GC-02.02 · Symbolic Model & Source

Companion symbolic Python construction for GC-02.02 · Michel Bataille’s Equilateral Triangle & External Square (GC-02.02).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-02.02")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_bataille_triangle_square() -> ConstructionResult:
            """Michel Bataille's 2011 equilateral triangle and external square sweep."""
            m = new_catalog_model("bataille-triangle-square")
            s3 = sp.sqrt(3)
            B = m.set_point(0, 0, classes=["given"], ID="B")
            C = m.set_point(2, 0, classes=["given"], ID="C")
            A = m.set_point(1, s3, ID="A")
            D = m.set_point(2, -2, ID="D")
            E = m.set_point(0, -2, ID="E")

            m.construct_polygon([A, B, C], ID="ABC")
            m.construct_polygon([B, C, D, E], ID="BCDE")
            ln_ab = m.construct_line(A, B, ID="AB")
            m.construct_segment(C, E, classes=["auxiliary"], ID="CE")

            c_ce = m.construct_circle(C, E, classes=["golden"], ID="cCE")
            m.intersect(c_ce, ln_ab, beyond=(A, B), classes=["golden"], ID="F")

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

Exact Symbolic Point Coordinates

All points in bataille-triangle-square are certified free of floating-point approximations over \(\mathbb{Q}(\sqrt{3}, \phi)\):

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

Certified Golden Ratio: AB / BF = GoldenRatio.