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P82 · Packing and covering · Classic · Hard

Equal circles in regular polygons · m = 8 · n = 26

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 = 26
ObjectiveMaximize circle radius
Best known, unproven0.163606136259Source-precision interval: 0.163606136258–0.16360613626 (values inside match)exhibited site certificate 0.163606135Packomania (compiled by Eckard Specht), m=8, n=26. 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 = 26, 0.163606135
Current leader

0.163606135

circle radius

Answer sourcePaolo Amore
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
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": "0.997247696",
      "y": "0.178226047"
    },
    {
      "turn": "0",
      "x": "0.527592545",
      "y": "0.372927962"
    },
    {
      "turn": "0",
      "x": "0.327033380",
      "y": "0.637995016"
    },
    {
      "turn": "0",
      "x": "0.841497852",
      "y": "0.465993067"
    },
    {
      "turn": "0",
      "x": "1.224893318",
      "y": "0.413269325"
    },
    {
      "turn": "0",
      "x": "0.642303641",
      "y": "0.725588016"
    },
    {
      "turn": "0",
      "x": "0.201814664",
      "y": "0.940299738"
    },
    {
      "turn": "0",
      "x": "1.552018946",
      "y": "0.405739740"
    },
    {
      "turn": "0",
      "x": "0.966716567",
      "y": "0.768297788"
    },
    {
      "turn": "0",
      "x": "0.517084926",
      "y": "1.027892737"
    },
    {
      "turn": "0",
      "x": "1.292074421",
      "y": "0.733510750"
    },
    {
      "turn": "0",
      "x": "0.283592032",
      "y": "1.257128293"
    },
    {
      "turn": "0",
      "x": "1.668477173",
      "y": "0.711526189"
    },
    {
      "turn": "0",
      "x": "0.841497852",
      "y": "1.070602509"
    },
    {
      "turn": "0",
      "x": "1.166855706",
      "y": "1.035815472"
    },
    {
      "turn": "0",
      "x": "0.642303641",
      "y": "1.330197459"
    },
    {
      "turn": "0",
      "x": "1.495869935",
      "y": "0.989509305"
    },
    {
      "turn": "0",
      "x": "0.408810748",
      "y": "1.559433014"
    },
    {
      "turn": "0",
      "x": "1.822913994",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.970127675",
      "y": "1.373205623"
    },
    {
      "turn": "0",
      "x": "1.370651220",
      "y": "1.291814026"
    },
    {
      "turn": "0",
      "x": "0.724081009",
      "y": "1.647026014"
    },
    {
      "turn": "0",
      "x": "1.697695279",
      "y": "1.302304721"
    },
    {
      "turn": "0",
      "x": "1.223837030",
      "y": "1.584240881"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.822913994"
    },
    {
      "turn": "0",
      "x": "1.550881089",
      "y": "1.594731576"
    }
  ],
  "radius": "0.163606135"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26,
  "sides": 8
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0",
      "x": "0.997247696",
      "y": "0.178226047"
    },
    {
      "turn": "0",
      "x": "0.527592545",
      "y": "0.372927962"
    },
    {
      "turn": "0",
      "x": "0.327033380",
      "y": "0.637995016"
    },
    {
      "turn": "0",
      "x": "0.841497852",
      "y": "0.465993067"
    },
    {
      "turn": "0",
      "x": "1.224893318",
      "y": "0.413269325"
    },
    {
      "turn": "0",
      "x": "0.642303641",
      "y": "0.725588016"
    },
    {
      "turn": "0",
      "x": "0.201814664",
      "y": "0.940299738"
    },
    {
      "turn": "0",
      "x": "1.552018946",
      "y": "0.405739740"
    },
    {
      "turn": "0",
      "x": "0.966716567",
      "y": "0.768297788"
    },
    {
      "turn": "0",
      "x": "0.517084926",
      "y": "1.027892737"
    },
    {
      "turn": "0",
      "x": "1.292074421",
      "y": "0.733510750"
    },
    {
      "turn": "0",
      "x": "0.283592032",
      "y": "1.257128293"
    },
    {
      "turn": "0",
      "x": "1.668477173",
      "y": "0.711526189"
    },
    {
      "turn": "0",
      "x": "0.841497852",
      "y": "1.070602509"
    },
    {
      "turn": "0",
      "x": "1.166855706",
      "y": "1.035815472"
    },
    {
      "turn": "0",
      "x": "0.642303641",
      "y": "1.330197459"
    },
    {
      "turn": "0",
      "x": "1.495869935",
      "y": "0.989509305"
    },
    {
      "turn": "0",
      "x": "0.408810748",
      "y": "1.559433014"
    },
    {
      "turn": "0",
      "x": "1.822913994",
      "y": "1.000000000"
    },
    {
      "turn": "0",
      "x": "0.970127675",
      "y": "1.373205623"
    },
    {
      "turn": "0",
      "x": "1.370651220",
      "y": "1.291814026"
    },
    {
      "turn": "0",
      "x": "0.724081009",
      "y": "1.647026014"
    },
    {
      "turn": "0",
      "x": "1.697695279",
      "y": "1.302304721"
    },
    {
      "turn": "0",
      "x": "1.223837030",
      "y": "1.584240881"
    },
    {
      "turn": "0",
      "x": "1.000000000",
      "y": "1.822913994"
    },
    {
      "turn": "0",
      "x": "1.550881089",
      "y": "1.594731576"
    }
  ],
  "radius": "0.163606135"
}

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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