Golden Angle and Golden Spiral Unify Phyllotaxis via Discrete Differential Geometry — E8 Intelligence Research

Abstract

FINDING: Phyllotaxis and the golden spiral are governed by the golden angle (137.51°) and the logarithmic spiral's growth factor φ, with discrete differential geometry providing a unifying integrable framework. MATH: - Golden angle: \( \theta = 360^\circ \times (1 - 1/\varphi) = 360^\circ \times (2 - \varphi) \approx 137.5077^\circ \). Equivalently, \( \theta = 2\pi / \varphi^2 \) radians (since \( 1/\varphi^2 = 2 - \varphi \approx 0.381966 \)). - Golden spiral: \( r(\theta) = a e^{b\theta} \), with growth factor per full turn \( e^{2\pi b} = \varphi \), so \( b = \ln(\varphi) / (2\pi) \approx 0.0053468 \). - Fibonacci recurrence: \( F_{n+1} = F_n + F_{n-1} \), with \( \lim_{n\to\infty} F_{n+1}/F_n = \varphi \). - Discrete differential geometry (arXiv:math/0504358v1): consistency conditions (e.g., quadrilateral lattices, circle patterns) yield integrable systems; phyllotaxis emerges as a discrete analogue of the logarithmic spiral via circle packing and curvature flow. CONN Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com