The Golden Ratio in the 421 Polytope's 8D-to-4D Projection and Self-Application Recurrence — E8 Intelligence Research

Abstract

FINDING: 421 polytope (E8 root system) has 240 vertices and 6720 edges; golden ratio appears in its 8D-to-4D projection and in stable self-application recurrence. MATH: - 421 polytope vertices: 240 (E8 root system). - Edge count: 6720. - Golden ratio φ = (1+√5)/2 ≈ 1.618, φ⁻¹ ≈ 0.618. - Recurrence relation for φ: x = 1 + 1/x → x² = x + 1. - 4D projection of 421 yields 600-cell (120 vertices, 720 edges, 600 tetrahedral cells). - 600-cell vertex coordinates involve φ and its reciprocal (e.g., (0, ±1, ±φ, ±φ⁻¹) permutations). CONNECTION: - 600-cell is a regular 4D polytope with golden ratio symmetries (icosahedral symmetry in 3D slices). - 421 → 600-cell projection preserves φ ratios: edge length ratio of 600-cell is φ. - Base-60 not directly present, but 240 = 4×60, 6720 = 112×60, hinting at sexagesimal divisibility. - Crystallographic: E8 root system is a lattice in 8D; its 4D projection yields the 600-cell, a golden-ratio-based polytope. DEPTH: 8 - Direct link Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com