GC-05.01 · Symbolic Model & Source

Companion symbolic Python construction for GC-05.01 · Claudius Ptolemy’s Regular Pentagon & Pentagram (Almagest I.10) (GC-05.01).

Programmatic Usage

from geometor.divine.catalog import build_construction

        model, cert = build_construction("GC-05.01")
        assert cert.ratio_expr == sp.GoldenRatio
        

Exact Builder Implementation

def build_ptolemy_pentagon_pentagram() -> ConstructionResult:
            """Claudius Ptolemy's Almagest I.10 regular pentagon & pentagram construction."""
            m = new_catalog_model("ptolemy-pentagon-pentagram")
            _phi, s36, c36, s72, c72 = _pentagon_radicals()

            O = m.set_point(0, 0, classes=["given"], ID="O")
            B = m.set_point(1, 0, classes=["given"], ID="B")
            A = m.set_point(-1, 0, ID="A")
            C = m.set_point(0, 1, ID="C")
            M = m.set_point(sp.Rational(1, 2), 0, ID="M")

            c_o = m.construct_circle(O, B, classes=["primary"], ID="cO")
            _ = c_o
            ln_ab = m.construct_line(A, B, classes=["auxiliary"], ID="AB")
            m.construct_segment(O, C, classes=["auxiliary"], ID="OC")

            c_mc = m.construct_circle(M, C, classes=["compass"], ID="cMC")
            E = m.intersect(c_mc, ln_ab, between=(A, O), classes=["golden"], ID="E")
            m.construct_segment(C, E, classes=["auxiliary"], ID="CE")

            # Inscribed regular pentagon C P1 P2 P3 P4 of side |CE| = 2 * s36
            P1 = m.set_point(-s72, c72, ID="P1")
            P2 = m.set_point(-s36, -c36, ID="P2")
            P3 = m.set_point(s36, -c36, ID="P3")
            P4 = m.set_point(s72, c72, ID="P4")
            m.construct_polygon([C, P1, P2, P3, P4], ID="CP1P2P3P4")

            # Inscribed regular pentagram diagonals
            ln_p1p4 = m.construct_line(P1, P4, classes=["golden"], ID="P1P4")
            ln_cp2 = m.construct_line(C, P2, classes=["auxiliary"], ID="CP2")
            ln_cp3 = m.construct_line(C, P3, classes=["auxiliary"], ID="CP3")
            m.construct_segment(P1, P3, classes=["auxiliary"], ID="P1P3")
            m.construct_segment(P2, P4, classes=["auxiliary"], ID="P2P4")

            K1 = m.intersect(ln_p1p4, ln_cp2, classes=["golden"], ID="K1")
            m.intersect(ln_p1p4, ln_cp3, classes=["golden"], ID="K2")
            _ = K1

            return make_result(m, ("P1", "K2"), ("K2", "P4"), ("P1", "K2", "P4"))
        

Exact Symbolic Point Coordinates

All points in ptolemy-pentagon-pentagram are certified free of floating-point approximations over \(\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})\):

Point	Exact \(x\)	Exact \(y\)
O	\(0\)	\(0\)
B	\(1\)	\(0\)
A	\(-1\)	\(0\)
C	\(0\)	\(1\)
M	\(\frac{1}{2}\)	\(0\)
E	\(\frac{1}{2} - \frac{\sqrt{5}}{2}\)	\(0\)
P1	\(\frac{- \sqrt{10} \sqrt{5 - \sqrt{5}} - \sqrt{2} \sqrt{5 - \sqrt{5}}}{8}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)
P2	\(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{4}\)	\(- \frac{\sqrt{5}}{4} - \frac{1}{4}\)
P3	\(\frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{4}\)	\(- \frac{\sqrt{5}}{4} - \frac{1}{4}\)
P4	\(\frac{\sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{5 - \sqrt{5}}}{8}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)
K1	\(\frac{- 3 \sqrt{2} \sqrt{5 - \sqrt{5}} + \sqrt{10} \sqrt{5 - \sqrt{5}}}{8}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)
K2	\(\frac{- \sqrt{10} \sqrt{5 - \sqrt{5}} + 3 \sqrt{2} \sqrt{5 - \sqrt{5}}}{8}\)	\(- \frac{1}{4} + \frac{\sqrt{5}}{4}\)

Certified Golden Ratio: P1K2 / K2P4 = GoldenRatio.