GC-06.05 · Symbolic Model & Source

Companion symbolic Python construction for GC-06.05 · Jerzy Kocik’s Six-Circle Apollonian Golden Window (2010) (GC-06.05).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-06.05")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_kocik_golden_window() -> ConstructionResult:
            """Jerzy Kocik's six-circle Golden Window Apollonian configuration (2010)."""
            m = new_catalog_model("kocik-golden-window")
            s5 = sp.sqrt(5)
            phi = sp.Rational(1, 2) + s5 / 2
            phi2 = phi + 1
            phi3 = 2 * phi + 1

            O = m.set_point(0, 0, classes=["given"], ID="O")
            P1 = m.set_point(-1, 0, ID="P1")
            P2 = m.set_point(1, 0, classes=["given"], ID="P2")
            C1 = m.set_point(-sp.Rational(1, 2), 0, guide=True, ID="C1")
            C2 = m.set_point(sp.Rational(1, 2), 0, guide=True, ID="C2")

            # Central unit-diameter circles
            m.construct_circle(C1, O, classes=["compass"], ID="cC1")
            m.construct_circle(C2, O, classes=["compass"], ID="cC2")

            # Flanking golden circles of radius phi centered at +-phi^2
            C3 = m.set_point(-phi2, 0, ID="C3")
            C4 = m.set_point(phi2, 0, classes=["golden"], ID="C4")
            A3 = m.set_point(-phi3, 0, ID="A3")
            A4 = m.set_point(phi3, 0, classes=["golden"], ID="A4")
            m.construct_circle(C3, P1, classes=["golden"], ID="cC3")
            m.construct_circle(C4, P2, classes=["golden"], ID="cC4")

            # Outer window circle of radius phi^3 = 2*phi + 1
            m.construct_circle(O, A4, classes=["primary"], ID="cO")
            m.construct_line(A3, A4, classes=["golden"], ID="A3A4")

            # Upper & lower circles tangent to all five circles: center (0, +-2*phi^2/sqrt(5)), top (0, +-phi^3)
            hu = (2 * phi2) / s5
            U1 = m.set_point(0, hu, ID="U1")
            U2 = m.set_point(0, -hu, ID="U2")
            T_top = m.set_point(0, phi3, ID="T1")
            T_bot = m.set_point(0, -phi3, ID="T2")
            m.construct_circle(U1, T_top, classes=["auxiliary"], ID="cU1")
            m.construct_circle(U2, T_bot, classes=["auxiliary"], ID="cU2")

            return make_result(m, ("O", "C4"), ("C4", "A4"), ("O", "C4", "A4"))
        

Exact Symbolic Point Coordinates

All points in kocik-golden-window are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
P1	\(-1\)	\(0\)
P2	\(1\)	\(0\)
C3	\(- \frac{3}{2} - \frac{\sqrt{5}}{2}\)	\(0\)
C4	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)	\(0\)
A3	\(- \sqrt{5} - 2\)	\(0\)
A4	\(2 + \sqrt{5}\)	\(0\)
U1	\(0\)	\(1 + \frac{3 \sqrt{5}}{5}\)
U2	\(0\)	\(- \frac{3 \sqrt{5}}{5} - 1\)
T1	\(0\)	\(2 + \sqrt{5}\)
T2	\(0\)	\(- \sqrt{5} - 2\)

Certified Golden Ratio: OC4 / C4A4 = GoldenRatio.