Hexagonal Dominance and Symmetry-Breaking in Poisson-Voronoi Tessellations — E8 Intelligence Research

FINDING: Poisson-Voronoi tessellations exhibit hexagonal dominance in Euclidean space, with symmetry-breaking transitions to square/cubic lattices under anisotropic or constrained conditions; hyperbolic variants show distinct percolation and tiling behavior. | MATH: Poisson-Voronoi cell area distribution (Euclidean): \( f_A(a) = \frac{343}{15} \sqrt{\frac{7}{2\pi}} \frac{a^{5/2}}{\lambda^{7/2}} e^{-7a/(2\lambda)} \) (exact, 2D); mean neighbor count = 6 (hexagonal); variance of neighbor count ≈ 1.78; hexagonal regularity index \( \eta = \frac{\text{perimeter}^2}{4\pi \cdot \text{area}} \) → 1 for perfect hexagon, >1 for disorder. Hyperbolic Poisson-Voronoi: critical percolation threshold \( p_c \) depends on curvature \( \kappa \), with \( p_c \to 1/2 \) as \( \kappa \to -\infty \) (tree-like limit). | CONNECTION: The Euclidean Voronoi cell's *average* shape is the regular hexagon — the unique space-filling polygon with 120° angles (crystallographic point group \( D_6 \)). The transitio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com