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

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: \( θ = 360^∘ × (1 - 1/φ) = 360^∘ × (2 - φ) ≈ 137.5077^∘ \). Equivalently, \( θ = 2π / φ^2 \) radians (since \( 1/φ^2 = 2 - φ ≈ 0.381966 \)). - Golden spiral: \( r(θ) = a ebθ \), with growth factor per full turn \( e2π b = φ \), so \( b = ln(φ) / (2π) ≈ 0.0053468 \). - Fibonacci recurrence: \( Fₙ₊₁ = F_n + Fₙ₋₁ \), with \( limn→∞ Fₙ₊₁/F_n = φ \). - 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