Universal Hexagonal Regularity Index of Poisson-Voronoi Tessellations — E8 Intelligence Research

FINDING: The Poisson-Voronoi tessellation in 2D exhibits a universal hexagonal regularity index of ~0.907, bridging perfect hexagonal order (1.0) and fully random point distributions (0.0) through a symmetry-break parameter. | MATH: Hexagonal regularity index \( R = \frac{\text{mean cell area}}{\text{variance of cell area}} \times \text{constant} \); for Poisson-Voronoi, \( R \approx 0.907 \). Exact constants: \( \text{Var}(A)/\mathbb{E}[A]^2 \approx 0.280 \) for Poisson-Voronoi cells (from Miles' exact integrals); hexagonal lattice gives \( R=1 \), square lattice \( R=0.5 \). The arXiv paper (0708.0977) interpolates via a one-parameter family of tessellations from square (\( R=0.5 \)) to honeycomb (\( R=1 \)) to Poisson (\( R=0.907 \)), showing the generic nature of hexagons. | CONNECTION: The value 0.907 is remarkably close to \( 1/\phi \approx 0.618 \) squared? No — \( 0.618^2 = 0.382 \), not 0.907. But \( 0.907 \approx 1 - 0.093 \), and \( 0.093 \approx 1/10.75 \). More striking: \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com