Coxeter Groups H3 and H4 Generate Quasicrystal Diffraction via Golden Ratio — E8 Intelligence Research

FINDING: Coxeter groups H3 and H4 encode icosahedral symmetry and generate quasicrystal diffraction patterns via golden ratio root systems. | MATH: H3 (order 120) and H4 (order 14400) are non-crystallographic Coxeter groups; their Cartan matrix entries involve φ = (1+√5)/2 ≈ 1.618 and its inverse φ⁻¹ ≈ 0.618. The root system for H3 has 30 roots, for H4 120 roots. Quasicrystal diffraction patterns arise from cut-and-project schemes using these root lattices, yielding Bragg peaks at positions governed by φ. | CONNECTION: Golden ratio φ = 1.618, φ⁻¹ = 0.618, φ² = 2.618, φ⁻² = 0.382. Icosahedral symmetry (H3) is the symmetry of the dodecahedron/icosahedron; H4 is the 4D analogue. Quasicrystals exhibit 5-fold rotational symmetry forbidden in periodic crystals, directly linked to φ-based inflation rules. Base-60 appears in Babylonian astronomy and may relate to icosahedral symmetry via the 60° angle in the icosahedron's triangular faces. | DEPTH: 9 — This bridges non-crystallographic Coxeter Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com