{
  "revision": 3,
  "rho": "1.32471795724474602596090885447809734073440405690173336453401505030282785125",
  "definition": "rho^3=rho+1, rho>1",
  "scale_anchor": "main-square side D=1000",
  "construction_rule": "Erect the main square upward from the horizontal side of T0; move inward through the same spiral and select the first candidate square erected upward from a horizontal triangle side that lies completely above the main square.",
  "selected_index": -3,
  "line_weight_rule": "main frame black area equals detached-square black area",
  "formulas": {
    "S": "D/rho^3",
    "G": "D*((sqrt(3)/2)*rho-1)",
    "O": "D*(rho^-1-rho^-2)/2",
    "C": "sqrt(D^2-S^2)/2",
    "W": "(D-sqrt(D^2-S^2))/3",
    "square_left": "D+O-S",
    "square_top": "-G-S"
  },
  "values": {
    "D": "1000",
    "S": "430.159709001946734088600041880431351160202560871059779455268920343252803878",
    "G": "147.239403823377908490116814824628778614609704933212814225720321254740034730",
    "O": "92.518687624319747069054469119480021527404589321926461722276445494666729506",
    "C": "451.376401895181186780878480478909600850566194212259069296625543932337529992",
    "W": "32.415732069879208812747679680726932766289203858493953802249637378441646672"
  },
  "triangle_formula": "q=rho*exp(i*pi/3), v_n=D*q^n/(q-1), v0=D/(q-1); z_screen=(D+iD)-(z-v0), for triangle vertices z=(v_n,v_n+D*q^n,v_n+D*exp(i*pi/3)*q^n)",
  "triangles": {
    "0": [
      [
        1000.0,
        1000.0
      ],
      [
        0.0,
        1000.0
      ],
      [
        499.9999999999999,
        133.9745962155614
      ]
    ],
    "-1": [
      [
        1377.4388331233463,
        346.2567642808532
      ],
      [
        1000.0,
        999.9999999999999
      ],
      [
        622.5611668766538,
        346.2567642808532
      ]
    ],
    "-2": [
      [
        1092.51868762432,
        -147.23940382337787
      ],
      [
        1377.4388331233463,
        346.2567642808531
      ],
      [
        807.5985421252933,
        346.2567642808531
      ]
    ],
    "-3": [
      [
        662.3589786223731,
        -147.23940382337787
      ],
      [
        1092.5186876243197,
        -147.23940382337787
      ],
      [
        877.4388331233465,
        225.28983185682955
      ]
    ]
  },
  "candidates": [
    {
      "index": -1,
      "left": 622.5611668766538,
      "right": 1377.4388331233463,
      "top": -408.62090196583927,
      "bottom": 346.2567642808532,
      "side": 754.8776662466925,
      "intersection_with_outer_main_square": 130690.74907123076,
      "fully_above": false
    },
    {
      "index": -2,
      "left": 807.5985421252933,
      "right": 1377.4388331233463,
      "top": -223.5835267171999,
      "bottom": 346.2567642808531,
      "side": 569.840290998053,
      "intersection_with_outer_main_square": 66620.3062466148,
      "fully_above": false
    },
    {
      "index": -3,
      "left": 662.3589786223731,
      "right": 1092.5186876243197,
      "top": -577.3991128253244,
      "bottom": -147.23940382337787,
      "side": 430.15970900194657,
      "intersection_with_outer_main_square": 0.0,
      "fully_above": true
    }
  ],
  "clear_space": "G on every side of complete ink bounds",
  "design_choices": [
    "retain four-cell frame and one floating square",
    "upward square erection on horizontal triangle edges",
    "select first fully detached inward candidate",
    "global orientation places detached square at upper right",
    "equal black areas determine W",
    "use pure black, sharp corners and gap G as export clear space"
  ]
}