GC-07.06 · Symbolic Model & Source

Companion symbolic Python construction for GC-07.06 · Exact φ-Limit Resolution of the Hooper–Curry Fibonacci Dissection (GC-07.06).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-07.06")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_curry_hooper_dissection() -> ConstructionResult:
            """Exact phi-limit resolution of the Hooper (1794) / Curry Fibonacci dissection."""
            m = new_catalog_model("curry-hooper-dissection")
            phi = sp.Rational(1, 2) + sp.sqrt(5) / 2
            inv_phi = phi - 1
            phi2 = phi + 1
            phi3 = 2 * phi + 1

            # Golden rectangle ABCD of width phi^3 and height phi (area = phi^4 = (phi^2)^2, zero gap!)
            A = m.set_point(0, 0, classes=["given"], ID="A")
            B = m.set_point(phi3, 0, classes=["given"], ID="B")
            C = m.set_point(phi3, phi, ID="C")
            D = m.set_point(0, phi, ID="D")
            m.construct_polygon([A, B, C, D], ID="ABCD")

            # Four-piece exact dissection cuts: E=(phi^2, 0), F=(phi, phi), P=(phi^2, 1), Q=(phi, 1/phi)
            E = m.set_point(phi2, 0, ID="E")
            F = m.set_point(phi, phi, ID="F")
            P = m.set_point(phi2, 1, classes=["golden"], ID="P")
            Q = m.set_point(phi, inv_phi, classes=["golden"], ID="Q")

            m.construct_line(A, C, classes=["golden"], ID="AC")
            m.construct_segment(E, P, classes=["auxiliary"], ID="EP")
            m.construct_segment(F, Q, classes=["auxiliary"], ID="FQ")
            m.construct_polygon([A, E, P], classes=["auxiliary"], ID="AEP")
            m.construct_polygon([C, F, Q], classes=["auxiliary"], ID="CFQ")

            return make_result(m, ("A", "Q"), ("Q", "P"), ("A", "Q", "P"))
        

Exact Symbolic Point Coordinates

All points in curry-hooper-dissection are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

Point	Exact \(x\)	Exact \(y\)
A	\(0\)	\(0\)
B	\(2 + \sqrt{5}\)	\(0\)
C	\(2 + \sqrt{5}\)	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
D	\(0\)	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
E	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)	\(0\)
F	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
P	\(\frac{\sqrt{5}}{2} + \frac{3}{2}\)	\(1\)
Q	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(- \frac{1}{2} + \frac{\sqrt{5}}{2}\)

Certified Golden Ratio: AQ / QP = GoldenRatio.