Penrose Tilings were discovered by Roger Penrose in 1974. Using just two rhombus shapes — a fat rhombus (angles 72°/108°) and a thin rhombus (angles 36°/144°) — the plane can be tiled without ever repeating. The key rule: matching constraints on edges prevent periodic arrangements.
The tiling is generated by inflation: each tile is subdivided into smaller copies using the golden ratio φ = (1+√5)/2 ≈ 1.618. The ratio of fat to thin rhombi in any infinite tiling converges exactly to φ. Despite being aperiodic, the tiling has perfect five-fold rotational symmetry in its diffraction pattern — a property impossible for any periodic crystal, discovered in real quasicrystals by Shechtman (Nobel 2011).