Golden Ratio and 5D Projection: Unifying Penrose Tilings via H3 Coxeter Eigenvalues — E8 Intelligence Research

FINDING: de Bruijn's cut-and-project method generates Penrose tilings via 5-fold symmetric projection from a higher-dimensional lattice, with the golden ratio emerging as the characteristic eigenvalue of the H3 Coxeter group's Cartan matrix. MATH: - Cut-and-project: \(\pi_\parallel(\mathbb{Z}^5 \cap \Omega)\) where \(\Omega = \mathbb{R}^5\) slab perpendicular to the 5-fold axis; Penrose tiling = projection of 5D hypercubic lattice points inside a strip. - H3 Coxeter group (icosahedral symmetry): Cartan matrix \(A\) has eigenvalues \(\{2, \varphi, \varphi^{-1}\}\) where \(\varphi = (1+\sqrt{5})/2 = 1.618...\) and \(\varphi^{-1} = 0.618...\). - The golden ratio satisfies \(\varphi^2 = \varphi + 1\), and the H3 root system's simple roots have inner products giving off-diagonal entries \(-1\) or \(-\varphi^{-1}\) (depending on convention). - Involution product result (arXiv:1405.3051): For finite Coxeter groups, every element is a product of two involutions; the minimal "defect" Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com