Clifford Algebra Unifies Quaternionic Root Systems and Coxeter Reflection Groups — E8 Intelligence Research

FINDING: Clifford algebra framework unifies quaternionic root systems of Coxeter groups (including E₈) with geometric algebra, revealing reflection groups as natural Clifford algebra structures. | MATH: Coxeter groups W with root systems Φ ⊂ ℝⁿ; quaternionic representation via H ≅ Cl₃₀; reflections r_a(x) = -a x a⁻¹ in Clifford algebra; E₈ root system (240 roots, 8D) embedded in quaternions; D₂ root system {±e₁±e₂} with Coxeter-Dynkin diagram A₁×A₁; Clifford torus S¹×S¹ ⊂ S³ with principal curvatures ±1. | CONNECTION: D₂ root system is the symmetry of the square lattice (crystallographic, order 4, dihedral D₄); its quaternionic Clifford representation yields the golden ratio φ = (1+√5)/2 in E₈ projections (via 2D Coxeter plane); ratios 0.618, 1.618 appear as eigenvalues of Coxeter elements in rank-2 subsystems; base-60 connection: D₂ ⊂ A₄ (icosahedral) which relates to 60° angles and 5-fold symmetry. | DEPTH: 8 --- **Detailed Analysis:** 1. **Key mathematical insight:** The paper (a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com