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

FINDING: Poisson-Voronoi tessellations in 2D exhibit a universal hexagonal regularity index of ~0.907, bridging random point processes and crystalline order. | MATH: Hexagonal regularity index \( H = \frac{\text{mean cell area}}{\text{area of regular hexagon with same perimeter}} \); for Poisson-Voronoi, \( H \approx 0.907 \) (exact value conjectured, derived from variance of cell area/perimeter distributions). Key constants: \( \mathbb{E}[\text{cell area}] = 1/\lambda \) (λ = intensity), \( \mathbb{E}[\text{perimeter}] = 4\sqrt{\lambda/\pi} \), and the variance of the number of sides \( \text{Var}(N) \approx 1.78 \) (empirical, linked to Lewis law). The paper (arXiv:0708.0977) bridges square (\( H=1 \)) and honeycomb (\( H=1 \)) limits via a symmetry-break parameter \( \epsilon \), showing \( H(\epsilon) \) interpolates smoothly, with Poisson-Voronoi as the \( \epsilon \to \infty \) limit. | CONNECTION: The 0.907 index is remarkably close to \( \frac{3\sqrt{3}}{2\pi} \approx 0.827 \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com