Continued Fractions, Quadratic Irrationals, and Optimal Fibonacci Approximations — E8 Intelligence Research

Abstract

FINDING: Continued fractions converge to quadratic irrationals (incl. √n and φ), with Fibonacci convergents providing optimal rational approximations; recent work links convergents to spanning-tree counts. MATH: - Simple continued fraction: \( a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}} \) - Convergents \( p_n/q_n \) satisfy \( p_n = a_n p_{n-1} + p_{n-2} \), \( q_n = a_n q_{n-1} + q_{n-2} \) - For φ = (1+√5)/2 = [1;1,1,1,…], convergents are \( F_{n+1}/F_n \) → φ, error ~ \( 1/(q_n q_{n+1}) \) - √n has periodic continued fraction (Lagrange), e.g., √2 = [1;2,2,2,…] - Best rational approximation property: \( |x - p/q| < 1/(2q^2) \) for convergents - Swee Hong Chan: spanning-tree counts of certain graphs expressed via continued fractions — connects combinatorial enumeration to rational approximation. CONNECTION: - φ = 1.618033…, its reciprocal 0.618033…, and φ−1 = 0.618, φ−2 = 0.382 — all appear as limits of Fibonacci convergents. - Phyllotaxis: divergence angle ≈ 1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com