Penrose tilings are aperiodic — they tile the plane without repeating. Built from two tiles (kite/dart or thick/thin rhombs), they exhibit 5-fold quasicrystalline symmetry. The deflation rule subdivides each tile into smaller copies, revealing self-similarity at every scale. The ratio of tile counts converges to φ = (1+√5)/2.
P3 Penrose tiling: thick rhombs (36°) and thin rhombs (72°). Inflation ratio = φ. The tiling has long-range 5-fold orientational order but no translational periodicity — a quasicrystal in 2D. Shechtman's Al-Mn quasicrystals (1984, Nobel 2011) show these diffraction patterns.