The Golden Ratio's Link to E8 and the 120-Cell: An Unverified Claim — E8 Intelligence Research

Abstract

FINDING: The search results are a mix of popular expositions (Numberphile, PBS) and one obscure combinatorics paper; no direct source links φ to E8 vertex coordinates, the icosian ring, or the 120-cell symmetry group of order 14400. The core mathematical link is implied but not evidenced in the provided abstracts. | MATH: φ = (1+√5)/2 ≈ 1.618; φ⁻¹ = φ−1 ≈ 0.618; φ² = φ+1 ≈ 2.618; φ⁻² = 2−φ ≈ 0.382. E8 root system: 240 vertices, Coxeter number 30, Weyl group order 696,729,600. 120-cell (4D): 600 vertices, 120 dodecahedral cells, symmetry group order 14400 = 120². Icosian ring: quaternion algebra generated by (1, i, j, k) with relations i²=j²=k²=ijk=−1, extended by φ and its conjugates to yield 120 units (the icosians). | CONNECTION: The 120-cell's symmetry group (order 14400) is the binary icosahedral group × ℤ₂; the binary icosahedral group is precisely the 120 units of the icosian ring, whose coordinates involve φ and 0.382, 0.618, 1.618, 2.618. The 600 vertices of the 120-cell can be Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com