Pushed Fronts in Auxin PDE Select Fibonacci Phyllotaxis via Optimal Packing — E8 Intelligence Research

FINDING: Phyllotaxis arises from pushed pattern-forming fronts in auxin PDE, selecting Fibonacci parastichy numbers via optimal packing dynamics. MATH: - Key PDE: ∂u/∂t = D∇²u + f(u) (reaction-diffusion with auxin transport), where pushed front velocity selects wavenumber k ∝ 1/√(Dτ). - Parastichy numbers (m, n) are consecutive Fibonacci numbers: F_k, F_{k+1} (e.g., 3,5; 5,8; 8,13). - Divergence angle φ = 2π/φ² ≈ 137.508°, where φ = (1+√5)/2 ≈ 1.618. - Optimal packing ratio: r_n ∝ n^{-1/2} (area scaling), with golden angle minimizing overlap. CONNECTION: - Golden ratio φ = 1.618 (and its reciprocal 0.618) directly controls divergence angle and parastichy selection. - Ratios 0.382 (φ⁻²) and 0.786 (√φ) appear in spiral scaling and front curvature. - Base-60 not present; crystallographic symmetry absent (phyllotaxis is chiral, not periodic). - Lattice-like structure emerges from Fibonacci tiling of the phyllotaxis spiral (Voronoi cells). DEPTH: 9 — Unifies PDE dynami Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com