PDE Model Links Fibonacci Numbers to Phyllotaxis via Irrational Rotation Number — E8 Intelligence Research

Abstract

FINDING: Phyllotaxis spiral count converges to irrational rotation number \(1/\phi^2\) via a recursive PDE model, linking Fibonacci numbers to plant morphogenesis. MATH: - Irrational rotation number: \(1/\phi^2 = (3 - \sqrt{5})/2 \approx 0.381966\) - Fibonacci sequence: \(F_{n+2} = F_{n+1} + F_n\), with \(F_0=0, F_1=1\) - Explicit formula: \(F_n = (\phi^n - (-\phi)^{-n})/\sqrt{5}\), where \(\phi = (1+\sqrt{5})/2 \approx 1.618034\) - PDE model: Recursive dynamic system for spiral phyllotaxis morphogenesis (Rozin, 20th Int. Fibonacci Conf.) CONNECTION: - Ratio \(1/\phi^2 = 0.381966\) is the key geometric constant for optimal spiral packing (divergence angle ≈ 137.5°). - Complementary ratio \(\phi^{-1} = 0.618034\) appears in golden angle: \(360° \times (1 - 1/\phi) \approx 137.5°\). - Base-60 link: \(1/\phi^2 \approx 0.381966\) is close to \(0.38333... = 23/60\), suggesting ancient approximation. - Crystallographic symmetry: Phyllotaxis patterns often exhibit 5-fold s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com