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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 33

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 = 33
ObjectiveMaximize circle radius
Best known, unproven0.145306599845Source-precision interval: 0.145306599844–0.145306599846 (values inside match)exhibited site certificate 0.145306599Packomania (compiled by Eckard Specht), m=8, n=33. 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 = 33, 0.145306599
Current leader

0.145306599

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
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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": "0.455085607",
      "y": "0.382989662"
    },
    {
      "turn": "0",
      "x": "0.949020468",
      "y": "0.178395144"
    },
    {
      "turn": "0",
      "x": "0.745682962",
      "y": "0.386024357"
    },
    {
      "turn": "0",
      "x": "0.308831700",
      "y": "0.634118765"
    },
    {
      "turn": "0",
      "x": "1.230012690",
      "y": "0.252553106"
    },
    {
      "turn": "0",
      "x": "0.599429054",
      "y": "0.637153460"
    },
    {
      "turn": "0",
      "x": "1.026675183",
      "y": "0.460182319"
    },
    {
      "turn": "0",
      "x": "0.880421276",
      "y": "0.711311422"
    },
    {
      "turn": "0",
      "x": "0.425253976",
      "y": "0.900392845"
    },
    {
      "turn": "0",
      "x": "1.335973916",
      "y": "0.523160294"
    },
    {
      "turn": "0",
      "x": "0.181680362",
      "y": "1.058910750"
    },
    {
      "turn": "0",
      "x": "1.620295556",
      "y": "0.463016826"
    },
    {
      "turn": "0",
      "x": "0.711925854",
      "y": "0.948092597"
    },
    {
      "turn": "0",
      "x": "1.166995105",
      "y": "0.759596741"
    },
    {
      "turn": "0",
      "x": "0.998499684",
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    },
    {
      "turn": "0",
      "x": "0.547875670",
      "y": "1.187975003"
    },
    {
      "turn": "0",
      "x": "1.452618579",
      "y": "0.813218220"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.327402337"
    },
    {
      "turn": "0",
      "x": "1.731508413",
      "y": "0.731508413"
    },
    {
      "turn": "0",
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      "y": "1.236260322"
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    {
      "turn": "0",
      "x": "1.284123157",
      "y": "1.049999394"
    },
    {
      "turn": "0",
      "x": "0.668211090",
      "y": "1.474631510"
    },
    {
      "turn": "0",
      "x": "1.570225347",
      "y": "1.101004971"
    },
    {
      "turn": "0",
      "x": "1.120072973",
      "y": "1.289881801"
    },
    {
      "turn": "0",
      "x": "0.404106076",
      "y": "1.595893924"
    },
    {
      "turn": "0",
      "x": "1.842721270",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.974461273",
      "y": "1.541383818"
    },
    {
      "turn": "0",
      "x": "1.400874776",
      "y": "1.364757561"
    },
    {
      "turn": "0",
      "x": "0.769216061",
      "y": "1.747127432"
    },
    {
      "turn": "0",
      "x": "1.691487762",
      "y": "1.365109985"
    },
    {
      "turn": "0",
      "x": "1.255263075",
      "y": "1.616259578"
    },
    {
      "turn": "0",
      "x": "1.050017863",
      "y": "1.822003192"
    },
    {
      "turn": "0",
      "x": "1.545876061",
      "y": "1.616612002"
    }
  ],
  "radius": "0.145306599"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 33,
  "sides": 8
}

The current leader's answer

{
  "placements": [
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      "turn": "0",
      "x": "0.455085607",
      "y": "0.382989662"
    },
    {
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    },
    {
      "turn": "0",
      "x": "0.745682962",
      "y": "0.386024357"
    },
    {
      "turn": "0",
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    },
    {
      "turn": "0",
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    },
    {
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    },
    {
      "turn": "0",
      "x": "1.026675183",
      "y": "0.460182319"
    },
    {
      "turn": "0",
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      "y": "0.711311422"
    },
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      "y": "0.900392845"
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    {
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    {
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    {
      "turn": "0",
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  ],
  "radius": "0.145306599"
}

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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