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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 30

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 = 30
ObjectiveMaximize circle radius
Best known, unproven0.152319579409Source-precision interval: 0.152319579408–0.15231957941 (values inside match)exhibited site certificate 0.152319578Packomania (compiled by Eckard Specht), m=8, n=30. Eckard Specht, program coc , 2020. 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 = 30, 0.152319578
Current leader

0.152319578

circle radius

Answer sourceEckard Specht
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": "0.738763877",
      "y": "0.273077070"
    },
    {
      "turn": "0",
      "x": "1.027626655",
      "y": "0.176312860"
    },
    {
      "turn": "0",
      "x": "0.495645041",
      "y": "0.478168596"
    },
    {
      "turn": "0",
      "x": "0.281449884",
      "y": "0.718550116"
    },
    {
      "turn": "0",
      "x": "1.309076539",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.805687506",
      "y": "0.570274386"
    },
    {
      "turn": "0",
      "x": "1.106170498",
      "y": "0.520124721"
    },
    {
      "turn": "0",
      "x": "0.580689196",
      "y": "0.775653979"
    },
    {
      "turn": "0",
      "x": "0.164869525",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.590526422",
      "y": "0.409473578"
    },
    {
      "turn": "0",
      "x": "1.387620382",
      "y": "0.636705080"
    },
    {
      "turn": "0",
      "x": "0.934400975",
      "y": "0.846386429"
    },
    {
      "turn": "0",
      "x": "0.464108837",
      "y": "1.057103863"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.690923461"
    },
    {
      "turn": "0",
      "x": "1.234294772",
      "y": "0.899946936"
    },
    {
      "turn": "0",
      "x": "0.765705228",
      "y": "1.100053064"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.309076539"
    },
    {
      "turn": "0",
      "x": "1.535891163",
      "y": "0.942896137"
    },
    {
      "turn": "0",
      "x": "1.065599025",
      "y": "1.153613571"
    },
    {
      "turn": "0",
      "x": "0.612379618",
      "y": "1.363294920"
    },
    {
      "turn": "0",
      "x": "0.409473578",
      "y": "1.590526422"
    },
    {
      "turn": "0",
      "x": "1.835130475",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.419310804",
      "y": "1.224346021"
    },
    {
      "turn": "0",
      "x": "0.893829502",
      "y": "1.479875279"
    },
    {
      "turn": "0",
      "x": "1.194312494",
      "y": "1.429725614"
    },
    {
      "turn": "0",
      "x": "0.690923461",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.718550116",
      "y": "1.281449884"
    },
    {
      "turn": "0",
      "x": "1.511866093",
      "y": "1.523568988"
    },
    {
      "turn": "0",
      "x": "0.972373345",
      "y": "1.823687140"
    },
    {
      "turn": "0",
      "x": "1.261236123",
      "y": "1.726922930"
    }
  ],
  "radius": "0.152319578"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 30,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.738763877",
      "y": "0.273077070"
    },
    {
      "turn": "0",
      "x": "1.027626655",
      "y": "0.176312860"
    },
    {
      "turn": "0",
      "x": "0.495645041",
      "y": "0.478168596"
    },
    {
      "turn": "0",
      "x": "0.281449884",
      "y": "0.718550116"
    },
    {
      "turn": "0",
      "x": "1.309076539",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.805687506",
      "y": "0.570274386"
    },
    {
      "turn": "0",
      "x": "1.106170498",
      "y": "0.520124721"
    },
    {
      "turn": "0",
      "x": "0.580689196",
      "y": "0.775653979"
    },
    {
      "turn": "0",
      "x": "0.164869525",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.590526422",
      "y": "0.409473578"
    },
    {
      "turn": "0",
      "x": "1.387620382",
      "y": "0.636705080"
    },
    {
      "turn": "0",
      "x": "0.934400975",
      "y": "0.846386429"
    },
    {
      "turn": "0",
      "x": "0.464108837",
      "y": "1.057103863"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.690923461"
    },
    {
      "turn": "0",
      "x": "1.234294772",
      "y": "0.899946936"
    },
    {
      "turn": "0",
      "x": "0.765705228",
      "y": "1.100053064"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.309076539"
    },
    {
      "turn": "0",
      "x": "1.535891163",
      "y": "0.942896137"
    },
    {
      "turn": "0",
      "x": "1.065599025",
      "y": "1.153613571"
    },
    {
      "turn": "0",
      "x": "0.612379618",
      "y": "1.363294920"
    },
    {
      "turn": "0",
      "x": "0.409473578",
      "y": "1.590526422"
    },
    {
      "turn": "0",
      "x": "1.835130475",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.419310804",
      "y": "1.224346021"
    },
    {
      "turn": "0",
      "x": "0.893829502",
      "y": "1.479875279"
    },
    {
      "turn": "0",
      "x": "1.194312494",
      "y": "1.429725614"
    },
    {
      "turn": "0",
      "x": "0.690923461",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.718550116",
      "y": "1.281449884"
    },
    {
      "turn": "0",
      "x": "1.511866093",
      "y": "1.523568988"
    },
    {
      "turn": "0",
      "x": "0.972373345",
      "y": "1.823687140"
    },
    {
      "turn": "0",
      "x": "1.261236123",
      "y": "1.726922930"
    }
  ],
  "radius": "0.152319578"
}

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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