Continued Fractions, Hyperbolic Geometry, and Fractal Dynamics Unify — E8 Intelligence Research

FINDING: Continued fraction expansions (including the golden ratio's all-ones form) connect to higher-dimensional hyperbolic geometry, torus translations, and fractal dynamical trajectories, while divergence/curl provides the calculus backbone for field theories. MATH: - Golden ratio: φ = [1;1,1,1,...] = (1+√5)/2 ≈ 1.618; its reciprocal φ⁻¹ = 0.618; φ⁻² = 0.382; φ⁻³ ≈ 0.236; φ⁻⁴ ≈ 0.146. - Iwasawa continued fractions: generalize to higher-dimensional hyperbolic spaces (SL(n,R)/SO(n) or related symmetric spaces), where the continued fraction map acts as a geodesic flow analogue. - Torus translations: symbolic coding via multidimensional continued fractions (e.g., Rauzy–Veech, Brun, Jacobi–Perron) — these generate substitutions with eigenvalues related to Pisot numbers (φ is the simplest Pisot). - Divergence theorem: ∮_∂V F·dA = ∫_V ∇·F dV; curl via Stokes: ∮_∂S F·dr = ∫_S (∇×F)·dA. - Fractal trajectory (Weierstrass function): W(t) = Σₙ₌₀^∞ a^n cos(b^n π t), with ab > 1 + Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com