GC-05.06 · Symbolic Model & Source

Companion symbolic Python construction for GC-05.06 · Tran Quang Hung’s Fivefold Wu Xing Equilateral Star (GC-05.06).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-05.06")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_tran_wu_xing_equilaterals() -> ConstructionResult:
            """Tran Quang Hung's fivefold Wu Xing congruent equilateral triangles & pentagram."""
            m = new_catalog_model("tran-wu-xing-equilaterals")
            _phi, s36, c36, s72, c72 = _pentagon_radicals()
            s3 = sp.sqrt(3)

            O = m.set_point(0, 0, classes=["given"], ID="O")
            v_coords = [
                (sp.Integer(0), sp.Integer(1)),
                (-s72, c72),
                (-s36, -c36),
                (s36, -c36),
                (s72, c72),
            ]

            v_pts = []
            w_pts = []
            for i, (vx, vy) in enumerate(v_coords, start=1):
                vp = m.set_point(vx, vy, classes=["given"] if i == 1 else [], ID=f"V{i}")
                wx = sp.expand(vx / 2 + (s3 * vy) / 2)
                wy = sp.expand(-(s3 * vx) / 2 + vy / 2)
                wp = m.set_point(wx, wy, ID=f"W{i}")
                v_pts.append(vp)
                w_pts.append(wp)
                m.construct_polygon([O, vp, wp], ID=f"OV{i}W{i}")

            # Pentagram star connecting V1..V5 skipping one vertex at a time
            ln_v2v5 = m.construct_line(v_pts[1], v_pts[4], classes=["golden"], ID="V2V5")
            for i in range(5):
                m.construct_segment(v_pts[i], v_pts[(i + 2) % 5], classes=["auxiliary"])

            ln_ow1 = m.construct_line(O, w_pts[0], classes=["golden"], ID="OW1")
            m.intersect(ln_v2v5, ln_ow1, classes=["golden"], ID="F")

            return make_result(m, ("O", "F"), ("F", "W1"), ("O", "F", "W1"))
        

Exact Symbolic Point Coordinates

All points in tran-wu-xing-equilaterals are certified free of floating-point approximations over \(\mathbb{Q}(\sqrt{3}, \phi, \sqrt{10 - 2\sqrt{5}})\):

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
V1	\(0\)	\(1\)
W1	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)
V2	\(\frac{- \sqrt{10} \sqrt{5 - \sqrt{5}} - \sqrt{2} \sqrt{5 - \sqrt{5}}}{8}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)
W2	\(\frac{- \sqrt{10} \sqrt{5 - \sqrt{5}} - 2 \sqrt{3} - \sqrt{2} \sqrt{5 - \sqrt{5}} + 2 \sqrt{15}}{16}\)	\(- \frac{1}{8} + \frac{\sqrt{6} \sqrt{5 - \sqrt{5}} + 2 \sqrt{5} + \sqrt{30} \sqrt{5 - \sqrt{5}}}{16}\)
V3	\(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{4}\)	\(- \frac{\sqrt{5}}{4} - \frac{1}{4}\)
W3	\(\frac{- \sqrt{15} - \sqrt{2} \sqrt{5 - \sqrt{5}} - \sqrt{3}}{8}\)	\(- \frac{1}{8} + \frac{- \sqrt{5} + \sqrt{6} \sqrt{5 - \sqrt{5}}}{8}\)
V4	\(\frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{4}\)	\(- \frac{\sqrt{5}}{4} - \frac{1}{4}\)
W4	\(\frac{- \sqrt{15} - \sqrt{3} + \sqrt{2} \sqrt{5 - \sqrt{5}}}{8}\)	\(\frac{- \sqrt{6} \sqrt{5 - \sqrt{5}} - \sqrt{5}}{8} - \frac{1}{8}\)
V5	\(\frac{\sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{5 - \sqrt{5}}}{8}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)
W5	\(\frac{- 2 \sqrt{3} + \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{5 - \sqrt{5}} + 2 \sqrt{15}}{16}\)	\(\frac{- \sqrt{30} \sqrt{5 - \sqrt{5}} - \sqrt{6} \sqrt{5 - \sqrt{5}} + 2 \sqrt{5}}{16} - \frac{1}{8}\)
F	\(\frac{- \sqrt{3} + \sqrt{15}}{4}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)

Certified Golden Ratio: OF / FW1 = GoldenRatio.