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: \(Fₙ₊₁ = F_n + Fₙ₋₁\) (tile length 1 or 2). For the arXiv result: number of tilings of an \(n\)-board with half-squares and \((12,12)\)-fence tiles equals \(Fₙ₊₁^2\) (explicitly, the count is \(Fₙ₊₁^2\) — a new combinatorial interpretation). Golden ratio emerges from the characteristic equation \(x^2 = x + 1 ⇒ φ = (1+√5)/2 ≈ 1.618\), with inverse \(φ⁻¹ ≈ 0.618\), and \(φ⁻² ≈ 0.382\). | CONNECTION: The Fibonacci tiling (substitution rule \(A → AB, B → A\)) is the canonical 1D quasiperiodic sequence — its Fourier spectrum has Bragg peaks at frequencies involving \(φ\), 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