Notation Bridge — Knott (\(\Phi, \phi\)) & SymPy

Dr. Ron Knott distinguishes the Golden Ratio \(\Phi = \frac{1+\sqrt{5}}{2}\) from its positive reciprocal \(\phi = \Phi - 1 = \frac{1}{\Phi} = \frac{\sqrt{5}-1}{2}\) strictly by letter case, whereas SymPy and geometor.model denote the larger root by sp.GoldenRatio (\(\phi_{\text{GEOMETOR}} = \Phi_{\text{Knott}}\)).

Exact Algebraic Bijection

A recurring source of confusion across Golden Ratio literature is whether \(\phi\) denotes \(1.61803\dots\) or \(0.61803\dots\). Throughout Fibonacci Numbers, the Golden Section and the Golden String (KnottPropsPhi), Dr. Ron Knott adopts a remarkably transparent convention in which uppercase \(\Phi\) and lowercase \(\phi\) share the exact same fractional tail \(0.6180339887\dots\):

Canonical Notation Concordance

Quantity

Dr. Ron Knott Notation

Exact \(\mathbb{Q}(\sqrt{5})\) Radical

SymPy / geometor.model

Golden Ratio (\(> 1\))

\(\Phi\) (geometor.knott.PHI)

\(\dfrac{1 + \sqrt{5}}{2} \approx 1.6180339887\dots\)

sp.GoldenRatio (\(\phi_{\text{GEOMETOR}}\))

Reciprocal Golden Ratio (\(< 1\))

\(\phi\) (geometor.knott.phi)

\(\dfrac{\sqrt{5} - 1}{2} \approx 0.6180339887\dots\)

1 / sp.GoldenRatio (\(\phi_{\text{GEOMETOR}} - 1\))

Galois Conjugate (\(< 0\))

\(-\phi = 1 - \Phi\) (geometor.knott.PSI)

\(\dfrac{1 - \sqrt{5}}{2} \approx -0.6180339887\dots\)

1 - sp.GoldenRatio

Fundamental Coupled Identities

Every algebraic verification in geometor.knott.constants checks the coupled Golden Ratio laws symbolically over \(\mathbb{Q}(\sqrt{5})\):

\[\Phi - \phi = 1, \qquad \Phi \cdot \phi = 1, \qquad \Phi + \phi = \sqrt{5}, \qquad \Phi^2 = \Phi + 1, \qquad \phi^2 = 1 - \phi = 2 - \Phi.\]

Lossless conversions between Dr. Knott’s \((\Phi, \phi)\) expressions and SymPy’s sp.GoldenRatio are provided by geometor.knott.constants.knott_to_geometor() and geometor.knott.constants.geometor_to_knott_radicals().

Rational Trigonometry Invariants

To avoid introducing transcendental acos or floating-point rounding into geometric proofs, geometor-knott pairs classical Euclidean ratios with Norman J. Wildberger’s Rational Trigonometry invariants (Wildberger2005):

  • Quadrance (quadrance): \(Q(A, B) = (x_B - x_A)^2 + (y_B - y_A)^2\) — exact squared Euclidean separation;

  • Spread (spread): \(s(\ell_1, \ell_2) = \sin^2\theta \in [0, 1]\) — exact rational/algebraic line separation (\(s = 0\) parallel, \(s = 1\) perpendicular, \(s(36^\circ) = \frac{5 - \sqrt{5}}{8}\), \(s(72^\circ) = \frac{5 + \sqrt{5}}{8}\));

  • Projective Cross-Ratio (cross-ratio): \((A, B; C, D) = \frac{(A - C)(B - D)}{(A - D)(B - C)}\) — exact projective harmonic and golden range invariant.