Aperiodic set of prototiles

rdf:langString Aperiodic set of prototiles

rdf:langString A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic. A given set of tiles, in the Euclidean plane or some other geometric setting, admits a tiling if non-overlapping copies of the tiles in the set can be fitted together to cover the entire space. A given set of tiles might admit periodic tilings — that is, tilings that remain invariant after being shifted by a translation (for example, a lattice of square tiles is periodic). It is not difficult to design a set of tiles that admits non-periodic tilings as well as periodic tilings. (For example, randomly arranged tilings using a 2×2 square and 2×1 rectangle are typically non-periodic.) However, an aperiodic set of tiles can only produce non-periodic tilings. Infinitely many distinct tilings may be obtained from a single aperiodic set of tiles. The best-known examples of an aperiodic set of tiles are the various Penrose tiles. The known aperiodic sets of prototiles are seen on the list of aperiodic sets of tiles. The underlying undecidability of the domino problem implies that there exists no systematic procedure for deciding whether a given set of tiles can tile the plane.