GC-07.02 · Symbolic Model & Source

Companion symbolic Python construction for GC-07.02 · John Molokach’s Golden Rhombus from Shifted 1×3 Rectangles (GC-07.02).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-07.02")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_molokach_golden_rhombus() -> ConstructionResult:
            """John Molokach's golden rhombus from two shifted 1x3 rectangles and a unit circle."""
            m = new_catalog_model("molokach-golden-rhombus")
            phi = sp.Rational(1, 2) + sp.sqrt(5) / 2
            inv_phi = phi - 1

            O = m.set_point(0, 0, classes=["given"], ID="O")
            T1 = m.set_point(0, 1, classes=["golden"], ID="T1")
            T2 = m.set_point(0, -1, ID="T2")
            _ = T2

            # Two shifted 3x1 rectangles [-2, 1] x [0, 1] and [-1, 2] x [-1, 0]
            R1 = m.set_point(-2, 1, ID="R1")
            K = m.set_point(1, 1, classes=["given"], ID="K")
            R2 = m.set_point(1, 0, guide=True, ID="R2")
            R3 = m.set_point(-2, 0, guide=True, ID="R3")
            m.construct_polygon([R1, K, R2, R3], classes=["auxiliary"], ID="RECT1")

            R4 = m.set_point(-1, 0, guide=True, ID="R4")
            R5 = m.set_point(2, 0, guide=True, ID="R5")
            R6 = m.set_point(2, -1, ID="R6")
            R7 = m.set_point(-1, -1, ID="R7")
            m.construct_polygon([R4, R5, R6, R7], classes=["auxiliary"], ID="RECT2")

            # Long diagonal R1 R6 and unit incircle cO
            m.construct_segment(R1, R6, classes=["auxiliary"], ID="R1R6")
            m.construct_circle(O, T1, classes=["primary"], ID="cO")

            # Golden rhombus ABCD circumscribed about cO via tangents at y=+-1 and 2x - y = +-sqrt(5)
            A = m.set_point(phi, 1, classes=["golden"], ID="A")
            B = m.set_point(-inv_phi, 1, classes=["golden"], ID="B")
            C = m.set_point(-phi, -1, ID="C")
            D = m.set_point(inv_phi, -1, ID="D")
            m.construct_polygon([A, B, C, D], classes=["golden"], ID="ABCD")
            m.construct_line(B, A, classes=["golden"], ID="BA")
            m.construct_segment(A, C, classes=["auxiliary"], ID="AC")
            m.construct_segment(B, D, classes=["auxiliary"], ID="BD")

            return make_result(m, ("T1", "K"), ("K", "A"), ("T1", "K", "A"))
        

Exact Symbolic Point Coordinates

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

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
T1	\(0\)	\(1\)
T2	\(0\)	\(-1\)
R1	\(-2\)	\(1\)
K	\(1\)	\(1\)
R6	\(2\)	\(-1\)
R7	\(-1\)	\(-1\)
A	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(1\)
B	\(\frac{1}{2} - \frac{\sqrt{5}}{2}\)	\(1\)
C	\(- \frac{\sqrt{5}}{2} - \frac{1}{2}\)	\(-1\)
D	\(- \frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(-1\)

Certified Golden Ratio: T1K / KA = GoldenRatio.