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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 17

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 = 17
ObjectiveMaximize circle radius
Best known, unproven0.198194125496Source-precision interval: 0.198194125495–0.198194125497 (values inside match)exhibited site certificate 0.198194124Packomania (compiled by Eckard Specht), m=8, n=17. 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 = 17, 0.198194124
Current leader

0.198194124

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": "0.865956010",
      "y": "0.270046614"
    },
    {
      "turn": "0",
      "x": "0.490730215",
      "y": "0.427225774"
    },
    {
      "turn": "0",
      "x": "1.258953104",
      "y": "0.321785663"
    },
    {
      "turn": "0",
      "x": "0.306612856",
      "y": "0.781765124"
    },
    {
      "turn": "0",
      "x": "0.848109436",
      "y": "0.666032909"
    },
    {
      "turn": "0",
      "x": "1.592964008",
      "y": "0.535234396"
    },
    {
      "turn": "0",
      "x": "1.241106531",
      "y": "0.717771958"
    },
    {
      "turn": "0",
      "x": "0.644462638",
      "y": "1.006108822"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.189200573"
    },
    {
      "turn": "0",
      "x": "1.037459732",
      "y": "1.057847871"
    },
    {
      "turn": "0",
      "x": "1.744655225",
      "y": "0.901449388"
    },
    {
      "turn": "0",
      "x": "0.796153854",
      "y": "1.372323814"
    },
    {
      "turn": "0",
      "x": "1.425860747",
      "y": "1.137020453"
    },
    {
      "turn": "0",
      "x": "0.444584435",
      "y": "1.555415565"
    },
    {
      "turn": "0",
      "x": "1.183131468",
      "y": "1.458184003"
    },
    {
      "turn": "0",
      "x": "0.915285556",
      "y": "1.750386353"
    },
    {
      "turn": "0",
      "x": "1.576921344",
      "y": "1.503496020"
    }
  ],
  "radius": "0.198194124"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.865956010",
      "y": "0.270046614"
    },
    {
      "turn": "0",
      "x": "0.490730215",
      "y": "0.427225774"
    },
    {
      "turn": "0",
      "x": "1.258953104",
      "y": "0.321785663"
    },
    {
      "turn": "0",
      "x": "0.306612856",
      "y": "0.781765124"
    },
    {
      "turn": "0",
      "x": "0.848109436",
      "y": "0.666032909"
    },
    {
      "turn": "0",
      "x": "1.592964008",
      "y": "0.535234396"
    },
    {
      "turn": "0",
      "x": "1.241106531",
      "y": "0.717771958"
    },
    {
      "turn": "0",
      "x": "0.644462638",
      "y": "1.006108822"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.189200573"
    },
    {
      "turn": "0",
      "x": "1.037459732",
      "y": "1.057847871"
    },
    {
      "turn": "0",
      "x": "1.744655225",
      "y": "0.901449388"
    },
    {
      "turn": "0",
      "x": "0.796153854",
      "y": "1.372323814"
    },
    {
      "turn": "0",
      "x": "1.425860747",
      "y": "1.137020453"
    },
    {
      "turn": "0",
      "x": "0.444584435",
      "y": "1.555415565"
    },
    {
      "turn": "0",
      "x": "1.183131468",
      "y": "1.458184003"
    },
    {
      "turn": "0",
      "x": "0.915285556",
      "y": "1.750386353"
    },
    {
      "turn": "0",
      "x": "1.576921344",
      "y": "1.503496020"
    }
  ],
  "radius": "0.198194124"
}

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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