Aperiodic set of prototiles

The Penrose tiles are an aperiodic set of tiles, since they admit only non-periodic tilings of the plane (see next image).

All of the infinitely many tilings by the Penrose tiles are aperiodic. That is, the Penrose tiles are an aperiodic set of prototiles.

A set of prototiles is aperiodic if copies of them can be assembled to create tilings, and all such tilings are non-periodic. Consequently, aperiodicity is a property of the set of prototiles; the tilings themselves are just non-periodic. Typically, distinct tilings may be obtained from a single aperiodic set of tiles..

The various Penrose tiles are the best-known examples of an aperiodic set of tiles. Today not many aperiodic sets of prototiles are known (here is a List of aperiodic sets of tiles). This is perhaps natural: 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.

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 which 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 (for example, randomly arranged tilings using a 2×2 square and 2×1 rectangle will typically be non-periodic). An aperiodic set of tiles, however, admits only non-periodic tilings.

History

The second part of Hilbert's eighteenth problem asked for a single polyhedron tiling Euclidean 3-space, such that no tiling by it is isohedral (an anisohedral tile). The problem as stated was solved by Karl Reinhardt in 1928, but sets of aperiodic tilies have been considered as a natural extension.

The specific question of aperiodic sets of tiles first arose in 1961, when logician Hao Wang tried to determine whether the Domino Problem is decidable — that is, whether there exists an algorithm for deciding if a given finite set of prototiles admits a tiling of the plane. Wang found algorithms to enumerate the tilesets that cannot tile the plane, and the tilesets that tile it periodically; by this he showed that such a decision algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling.

The above Wang tiles will yield only non-periodic tilings of the plane and so are aperiodic.

Hence, when in 1966 Robert Berger found an aperiodic set of prototiles this demonstrated that the tiling problem is in fact not decidable. (Thus Wang's procedures do not work on all tile sets, although that does not render them useless for practical purposes.) This first such set, used by Berger in his proof of undecidability, required 20,426 Wang tiles. Berger later reduced his set to 104, and Hans Läuchli subsequently found an aperiodic set requiring only 40 Wang tiles. The set of 13 tiles given in the illustration on the right is an aperiodic set published by Karel Culik, II, in 1996.

However, a smaller aperiodic set, of six non-Wang tiles, was discovered by Raphael M. Robinson in 1971. Roger Penrose discovered three more sets in 1973 and 1974, reducing the number of tiles needed to two, and Robert Ammann discovered several new sets in 1977.

In 1988, Peter Schmitt discovered a single aperiodic prototile in 3-dimensional Euclidean space. While no tiling by this prototile admits a translation as a symmetry, it has tilings with a screw symmetry, the combination of a translation and a rotation through an irrational multiple of π. This was subsequently extended by John Horton Conway and Ludwig Danzer to a convex aperiodic prototile, the Schmitt–Conway–Danzer tile. Because of the screw axis symmetry, this resulted in a reevaluation of the requirements for periodicity. Chaim Goodman-Strauss suggested that a protoset be considered strongly aperiodic if it admits no tiling with an infinite cyclic group of symmetries, and that other aperiodic protosets (such as the SCD tile) be called weakly aperiodic.

In 1996 Petra Gummelt showed that a single-marked decagonal tile, with two kinds of overlapping allowed, can force aperiodicity; this overlapping goes beyond the normal notion of tiling. An aperiodic protoset consisting of just one tile in the Euclidean plane, with no overlapping allowed, was proposed in early 2010 by Joshua Socolar; this example requires either matching conditions relating tiles that do not touch, or a disconnected but unmarked tile. The existence of a strongly aperiodic protoset consisting of just one tile in a higher dimension, or of a single simply connected tile in two dimensions without matching conditions, is an unsolved problem.

Constructions

There are a few constructions of aperiodic tilings known, even forty years after Berger's groundbreaking construction. Some constructions are of infinite families of aperiodic sets of tiles. Those constructions which have been found are mostly constructed in a few ways, primarily by forcing some sort of non-periodic hierarchical structure. Despite this, the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction, and that in fact, there exist aperiodic sets of tiles for which there can be no proof of their aperiodicity.

It is worth noting that there can be no aperiodic set of tiles in one dimension: it is a simple exercise to show that any set of tiles in the line either cannot be used to form a complete tiling, or can be used to form a periodic tiling. Aperiodicity of prototiles requires two or more dimensions.