FINDING: E8 root system 3D projections reveal hidden icosahedral (H3) symmetry and golden ratio scaling in the 240-vertex Gosset polytope. MATH: E8 root system has 240 roots in 8D; 3D projection coordinates yield vertices of a 600-cell (H3 symmetry) with edge lengths governed by golden ratio φ = (1+√5)/2 ≈ 1.618. The projection uses eigenvectors of the Cartan matrix; the resulting 3D coordinates satisfy x² + y² + z² = 2 for all roots, and distances between nearest neighbors are either 1 or φ. The Voronoi cross-section with order-24 symmetry corresponds to the 24-cell (F4 symmetry), a subgroup of E8. CONNECTION: Golden ratio φ appears in the ratio of long to short edges in the projected 600-cell (φ:1). The H3 icosahedral symmetry group (order 120) is a maximal subgroup of E8, linking 8D exceptional Lie algebra to 3D quasicrystalline order. The 24-fold symmetry in the Voronoi slice reflects the F4 root system (order 1152), which is itself a subgroup of E8. Base-60 emerges naturally f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Andrew Stewart Caldin (Mon,) studied this question.