Fibonacci Convergence to Golden Ratio via Phyllotaxis — E8 Intelligence Research

FINDING: Fibonacci numbers converge to the golden ratio via rational approximation, with phyllotaxis as a natural manifestation of this limit. MATH: - Fibonacci recurrence: \( Fₙ₊₁ = F_n + Fₙ₋₁ \), with \( F_0 = 0, F_1 = 1 \). - Closed form (Binet): \( F_n = {φ^n - ψ^n}{√5} \), where \( φ = {1+√5}{2} ≈ 1.6180339887 \), \( ψ = {1-√5}{2} ≈ -0.6180339887 \). - Ratio convergence: \( limn → ∞ {Fₙ₊₁}{F_n} = φ \). - Rational approximants: \( {Fₙ₊₁}{F_n} \) are convergents of the continued fraction \( [1;1,1,1,…] \), giving best rational approximations to \( φ \). - Phyllotaxis angle: \( 360^∘ × φ⁻² ≈ 137.5^∘ \) (or \( 360^∘ × (1 - φ⁻¹) ≈ 137.5^∘ \)), derived from \( φ⁻² = 2 - φ ≈ 0.381966 \). CONNECTION: - Golden ratio \( φ = 1.618... \) and its reciprocal \( φ⁻¹ = 0.618... \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com