P82 · Packing and covering · Classic · Hard

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

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 = 8

ObjectiveMaximize circle radius

Best known, unproven0.285852433677Source-precision interval: 0.285852433676–0.285852433678 (values inside match)exhibited site certificate 0.285852433Packomania (compiled by Eckard Specht), m=8, n=8. 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.

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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

Current leader

0.285852433

circle radius

Answer sourceEckard Specht

Solution methodPublished reference construction

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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": "0.634756985",
      "y": "0.460693055"
    },
    {
      "turn": "0",
      "x": "1.202472367",
      "y": "0.393271245"
    },
    {
      "turn": "0",
      "x": "0.331933533",
      "y": "0.945609968"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.606728755",
      "y": "0.797527633"
    },
    {
      "turn": "0",
      "x": "0.511675200",
      "y": "1.488324800"
    },
    {
      "turn": "0",
      "x": "1.539306945",
      "y": "1.365243015"
    },
    {
      "turn": "0",
      "x": "1.054390032",
      "y": "1.668066467"
    }
  ],
  "radius": "0.285852433"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

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

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.634756985",
      "y": "0.460693055"
    },
    {
      "turn": "0",
      "x": "1.202472367",
      "y": "0.393271245"
    },
    {
      "turn": "0",
      "x": "0.331933533",
      "y": "0.945609968"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "1.606728755",
      "y": "0.797527633"
    },
    {
      "turn": "0",
      "x": "0.511675200",
      "y": "1.488324800"
    },
    {
      "turn": "0",
      "x": "1.539306945",
      "y": "1.365243015"
    },
    {
      "turn": "0",
      "x": "1.054390032",
      "y": "1.668066467"
    }
  ],
  "radius": "0.285852433"
}

DISCUSSION

Discussion

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