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: \( F_{n+1} = F_n + F_{n-1} \), with \( F_0 = 0, F_1 = 1 \). - Closed form (Binet): \( F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} \), where \( \varphi = \frac{1+\sqrt{5}}{2} \approx 1.6180339887 \), \( \psi = \frac{1-\sqrt{5}}{2} \approx -0.6180339887 \). - Ratio convergence: \( \lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \varphi \). - Rational approximants: \( \frac{F_{n+1}}{F_n} \) are convergents of the continued fraction \( [1;1,1,1,\ldots] \), giving best rational approximations to \( \varphi \). - Phyllotaxis angle: \( 360^\circ \times \varphi^{-2} \approx 137.5^\circ \) (or \( 360^\circ \times (1 - \varphi^{-1}) \approx 137.5^\circ \)), derived from \( \varphi^{-2} = 2 - \varphi \approx 0.381966 \). CONNECTION: - Golden ratio \( \varphi = 1.618... \) and its reciprocal \( \varphi^{-1} = 0.618... \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com