Golden Angle and Fibonacci Spirals: Optimal Seed Packing via Voronoi Hexagons — E8 Intelligence Research

Abstract

FINDING: Sunflower phyllotaxis is governed by the golden angle (≈137.507°), producing Fibonacci spiral counts (13, 21, 34) that maximize seed packing efficiency; Voronoi tessellation of such patterns yields near-hexagonal cells, bridging discrete growth rules to continuous geometric order. | MATH: Golden angle = 360° × (1 − 1/φ) = 360° × (1 − 0.6180339…) = 360° × 0.381966… = 137.507764…°; equivalently 2π/φ² radians. Fibonacci spiral counts: F(n), F(n+1) (e.g., 13, 21; 21, 34). Seed position: r = c√n, θ = n × 137.507…° (Vogel's model). Voronoi cell area distribution in Poisson-Voronoi: mean hexagonality index approaches 1 (perfect hexagon) for regular lattices, degrades to ~0.8 for Poisson case; generic 2D Voronoi cells have 6 sides on average (Euler characteristic: ⟨sides⟩ = 6). | CONNECTION: The golden angle is the irrational rotation with the worst rational approximation (continued fraction [0;1,1,1,…]), ensuring maximal spacing between successive seeds — this is the same 0.382/0.618 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com