Exact Site Percolation Threshold for Hexagonal Lattice: \(p_c = 1 - 2\sin(\pi/18)\) — E8 Intelligence Research

FINDING: Site percolation threshold for hexagonal lattice is exactly \(1 - 2\sin(\pi/18) \approx 0.652703...\), equal to the golden ratio conjugate squared plus a small term, but the exact relation is \(p_c = 1 - 2\sin(\pi/18)\). | MATH: \(p_c = 1 - 2\sin(\pi/18)\); \(\sin(\pi/18) = \sin(10^\circ) \approx 0.173648\); \(p_c \approx 0.652703\). The golden ratio conjugate \(\phi^{-1} = 0.618034\); \(\phi^{-2} = 0.381966\); note \(p_c - \phi^{-1} = 0.034669\), not a simple rational multiple. No direct exact equality with golden ratio or its powers. | CONNECTION: Hexagonal lattice is dual to triangular lattice; site percolation on triangular lattice has exact threshold \(p_c = 1/2\). The hexagonal threshold involves \(\sin(\pi/18)\), which relates to the 18-fold symmetry of the lattice's fundamental domain (hexagon angle \(120^\circ\), but \(\pi/18 = 10^\circ\) appears from trigonometric solution of the critical polynomial). This connects to crystallographic symmetry (hexagonal, point group Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com