PenroseTilings

Penrose tilings are a family of non-periodic tilings of the plane devised by Roger Penrose in the 1970s. They use a small set of prototiles and matching rules that enforce aperiodicity. Two widely studied versions are the kite-and-dart tiling (P2) and the rhomb tiling with thick and thin rhombs (P3); a third version, P1, uses pentagonal prototiles. In all cases, the tiles are equipped with markings or constraints that restrict how they may meet, ensuring the resulting tiling cannot repeat translationally.

A key feature of Penrose tilings is their inflation or substitution property: tiles can be grouped into larger clusters and replaced by a new set of tiles scaled by the golden ratio, allowing the tiling to be generated recursively at multiple scales. The tilings exhibit long-range order despite the lack of periodicity, characterized by hierarchical structure and local configurations that recur throughout the plane. They also display a form of fivefold quasi-symmetry in large regions, though the overall tiling does not possess rotational symmetry.

Penrose tilings have had a significant impact beyond mathematics, notably as a simple mathematical model for quasicrystals—structures that show sharp diffraction patterns without translational symmetry. The study of these tilings contributed to tiling theory, including the concept of tiling spaces and connections to dynamical systems, and inspired alternative construction methods such as projections from higher-dimensional lattices.