Fibonacci Polyrhythms as Temporal Golden Ratio Self-Similarity — E8 Intelligence Research

FINDING: Fibonacci-based polyrhythms (1:2:3:5:8:13) and their inharmonic/golden-ratio variants reveal a direct temporal analogue of the golden ratio's spatial self-similarity, with combinatorial tiling interpretations of Fibonacci-squared numbers. MATH: - Core sequence: \(F_n = Fₙ₋₁ + Fₙ₋₂\), \(F_1=1, F_2=1\) → ratios \(Fₙ₊₁/F_n → φ = 1.618...\) - Polyrhythm ratios: 1:2, 2:3, 3:5, 5:8, 8:13 — each pair \((F_n, Fₙ₊₁)\) defines a rational approximation to \(φ\). - Inharmonic "Golden Rhythmicon": pitches and rhythms both follow \(F_n\), creating a simultaneous frequency and duration ratio of \(φ\) (or its inverse \(1/φ = 0.618...\)). - Combinatorial result (arXiv:1907.06517): Number of tilings of an \(n\)-board with half-squares and \((12,12)\)-fence tiles equals \(F_n^2\) — i.e., \(F_n^2\) counts these tilings, linking Fibonacci squares to a 2D lattice-like tiling structure. CONNECTION: - The ratio \(Fₙ₊₁/F_n\) converges to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com