KNOTT-6.5 — Golden Angle (360° / Φ² ≈ 137.50776°) & Parastichy Bridges

Dividing a full turn in extreme and mean ratio yields the Golden Angle 2π / Φ² = 2π φ² ≈ 137.50776°, whose cylindrical primordia satisfy Fₖ φ² − Fₖ₋₂ = (−φ)ᵏ, pairing opposing left- and right-handed parastichy spirals in consecutive Fibonacci numbers (Fₖ, Fₖ₊₁).
O S P2 F2 F3 F5 F8

Curated Source & Catalog Provenance

Catalog ID

KNOTT-6.5

Curated Source

Dr. Ron Knott, Fibonacci Numbers and Nature (#seeds)

Source Attribution

© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition

Exact SymPy Validator

geometor.knott.identities.construct_parastichy_lattice_model() (0 sp.Float atoms)

GEOMETOR Concordance

geometor/phyllotaxis, geometor/thompson, geometor/fibonacci, geometor/pacioli, geometor/divine

Constructive Synthesis & Exact Proof

Concluding the catalog on fibnat.html (KnottNature), Dr. Ron Knott connects the Golden Angle \(2\pi / \Phi^2 = 2\pi\phi^2 \approx 137.50776^\circ\) to the opposing Fibonacci spiral families (parastichies) studied by Luca Pacioli, D'Arcy Wentworth Thompson (Thompson1917), and H. S. M. Coxeter (Coxeter1973).

  • Unrolled Cylindrical Primordia & Alternating Convergence: Unrolling a cylindrical stem around central axis OS (O to S), the \(F_k\)-th primordium lands at signed angular offset \(F_k \phi^2 - F_{k-2} = (-\phi)^k\), placing P2 (\(F_2 = 1\)), F2 (\(F_3 = 2\)), F3 (\(F_4 = 3\)), F5 (\(F_5 = 5\)), and F8 (\(F_6 = 8\)) on alternating sides of OS along vectors OP2 – OF8.

  • Unimodular Parastichy Parallelograms: Because \((-\phi)^k + (-\phi)^{k+1} = (-\phi)^{k+2}\) and \(F_k + F_{k+1} = F_{k+2}\), vector F5 is the exact 2D vector sum of F2 and F3 (edges F2F5 and F3F5), and F8 is the vector sum of F3 and F5 (edges F3F8 and F5F8). Moreover, \(\phi^k (F_k \phi + F_{k+1}) = \phi^k \Phi^k = 1\), proving that consecutive parastichy cells OF3F5F2 and OF3F8F5 are equal unimodular fundamental domains of normalized area \(1\).

Machine-Verified Exact Symbolic Invariants

Invariant

Exact Symbolic Value

minor turn fraction

\(\frac{3 - \sqrt{5}}{2}\)

major turn fraction

\(\frac{-1 + \sqrt{5}}{2}\)

golden angle radians

\(- \sqrt{5} \pi + 3 \pi\)

golden angle degrees

\(540 - 180 \sqrt{5}\)

reflex angle degrees

\(-180 + 180 \sqrt{5}\)

fundamental parastichy cell area

\(1\)