Cut-and-Project Method Links Penrose Tilings to Quasicrystals and Coxeter Groups — E8 Intelligence Research

FINDING: Cut-and-project method generates Penrose tilings via root lattice projections, linking quasicrystals to Coxeter groups and golden ratio. | MATH: Cut-and-project scheme: \( \pi_\parallel(\mathcal{L} \cap \mathcal{W}) \) where \( \mathcal{L} \) is a lattice (e.g., \( \mathbb{Z}^5 \) or \( \mathbb{Z}^4 \)), \( \mathcal{W} \) is a window in perpendicular space; Penrose tiling uses 5-fold symmetry from \( H_2 \) Coxeter group (order 10) embedded in \( \mathbb{R}^4 \) or \( \mathbb{R}^5 \); golden ratio \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) appears as inflation factor; diffraction pattern shows Bragg peaks with \( \phi \)-scaled indices. | CONNECTION: Golden ratio \( \phi \) and its inverse \( 1/\phi \approx 0.618 \) are intrinsic to Penrose tiling edge lengths and vertex configurations; \( H_3 \) (icosahedral) and \( H_4 \) (600-cell) Coxeter groups yield 3D and 4D quasicrystals with \( \phi \)-based symmetry; root lattice \( A_4 \) or \( D_6 \) used in cut-and-project produces Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com