Penrose P2 tiling (Roger Penrose, 1974): two tile shapes — kite and dart —
tile the plane aperiodically. No translation maps the tiling to itself.
The ratio of kites to darts → φ = (1+√5)/2 (golden ratio).
Inflation: each tile subdivides into smaller kites/darts by a factor of 1/φ.
Discovered to describe real materials: quasicrystals (Shechtman 1984, Nobel 2011) have
icosahedral symmetry — Penrose-like long-range order without periodicity.
The tiling has 5-fold symmetry but no 5-fold periodic lattice.