The Fibonacci sequence A→AB, B→A generates tiles of two lengths L and S in ratio φ = (1+√5)/2.
The diffraction pattern (Fourier transform of the scatterer positions) shows Bragg-like peaks — sharp peaks at irrational spacings, not periodic.
Peak positions: q = (2π/a)(m + nφ) for integers m, n. The pattern is dense but not periodic — a quasiperiodic structure.
This 1D model captures the essential physics of Shechtman's (Nobel 2011) quasicrystal discovery: icosahedral Al-Mn alloy showed sharp diffraction spots with 5-fold symmetry forbidden by crystallographic periodicity.
Bottom panel: the cut-and-project construction — the Fibonacci tiling as a 1D slice of a 2D square lattice projected at angle arctan(1/φ).