Fibonacci Quasicrystal

1D quasiperiodic chain: projection from 2D lattice, diffraction, Cantor spectrum
Chain
Diffraction
Spectrum (IDOS)
Projection
Generation: 10
α (angle): 36.0°
Coupling ratio (t_A/t_B): 2.0
φ = (1+√5)/2 ≈ 1.6180… | Substitution: A→AB, B→A | F_n = F_{n-1}+F_{n-2}
The Fibonacci chain is the canonical 1D quasicrystal, obtained by projecting a 2D square lattice onto a line at the golden angle arctan(1/φ) ≈ 31.7°. Long (L) and Short (S) tiles satisfy L/S=φ. Tiles follow the substitution L→LS, S→L. The diffraction pattern shows sharp Bragg peaks at positions m+nφ (dense, but every peak is sharp — true long-range order). The energy spectrum of the associated Schrödinger operator is a Cantor set of measure zero (Sütő 1989).