Golden Ratio Links E8 Root System to 120-Cell and 600-Cell Symmetries — E8 Intelligence Research

FINDING: The E8 root system (Gosset 4_21 polytope) and the 4D 120-cell/600-cell are deeply interlinked through golden-ratio-based projection symmetries, with the 120-cell being a 4D regular polytope whose 3D shadow reveals dodecahedral (icosahedral) symmetry. MATH: - E8: 240 vertices, 6720 edges, root system rank 8, Coxeter number 30. - 120-cell: {5,3,3}, 120 dodecahedral cells, 600 vertices, 1200 edges, 720 pentagonal faces. - 600-cell: {3,3,5}, 600 tetrahedral cells, 120 vertices, 720 edges, 1200 triangular faces. - Key ratio: Both 120-cell and 600-cell have vertex coordinates involving φ = (1+√5)/2 ≈ 1.618 and its inverse φ⁻¹ = (√5−1)/2 ≈ 0.618. - Projection of E8 to 4D yields two concentric 600-cells (or 120-cells) scaled by φ and φ⁻¹ — a direct golden-ratio nesting. - The 120-cell's 3D shadow (vertex-first) produces a dodecahedral shell with edge lengths in ratio 1 : φ : φ². - The 600-cell's 3D shadow (cell-first) yields an icosahedral arrangement with radii in rati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com