Aperiodic order, 10-fold symmetry, and quasicrystal diffraction
8
Penrose
12
25
Quasicrystals (Shechtman 1984, Nobel 2011): long-range order without periodicity. Fibonacci sequence (substitution S→SL, L→SL) generates 1D quasicrystal with ratio φ=(1+√5)/2 ≈ 1.618. 2D Penrose tiling has 5-fold/10-fold symmetry. Diffraction: δ-function peaks at positions n+mφ (dense but countable), giving apparent 5-fold pattern impossible in periodic crystals.