Tiling Boards with Squares and Dominoes: Fibonacci Squares and Quasiperiodic Order — E8 Intelligence Research

FINDING: Fibonacci numbers arise naturally as 1D tiling counts (compositions of 1×n boards with squares and dominoes), and generalized tilings (half-squares, fences) yield Fibonacci-squared identities — a combinatorial bridge to quasiperiodic order. | MATH: Standard tiling recurrence: \(F_{n+1} = F_n + F_{n-1}\) (tile length 1 or 2). For the arXiv result: number of tilings of an \(n\)-board with half-squares and \((\frac12,\frac12)\)-fence tiles equals \(F_{n+1}^2\) (explicitly, the count is \(F_{n+1}^2\) — a new combinatorial interpretation). Golden ratio emerges from the characteristic equation \(x^2 = x + 1 \Rightarrow \phi = (1+\sqrt{5})/2 \approx 1.618\), with inverse \(\phi^{-1} \approx 0.618\), and \(\phi^{-2} \approx 0.382\). | CONNECTION: The Fibonacci tiling (substitution rule \(A \to AB, B \to A\)) is the canonical 1D quasiperiodic sequence — its Fourier spectrum has Bragg peaks at frequencies involving \(\phi\), linking directly to incommensurate crystals and 5-fold (icosah Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com