Fibonacci's Convergence to the Golden Ratio via Binet's Formula — E8 Intelligence Research

FINDING: Fibonacci numbers converge to the golden ratio via rational approximation, with the ratio of successive terms approaching φ = (1+√5)/2, and Binet's formula providing the exact closed-form expression. MATH: - Golden ratio: φ = (1+√5)/2 ≈ 1.6180339887... - Conjugate: ψ = (1-√5)/2 ≈ -0.6180339887... - Binet's formula: F_n = (φ^n - ψ^n)/√5 - Ratio convergence: limn→∞ Fₙ₊₁/F_n = φ - Rational approximations: Fₙ₊₁/F_n ∈ {1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, ...} → φ - Related ratios: 1/φ = φ-1 ≈ 0.618, 1/φ² ≈ 0.382, φ² ≈ 2.618, φ³ ≈ 4.236 CONNECTION: - 0.618 and 0.382 are the fundamental division ratios of the golden section (1:φ :: φ-1:1). - 0.786 = √(1/φ) ≈ √0.618, appears in phyllotaxis angle (≈137.5° = 360° × 0.382). - The convergence is a rational approximation sequence (continued fraction [1;1,1,1,...]), directly linking to base-60 sexagesimal approximations (e.g., 1.616... in Babylonian clay tablets). - Phyllotaxis patterns (e.g., sunflower seed Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com