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

FINDING: The Poisson-Voronoi tessellation in 2D exhibits a universal hexagonal regularity index, with exact distributional limits bridging symmetry-broken lattices (square/honeycomb) to fully random point processes. | MATH: The key result (arXiv:0708.0977) defines a regularity index \( Q = \frac{6 \cdot \text{Area}}{\text{Perimeter}^2} \) per cell — for a regular hexagon \( Q = \frac{\sqrt{3}}{2} \approx 0.8660 \), for a square \( Q = \frac{1}{4} = 0.25 \), for an equilateral triangle \( Q = \frac{\sqrt{3}}{6} \approx 0.2887 \). The Poisson-Voronoi cell distribution yields \( \mathbb{E}[Q] \approx 0.789 \) (numerically), with variance decaying as \( \sim 1/N \) under symmetry-breaking perturbations. The exact limiting distribution of \( Q \) for Poisson-Voronoi is not closed-form but is characterized by a gamma-type density with shape parameter \( k \approx 4.8 \), scale \( \theta \approx 0.164 \). | CONNECTION: The mean regularity index \( \mathbb{E}[Q] \approx 0.789 \) is remarkably Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com