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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 12

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 = 12
ObjectiveMaximize circle radius
Best known, unproven0.234100964947Source-precision interval: 0.234100964946–0.234100964948 (values inside match)exhibited site certificate 0.234100964Packomania (compiled by Eckard Specht), m=8, n=12. 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 = 12, 0.234100964
Current leader

0.234100964

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
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Record holder's solver noteNo solver note yet (expand)＋

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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": "0.641616747",
      "y": "0.401836263"
    },
    {
      "turn": "0",
      "x": "1.097142557",
      "y": "0.293626823"
    },
    {
      "turn": "0",
      "x": "0.349572865",
      "y": "0.767791750"
    },
    {
      "turn": "0",
      "x": "1.528969018",
      "y": "0.474565916"
    },
    {
      "turn": "0",
      "x": "1.156357937",
      "y": "0.758069054"
    },
    {
      "turn": "0",
      "x": "0.755828805",
      "y": "1.000534437"
    },
    {
      "turn": "0",
      "x": "0.351139755",
      "y": "1.235991058"
    },
    {
      "turn": "0",
      "x": "1.166074552",
      "y": "1.226170149"
    },
    {
      "turn": "0",
      "x": "1.703232941",
      "y": "1.018527687"
    },
    {
      "turn": "0",
      "x": "0.645756461",
      "y": "1.599878463"
    },
    {
      "turn": "0",
      "x": "1.531883793",
      "y": "1.518397195"
    },
    {
      "turn": "0",
      "x": "1.102167343",
      "y": "1.704291842"
    }
  ],
  "radius": "0.234100964"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.641616747",
      "y": "0.401836263"
    },
    {
      "turn": "0",
      "x": "1.097142557",
      "y": "0.293626823"
    },
    {
      "turn": "0",
      "x": "0.349572865",
      "y": "0.767791750"
    },
    {
      "turn": "0",
      "x": "1.528969018",
      "y": "0.474565916"
    },
    {
      "turn": "0",
      "x": "1.156357937",
      "y": "0.758069054"
    },
    {
      "turn": "0",
      "x": "0.755828805",
      "y": "1.000534437"
    },
    {
      "turn": "0",
      "x": "0.351139755",
      "y": "1.235991058"
    },
    {
      "turn": "0",
      "x": "1.166074552",
      "y": "1.226170149"
    },
    {
      "turn": "0",
      "x": "1.703232941",
      "y": "1.018527687"
    },
    {
      "turn": "0",
      "x": "0.645756461",
      "y": "1.599878463"
    },
    {
      "turn": "0",
      "x": "1.531883793",
      "y": "1.518397195"
    },
    {
      "turn": "0",
      "x": "1.102167343",
      "y": "1.704291842"
    }
  ],
  "radius": "0.234100964"
}

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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