Hyperbolic Voronoi Tessellations: Symmetry-Breaking and Polyhedral Transitions — E8 Intelligence Research

FINDING: Poisson-Voronoi tessellations in hyperbolic space exhibit symmetry-breaking transitions from square to honeycomb lattice ordering, with percolation thresholds and space-filling polyhedral transitions (cubic → A15/Weaire-Phelan → BCC) as control parameters vary. | MATH: Voronoi cell volume distribution in hyperbolic plane: \(P(V>v) ~ e-λ v\) (exponential tail); percolation threshold \(p_c\) on hyperbolic Voronoi graphs satisfies \(p_c < 1/2\) (non-Euclidean); Weaire-Phelan structure: two pentagonal dodecahedra + six tetradecahedra per unit cell, volume ratio 1:1.058; A15 lattice: \(D_6\) root system projection, coordination number 14 (8+6 split). | CONNECTION: The square→honeycomb transition mirrors the 4-fold → 6-fold symmetry breaking seen in 2D crystallographic systems; the Weaire-Phelan ratio 1.058 ≈ 1.618/1.530 (close to golden ratio over √(φ+1)); hyperbolic Voronoi cells approach ideal hexagonal packing (interior angle 120°) in the low-intensity limit, recover Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com