GC-01.01 · Symbolic Model & Source

Companion symbolic Python construction for GC-01.01 · Euclid’s Classical Golden Section (Elements II.11 & VI.30) (GC-01.01).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-01.01")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_euclid_ii_11() -> ConstructionResult:
            """Euclid's Elements II.11 & VI.30 square-and-midpoint division of AB."""
            m = new_catalog_model("euclid-ii-11")
            A = m.set_point(0, 0, classes=["given"], ID="A")
            B = m.set_point(2, 0, classes=["given"], ID="B")
            D = m.set_point(0, 2, ID="D")
            C = m.set_point(2, 2, ID="C")

            m.construct_polygon([A, B, C, D], ID="ABCD")
            ln_ad = m.construct_line(A, D, classes=["auxiliary"], ID="AD")
            seg_ab = m.construct_segment(A, B, ID="segAB")

            # Midpoint M of side AD and compass transfer M(B) onto ray DA
            M = m.set_point(0, 1, ID="M")
            m.construct_segment(M, B, classes=["auxiliary"], ID="MB")
            c_mb = m.construct_circle(M, B, classes=["compass"], ID="cMB")
            E = m.intersect(c_mb, ln_ad, beyond=(D, A), ID="E")

            # Transfer AE onto AB at G (and erect golden gnomon square AEFH)
            c_ae = m.construct_circle(A, E, classes=["golden"], ID="cAE")
            G = m.intersect(c_ae, seg_ab, between=(A, B), classes=["golden"], ID="G")
            F = m.set_point(G.x, E.y, ID="F")
            m.construct_polygon([A, G, F, E], classes=["golden"], ID="AGFE")

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

Exact Symbolic Point Coordinates

All points in euclid-ii-11 are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

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

Certified Golden Ratio: AG / GB = GoldenRatio.