": "0.710300269"
    },
    {
      "turn": "0",
      "x": "0.222184047",
      "y": "1.095688566"
    },
    {
      "turn": "0",
      "x": "1.618191957",
      "y": "0.518945017"
    },
    {
      "turn": "0",
      "x": "0.929886258",
      "y": "0.827546183"
    },
    {
      "turn": "0",
      "x": "1.343041267",
      "y": "0.714053325"
    },
    {
      "turn": "0",
      "x": "0.555273408",
      "y": "1.042522491"
    },
    {
      "turn": "0",
      "x": "1.747956717",
      "y": "0.832908193"
    },
    {
      "turn": "0",
      "x": "1.073203376",
      "y": "1.132890837"
    },
    {
      "turn": "0",
      "x": "0.361365870",
      "y": "1.428962044"
    },
    {
      "turn": "0",
      "x": "0.717546286",
      "y": "1.338229555"
    },
    {
      "turn": "0",
      "x": "1.469887316",
      "y": "1.026599696"
    },
    {
      "turn": "0",
      "x": "1.029176144",
      "y": "1.467310868"
    },
    {
      "turn": "0",
      "x": "1.340806003",
      "y": "1.338229555"
    },
    {
      "turn": "0",
      "x": "0.628486666",
      "y": "1.663565594"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "1.266395609"
    },
    {
      "turn": "0",
      "x": "0.940116524",
      "y": "1.792646908"
    },
    {
      "turn": "0",
      "x": "1.266395609",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.578025467",
      "y": "1.578025467"
    }
  ],
  "radius": "0.168652863"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "1.010852103",
      "y": "0.187043633"
    },
    {
      "turn": "0",
      "x": "0.415677701",
      "y": "0.437176430"
    },
    {
      "turn": "0",
      "x": "0.751177710",
      "y": "0.402321045"
    },
    {
      "turn": "0",
      "x": "0.280523993",
      "y": "0.763466344"
    },
    {
      "turn": "0",
      "x": "1.342601305",
      "y": "0.324458652"
    },
    {
      "turn": "0",
      "x": "1.067450614",
      "y": "0.519566959"
    },
    {
      "turn": "0",
      "x": "0.613613354",
      "y": "0.710300269"
    },
    {
      "turn": "0",
      "x": "0.222184047",
      "y": "1.095688566"
    },
    {
      "turn": "0",
      "x": "1.618191957",
      "y": "0.518945017"
    },
    {
      "turn": "0",
      "x": "0.929886258",
      "y": "0.827546183"
    },
    {
      "turn": "0",
      "x": "1.343041267",
      "y": "0.714053325"
    },
    {
      "turn": "0",
      "x": "0.555273408",
      "y": "1.042522491"
    },
    {
      "turn": "0",
      "x": "1.747956717",
      "y": "0.832908193"
    },
    {
      "turn": "0",
      "x": "1.073203376",
      "y": "1.132890837"
    },
    {
      "turn": "0",
      "x": "0.361365870",
      "y": "1.428962044"
    },
    {
      "turn": "0",
      "x": "0.717546286",
      "y": "1.338229555"
    },
    {
      "turn": "0",
      "x": "1.469887316",
      "y": "1.026599696"
    },
    {
      "turn": "0",
      "x": "1.029176144",
      "y": "1.467310868"
    },
    {
      "turn": "0",
      "x": "1.340806003",
      "y": "1.338229555"
    },
    {
      "turn": "0",
      "x": "0.628486666",
      "y": "1.663565594"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "1.266395609"
    },
    {
      "turn": "0",
      "x": "0.940116524",
      "y": "1.792646908"
    },
    {
      "turn": "0",
      "x": "1.266395609",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.578025467",
      "y": "1.578025467"
    }
  ],
  "radius": "0.168652863"
}

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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