Penrose tiling

A tiling covers a flat plane with shapes without gaps or overlaps. In a periodic tiling, the whole arrangement can be shifted by a fixed distance in one or more directions and match itself again. A Penrose tiling is non-periodic: although its pieces obey strict geometric rules and form an ordered pattern, the pattern has no translational period. Crucially, the tiles are not simply free to meet in any arrangement. Matching rules constrain how neighboring edges may fit, and those constraints prevent periodic repetition while still allowing the plane to be covered.

One widely used Penrose system consists of two quadrilateral shapes known as the kite and the dart. They can be derived by dividing a rhombus with angles of 72 and 108 degrees according to the golden ratio. Additional markings, such as colored arcs or edge rules, specify legal joins; matching only the lengths of edges is not enough. Other equivalent descriptions use two rhombi, or subdivide the tiles into triangles. These forms express the same underlying non-periodic structure in different geometric languages.

Penrose patterns can display local fivefold arrangements and may have reflection or rotational symmetries in particular examples, but they do not repeat by translation. A fivefold motif can therefore appear locally without making the whole plane a periodic crystal lattice. The ratio of tile types in large regions is governed by the golden ratio, an irrational number. This limiting proportion is one indication of why a periodic repetition cell cannot describe the pattern: a finite repeating cell would contain rational proportions of its tiles.

A Penrose tiling is self-similar. Through an inflation rule, a collection of smaller tiles can be grouped into larger kite-and-dart tiles; deflation applies the construction in reverse. Repeating either procedure produces related patterns at successively different scales. These substitution rules give a practical way to construct large patches and reveal how local edge constraints generate structure over long distances. Any finite patch that appears in one Penrose tiling recurs elsewhere, even though no whole tiling repeats periodically.

Roger Penrose developed small sets of tiles that force non-periodic tilings during the 1970s, reducing a six-tile construction to two shapes, the kite and dart. The mathematical patterns later became relevant to materials science. The Royal Swedish Academy of Sciences' scientific background for the 2011 Chemistry Nobel Prize notes that Penrose tilings helped establish the credibility of quasicrystals: a Penrose pattern has long-range order without translational periodicity, a combination useful for understanding how ordered physical structures can lack an ordinary repeating unit cell. Dan Shechtman received the 2011 Nobel Prize in Chemistry for discovering quasicrystals; a Penrose tiling is a mathematical model, not itself a physical quasicrystal.