Fourier Transform Links H3 Icosahedral Symmetry to Quasicrystal Diffraction — E8 Intelligence Research

FINDING: Fourier transform bridges Coxeter group H3 icosahedral symmetry and quasicrystal diffraction, revealing that non-crystallographic root systems encode forbidden rotational symmetries in reciprocal space. MATH: - Coxeter group H3: order 120, generators (s₁,s₂,s₃) with relations (s₁s₂)³ = (s₂s₃)⁵ = (s₁s₃)² = 1. - Icosahedral symmetry: golden ratio φ = (1+√5)/2 = 1.618…, and its inverse φ⁻¹ = 0.618…, φ⁻² = 0.382…, φ⁻³ = 0.236… - Quasicrystal diffraction: Fourier transform of a quasiperiodic tiling yields Bragg peaks at positions k = Σ nᵢ aᵢ* where aᵢ* are reciprocal lattice vectors in 6D embedding (projected to 3D). - Involution product: for w ∈ W (finite Coxeter), w = xy with x²=y²=1, minimal ℓ(x)+ℓ(y)−ℓ(w) relates to root system depth. CONNECTION: - H3 contains 15 great circles (mirror planes) whose normals are icosahedron vertices; their pairwise angles give cos⁻¹(1/√5) ≈ 63.43° and cos⁻¹(1/φ√5) ≈ 31.72° — both expressible via φ. - The 6D embedding of icosahedra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com