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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 23

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 = 23
ObjectiveMaximize circle radius
Best known, unproven0.172561559662Source-precision interval: 0.172561559661–0.172561559663 (values inside match)exhibited site certificate 0.172561559Packomania (compiled by Eckard Specht), m=8, n=23. 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 = 23, 0.172561559
Current leader

0.172561559

circle radius

Answer sourceEckard Specht
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": "0.599898627",
      "y": "0.352506701"
    },
    {
      "turn": "0",
      "x": "0.918750813",
      "y": "0.220433801"
    },
    {
      "turn": "0",
      "x": "0.352506701",
      "y": "0.599898627"
    },
    {
      "turn": "0",
      "x": "1.256181695",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.686569707",
      "y": "0.686569707"
    },
    {
      "turn": "0",
      "x": "1.005421893",
      "y": "0.554496807"
    },
    {
      "turn": "0",
      "x": "0.220433801",
      "y": "0.918750813"
    },
    {
      "turn": "0",
      "x": "1.575033881",
      "y": "0.424966119"
    },
    {
      "turn": "0",
      "x": "1.330994983",
      "y": "0.669005017"
    },
    {
      "turn": "0",
      "x": "0.554496807",
      "y": "1.005421893"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.256181695"
    },
    {
      "turn": "0",
      "x": "1.011552730",
      "y": "1.023691546"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.743818305"
    },
    {
      "turn": "0",
      "x": "1.445503193",
      "y": "0.994578107"
    },
    {
      "turn": "0",
      "x": "0.669005017",
      "y": "1.330994983"
    },
    {
      "turn": "0",
      "x": "0.424966119",
      "y": "1.575033881"
    },
    {
      "turn": "0",
      "x": "1.779566199",
      "y": "1.081249187"
    },
    {
      "turn": "0",
      "x": "0.994578107",
      "y": "1.445503193"
    },
    {
      "turn": "0",
      "x": "1.313430293",
      "y": "1.313430293"
    },
    {
      "turn": "0",
      "x": "0.743818305",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.647493299",
      "y": "1.400101373"
    },
    {
      "turn": "0",
      "x": "1.081249187",
      "y": "1.779566199"
    },
    {
      "turn": "0",
      "x": "1.400101373",
      "y": "1.647493299"
    }
  ],
  "radius": "0.172561559"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 23,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.599898627",
      "y": "0.352506701"
    },
    {
      "turn": "0",
      "x": "0.918750813",
      "y": "0.220433801"
    },
    {
      "turn": "0",
      "x": "0.352506701",
      "y": "0.599898627"
    },
    {
      "turn": "0",
      "x": "1.256181695",
      "y": "0.292893219"
    },
    {
      "turn": "0",
      "x": "0.686569707",
      "y": "0.686569707"
    },
    {
      "turn": "0",
      "x": "1.005421893",
      "y": "0.554496807"
    },
    {
      "turn": "0",
      "x": "0.220433801",
      "y": "0.918750813"
    },
    {
      "turn": "0",
      "x": "1.575033881",
      "y": "0.424966119"
    },
    {
      "turn": "0",
      "x": "1.330994983",
      "y": "0.669005017"
    },
    {
      "turn": "0",
      "x": "0.554496807",
      "y": "1.005421893"
    },
    {
      "turn": "0",
      "x": "0.292893219",
      "y": "1.256181695"
    },
    {
      "turn": "0",
      "x": "1.011552730",
      "y": "1.023691546"
    },
    {
      "turn": "0",
      "x": "1.707106781",
      "y": "0.743818305"
    },
    {
      "turn": "0",
      "x": "1.445503193",
      "y": "0.994578107"
    },
    {
      "turn": "0",
      "x": "0.669005017",
      "y": "1.330994983"
    },
    {
      "turn": "0",
      "x": "0.424966119",
      "y": "1.575033881"
    },
    {
      "turn": "0",
      "x": "1.779566199",
      "y": "1.081249187"
    },
    {
      "turn": "0",
      "x": "0.994578107",
      "y": "1.445503193"
    },
    {
      "turn": "0",
      "x": "1.313430293",
      "y": "1.313430293"
    },
    {
      "turn": "0",
      "x": "0.743818305",
      "y": "1.707106781"
    },
    {
      "turn": "0",
      "x": "1.647493299",
      "y": "1.400101373"
    },
    {
      "turn": "0",
      "x": "1.081249187",
      "y": "1.779566199"
    },
    {
      "turn": "0",
      "x": "1.400101373",
      "y": "1.647493299"
    }
  ],
  "radius": "0.172561559"
}

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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