FINDING: E8 lattice projects to 3D icosahedral quasicrystal via icosagrid with golden ratio plane spacing, linking 8D root system to 3D quasiperiodic order. MATH: - E8 root system: 240 vertices, 6720 edges, Coxeter group E8 (order 696729600). - Icosagrid: 10 plane sets with icosahedral symmetry; plane spacing governed by Fibonacci chain (golden ratio φ = (1+√5)/2 ≈ 1.618). - Golden ratio scaling: φ, 1/φ ≈ 0.618, φ² ≈ 2.618, φ⁻² ≈ 0.382. - Projection: 8D → 3D via cut-and-project method using Coxeter plane (order 24 symmetry) or icosagrid multigrid. CONNECTION: - Golden ratio φ appears as fundamental scaling in plane spacing (Fibonacci chain), linking to icosahedral symmetry (order 5, 3, 2). - Rhombic dodecahedron and icosidodecahedron emerge as Voronoi cell cross-sections, reflecting E8's 240 vertices in 3D as 120 vertices of icosidodecahedron (ratio 2:1). - Crystallographic symmetry: icosahedral (non-crystallographic in 3D) arises from E8's 8D lattice (crystallographi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Andrew Stewart Caldin (Thu,) studied this question.