Coxeter Groups Unify Reflection Symmetries, Quasicrystals, and Vieta Jumping — E8 Intelligence Research

Abstract

FINDING: Coxeter groups encode reflection symmetries whose dihedral angles and Cartan matrices generate root systems — the same combinatorial skeleton appears in quasicrystalline order and olympiad "Vieta jumping" number theory. | MATH: Coxeter system (W,S) with m(s,t) ∈ {2,3,4,5,6,∞} defining dihedral angle π/m(s,t). Cartan matrix A_ij = -2cos(π/m_ij) for i≠j, A_ii=2. Finite Coxeter groups classified by crystallographic condition (A_ij ∈ {0,-1,-2,-3}) → root systems A_n–G_2. Involution product theorem: every finite W element = product of two involutions; minimal length defect ℓ(x)+ℓ(y)−ℓ(w) is even and bounded by rank. | CONNECTION: Dihedral angles π/5 (m=5) → cos(π/5)=φ/2=0.809 → Cartan entry −φ ≈ −1.618 (non-crystallographic H_2,H_3,H_4 — icosahedral symmetry, quasicrystal Penrose tilings). m=3 → cos(π/3)=0.5 → entry −1 (crystallographic). m=4 → cos(π/4)=√2/2≈0.707 → entry −√2. m=6 → cos(π/6)=√3/2≈0.866 → entry −√3. The golden ratio φ=1.618 appears directly as Cartan entry magnitude Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com