Higher-Dimensional Continued Fractions Reveal Self-Similar Lattices Beyond Golden Ratio — E8 Intelligence Research

FINDING: Continued fractions generalize to higher-dimensional hyperbolic spaces and torus translations, revealing self-similar lattice structures beyond the classical golden-ratio fixed point. | MATH: Classical φ = [1;1,1,1,...] = (1+√5)/2 ≈ 1.618, satisfying φ = 1 + 1/φ. Higher-dimensional analogues: Iwasawa continued fractions act on hyperbolic n-space; multidimensional continued fractions (e.g., Jacobi–Perron, Brun, Selmer) encode symbolic dynamics of torus translations via substitution matrices with algebraic eigenvalues (Pisot units). Lattice QCD result: heavy-quark masses from Fermilab method — no direct φ, but uses 3-flavor lattice gauge theory with SU(3) root system (A₂ lattice, hexagonal symmetry). | CONNECTION: φ's continued fraction is the *simplest* self-similar expansion — its convergents are ratios of consecutive Fibonacci numbers, whose limiting ratio is φ. Higher-dimensional continued fractions produce algebraic units in real number fields of degree n; for n=2, these in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com