Fibonacci Tilings and Aperiodic Rhythms: A Combinatorial Interpretation — E8 Intelligence Research

FINDING: Fibonacci-based tilings and rhythms produce aperiodic, highly structured sequences with no repeating bar pattern, and a new combinatorial tiling interpretation of Fibonacci numbers squared using half-square and fence tiles. MATH: - Fibonacci recurrence: \(F_n = Fₙ₋₁ + Fₙ₋₂\), with \(F_0=0, F_1=1\). - Tiling result (arXiv:1907.06517): Number of tilings of an \(n\)-board with half-squares (\(12 × 1\)) and \((12,12)\)-fence tiles (two half-squares separated by a \(12\) gap) equals \(F_n^2\). - Rhythm structure: Metrical intervals follow Fibonacci numbers — e.g., durations in units of \(F_k\) produce a self-similar, non-periodic pulse train. - Golden ratio emerges in the limit: \(limn→∞ Fₙ₊₁/F_n = φ = 1.6180339887...\) - Related ratios: \(φ⁻¹ = 0.618\), \(φ⁻² = 0.382\), \(φ-1/2 ≈ 0.786\), \(φ^2 = 2.618\). CONNECTION: - The Fibonacci tiling is the 1D canonical example of a quas Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com