GC-03.07 · Symbolic Model & Source

Companion symbolic Python construction for GC-03.07 · Bùi Quang Tuân & Alexandre Borovik’s Greek Cross & Square (GC-03.07).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-03.07")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_bui_cross_and_square() -> ConstructionResult:
            """Bùi Quang Tuân & Alexandre Borovik's Greek cross and square in a circle."""
            m = new_catalog_model("bui-cross-and-square")
            s5 = sp.sqrt(5)

            O = m.set_point(0, 0, classes=["given"], ID="O")
            A = m.set_point(1, 3, classes=["given"], ID="A")
            m.construct_circle(O, A, classes=["primary"], ID="cO")

            # Greek cross of five 2x2 squares inscribed in cO (radius sqrt(10))
            X1 = m.set_point(-1, 3, ID="X1")
            C = m.set_point(1, 1, ID="C")
            X2 = m.set_point(3, 1, ID="X2")
            X3 = m.set_point(3, -1, ID="X3")
            D = m.set_point(1, -1, ID="D")
            X4 = m.set_point(1, -3, ID="X4")
            X5 = m.set_point(-1, -3, ID="X5")
            X6 = m.set_point(-1, -1, ID="X6")
            X7 = m.set_point(-3, -1, ID="X7")
            X8 = m.set_point(-3, 1, ID="X8")
            X9 = m.set_point(-1, 1, ID="X9")
            m.construct_polygon([X1, A, C, X2, X3, D, X4, X5, X6, X7, X8, X9], ID="CROSS")

            # Inscribed concentric square S1 S2 S3 S4 of side 2*sqrt(5)
            S1 = m.set_point(-s5, s5, ID="S1")
            S2 = m.set_point(s5, s5, ID="S2")
            S3 = m.set_point(s5, -s5, ID="S3")
            S4 = m.set_point(-s5, -s5, ID="S4")
            m.construct_polygon([S1, S2, S3, S4], classes=["golden"], ID="S1S2S3S4")

            ln_ac = m.construct_line(A, C, classes=["golden"], ID="AC")
            ln_s1s2 = m.construct_line(S1, S2, classes=["auxiliary"], ID="S1S2")
            ln_s4s3 = m.construct_line(S4, S3, classes=["auxiliary"], ID="S4S3")
            m.intersect(ln_ac, ln_s1s2, classes=["golden"], ID="B")
            E = m.intersect(ln_ac, ln_s4s3, classes=["golden"], ID="E")
            _ = E

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

Exact Symbolic Point Coordinates

All points in bui-cross-and-square are certified free of floating-point approximations over \(\mathbb{Q}(\phi)\):

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
A	\(1\)	\(3\)
X1	\(-1\)	\(3\)
C	\(1\)	\(1\)
X2	\(3\)	\(1\)
X3	\(3\)	\(-1\)
D	\(1\)	\(-1\)
X4	\(1\)	\(-3\)
X5	\(-1\)	\(-3\)
X6	\(-1\)	\(-1\)
X7	\(-3\)	\(-1\)
X8	\(-3\)	\(1\)
X9	\(-1\)	\(1\)
S1	\(- \sqrt{5}\)	\(\sqrt{5}\)
S2	\(\sqrt{5}\)	\(\sqrt{5}\)
S3	\(\sqrt{5}\)	\(- \sqrt{5}\)
S4	\(- \sqrt{5}\)	\(- \sqrt{5}\)
B	\(1\)	\(\sqrt{5}\)
E	\(1\)	\(- \sqrt{5}\)

Certified Golden Ratio: BC / AB = GoldenRatio.