Coxeter Groups, Dynkin Diagrams, and Euler Characteristic in Regular Polytopes — E8 Intelligence Research

FINDING: Coxeter groups and Dynkin diagrams encode the symmetry and classification of all regular polytopes in any dimension, with Euler characteristic as a topological invariant. | MATH: Euler characteristic for convex polyhedra: V - E + F = 2 (generalized to higher dimensions via alternating sum of face counts). Coxeter groups are generated by reflections; Dynkin diagrams encode Cartan matrices with entries a_ij = 2 cos(π / p_ij) where p_ij is the order of the product of two generators. For regular polytopes, the symmetry group is a finite Coxeter group (e.g., A_n, B_n, D_n, E_6, E_7, E_8, F_4, H_3, H_4, I_2(p)). | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in Coxeter groups H_3 (icosahedral symmetry) and H_4 (600-cell/120-cell). Ratios 0.618 (1/φ), 0.382 (1/φ²), 0.786 (√(φ-1)?), 2.618 (φ²) emerge from the geometry of these polytopes. The E_8 root system (240 roots) has connections to the golden ratio and icosian quaternions. Base-60 appears in Babylonian astronomy but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com