GC-01.04 · Symbolic Model & Source

Companion symbolic Python construction for GC-01.04 · Hofstetter’s 2004 Five-Step Secant Division (GC-01.04).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-01.04")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_hofstetter_5step_2004() -> ConstructionResult:
            """Kurt Hofstetter's 2004 5-step straightedge-and-compass division."""
            m = new_catalog_model("hofstetter-5step-2004")
            A = m.set_point(0, 0, classes=["given"], ID="A")
            B = m.set_point(1, 0, classes=["given"], ID="B")

            # 5 elementary Euclidean steps: line AB, circles A(B), B(A), E(B), line DF
            ln_ab = m.construct_line(A, B, ID="AB")
            c_ab = m.construct_circle(A, B, classes=["compass"], ID="cAB")
            E = m.intersect(c_ab, ln_ab, beyond=(B, A), ID="E")
            c_ba = m.construct_circle(B, A, classes=["compass"], ID="cBA")
            C = m.intersect(c_ab, c_ba, side_of=(A, B, 1), ID="C")
            D = m.intersect(c_ab, c_ba, side_of=(A, B, -1), ID="D")
            _ = C

            c_eb = m.construct_circle(E, B, classes=["compass"], ID="cEB")
            F = m.intersect(c_eb, c_ba, side_of=(A, B, 1), ID="F")
            ln_df = m.construct_line(D, F, classes=["golden"], ID="DF")
            m.intersect(ln_df, ln_ab, classes=["golden"], ID="G")

            return make_result(m, ("A", "G"), ("G", "B"), ("A", "G", "B"))
        

Exact Symbolic Point Coordinates

All points in hofstetter-5step-2004 are certified free of floating-point approximations over \(\mathbb{Q}(\sqrt{3}, \phi)\):

Point	Exact \(x\)	Exact \(y\)
A	\(0\)	\(0\)
B	\(1\)	\(0\)
E	\(-1\)	\(0\)
C	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)
D	\(\frac{1}{2}\)	\(- \frac{\sqrt{3}}{2}\)
F	\(\frac{3}{4}\)	\(\frac{\sqrt{15}}{4}\)
G	\(- \frac{1}{2} + \frac{\sqrt{5}}{2}\)	\(0\)

Certified Golden Ratio: AG / GB = GoldenRatio.