E8 Weyl Group Order 696,729,600: Finite Reflection Group in Lie Theory and Quantum Symmetry — E8 Intelligence Research

FINDING: The E8 Weyl group order 696729600 is a finite reflection group intimately tied to the E8 root lattice, a fundamental structure in Lie theory and quantum symmetry. MATH: - E8 Weyl group order = 696,729,600 = 2^14 × 3^5 × 5^2 × 7 - E8 root system: 240 roots in 8-dimensional space, with Cartan matrix determinant = 1 (unimodular lattice) - Base-60 decomposition: 696,729,600 / 60^4 = 5.376 (not an integer, but 60^3 = 216,000 divides 696,729,600 exactly 3225.6 times — no clean sexagesimal ratio) - Key constant: The Coxeter number of E8 is 30, and the dual Coxeter number is 30. The ratio of the longest to shortest root length is √2. CONNECTION: - The E8 root lattice is a crystallographic root system of type E8, the largest exceptional simple Lie group. Its Weyl group is a finite Coxeter group, order 696,729,600. - Geometric harmony: The E8 lattice is intimately connected to the golden ratio φ = (1+√5)/2 ≈ 1.618. Specifically, the E8 root system can be constructed from t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com