Projecting E8 and Gosset 4₂₁ Polytope: 3D Coxeter-Plane Cross-Sections and Voronoi Degenerations — E8 Intelligence Research

FINDING: The E8 root system (240 vectors) and Gosset 4₂₁ polytope (6720 edges) project onto 3D with Voronoi-cell cross-sections revealing Coxeter-plane symmetries; central degeneration of affine Grassmannian connects to convex polytopes. MATH: - E8 roots: 240 vectors in ℝ⁸, norm² = 2; 4₂₁ polytope has 240 vertices, 6720 edges (each vertex degree 56). - E8 lattice Voronoi cell: 8D polytope with 240 facets (each a 7-simplex); 3D cross-sections via Coxeter plane (order-8 rotation) show hexagonal/cuboctahedral traces. - Coxeter plane projection: uses eigenvalues exp(2πi/h) with h=30 (Coxeter number of E8); 3D sections yield distance maps of squared distances (LPVT8aDK2pc). - Central degeneration (arXiv:1604.08641): type A affine Grassmannian → polytopes are **hypersimplices** (Δ(k,n)) with volumes given by Eulerian numbers; central fiber is a toric variety with moment polytope = permutohedron. CONNECTION: - E8 root system is **crystallographic** (all Cartan integers ∈ {0,±1,± Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com