Quaternionic Snub 24-Cell from E₈ via D₄ Symmetry Group of Order 576 — E8 Intelligence Research

FINDING: Quaternionic representation of snub 24-cell derived from E₈ root system via D₄ symmetry group of order 576. | MATH: Symmetry group W(D₄):C₃ of order 576; quaternionic unit norm representation; E₈ root system as parent structure; snub 24-cell vertices expressed as quaternions; H₄ root system connection implied. | CONNECTION: D₄ root system is 4D crystallographic; 24-cell is self-dual regular polytope with 24 vertices; snub 24-cell has 96 vertices; quaternions encode 3D rotations and 4D symmetries; E₈ (240 roots) contains H₄ (120 roots) as subgroup; golden ratio φ = (1+√5)/2 ≈ 1.618 appears in H₄ coordinates; icosian representation links to icosahedral symmetry (φ, 1/φ). | DEPTH: 8 — Directly ties quaternionic algebra, exceptional Lie groups (E₈), 4D polytopes (24-cell), and crystallographic root systems (D₄) into a unified geometric structure; fundamental for understanding higher-dimensional symmetry and topological invariants (knot polynomials via braid groups from Coxeter-Dyn Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com