An aperiodic tiling is a tiling obtained from an aperiodic set of tiles. A set of tile-types (or prototiles) is aperiodic if there are some tilings using only these types, and all such tilings are non-periodic. Properly speaking, aperiodicity is a property of the set of prototiles; the tilings themselves are merely non-periodic. Typically, distinct tilings may be obtained from a single aperiodic set of tiles.
Quasicrystals — physical materials with the apparent structure of the Penrose tilings — were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood.
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.
Few methods for constructing aperiodic tilings are known. 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.
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 aperiodic tilings have been considered as a natural extension.
The specific question of aperiodic tiling 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.
Hence, when in 1966 Robert Berger demonstrated that the tiling problem is in fact not decidable, it followed logically that there must exist an aperiodic set of prototiles. (Thus Wang's procedures do not work on all tile sets, although that does not render them useless for practical purposes.) The first such set, presented by Berger and used 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.
There are remarkably few constructions of aperiodic sets of tiles 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 requires two or more dimensions.
Aperiodic hierarchical tilings
To date, there is not a formal definition describing when a tiling has a hierarchical structure; nonetheless, it is clear that substitution tilings have them, as do the tilings of Berger, Knuth, Läuchli and Robinson. As with the term "aperiodic tiling" itself, the term "aperiodic hierarchical tiling" is a convenient shorthand, meaning something along the lines of "a set of tiles admitting only non-periodic tilings with a hierarchical structure".
Each of these sets of tiles, in any tiling they admit, forces a particular hierarchical structure. (In many later examples, this structure can be described as a substitution tiling system; this is described below). No tiling admitted by such a set of tiles can be periodic, simply because no single translation can leave the entire hierarchical structure invariant. Consider Robinson's 1971 tiles:
Any tiling by these tiles can only exhibit a hierarchy of square lattices: each orange square is at the corner of a larger orange square, ad infinitum. Any translation must be smaller than some size of square, and so cannot leave any such tiling invariant.
Robinson proves these tiles must form this structure inductively; in effect, the tiles must form blocks which themselves fit together as larger versions of the original tiles, and so on. This idea — of finding sets of tiles that can only admit hierarchical structures — has been used in the construction of most known aperiodic sets of tiles to date.
Substitution tiling systems provide a rich source of hierarchical non-periodic structures; however the substituted tiles themselves are not typically aperiodic. A set of tiles that forces a substitution structure to emerge is said to enforce the substitution structure. For example, the chair tiles shown below admit a substitution, and a portion of a substitution tiling is shown at right below. These substitution tilings are necessarily non-periodic, in precisely the same manner as described above, but the chair tile itself is not aperiodic—it is easy to find periodic tilings by unmarked chair tiles.
However, the tiles shown below force the chair substitution structure to emerge, and so are themselves aperiodic.
The Penrose tiles, and shortly thereafter Amman's several different sets of tiles, were the first example based on explicitly forcing a substitution tiling structure to emerge. Joshua Socolar, Roger Penrose, Ludwig Danzer, and Chaim Goodman-Strauss  have found several subsequent sets. Shahar Mozes gave the first general construction, showing that every product of one-dimensional substitution systems can be enforced by matching rules. Charles Radin found rules enforcing the Conway-pinwheel substitution tiling system. In 1998, Goodman-Strauss showed that local matching rules can be found to force any substitution tiling structure, subject to some mild conditions.
Non-periodic tilings can also be obtained by projection of higher-dimensional structures into spaces with lower dimensionality and under some circumstances there can be tiles that enforce this non-periodic structure and so are aperiodic. The Penrose tiles are the first and most famous example of this, as first noted in the pioneering work of de Bruijn. There is yet no complete (algebraic) characterization of cut and project tilings that can be enforced by matching rules, although numerous necessary or sufficient conditions are known.
Only a few different kinds of constructions have been found. Notably, Jarkko Kari gave an aperiodic set of Wang tiles based on multiplications by 2 or 2/3 of real numbers encoded by lines of tiles (the encoding is related to Sturmian sequences made as the differences of consecutive elements of Beatty sequences), with the aperiodicity mainly relying on the fact that 2^n/3^m is never equal to 1 for any positive integers n and m. This method was later adapted by Goodman-Strauss to give a strongly aperiodic set of tiles in the hyperbolic plane. Shahar Mozes has found many alternative constructions of aperiodic sets of tiles, some in more exotic settings; for example in semi-simple Lie Groups. Block and Weinberger used homological methods to construct aperiodic sets of tiles for all non-amenable manifolds. Joshua Socolar also gave another way to enforce aperiodicity, in terms of alternating condition. This generally leads to much smaller tile sets that the one derived from substitutions.
Physics of aperiodic tilings
Aperiodic tilings were considered as mathematical artefacts until 1984, when physicist Dan Shechtman announced the discovery of a phase of an aluminium-manganese alloy which produced a sharp diffractogram with an unambiguous fivefold symmetry – so it had to be a crystalline substance with icosahedral symmetry. In 1975 Robert Ammann had already extended the Penrose construction to a three dimensional icosahedral equivalent. In such cases the term 'tiling' is taken to mean 'filling the space'. Photonic devices are currently built as aperiodical sequences of different layers, being thus aperiodic in one direction and periodic in the other two. Quasicrystal structures of Cd-Te appear to consist of atomic layers in which the atoms are arranged in a planar aperiodic pattern. Sometimes an energetical minimum or a maximum of entropy occur for such aperiodic structures. Steinhardt has shown that Gummelt's overlapping decagons allow the application of an extremal principle and thus provide the link between the mathematics of aperiodic tiling and the structure of quasicrystals. Faraday waves have been observed to form large patches of aperiodic patterns. The physics of this discovery has revived the interest in incommensurate structures and frequencies suggesting to link aperiodic tilings with interference phenomena.
Confusion regarding terminology
The terms non-periodic, quasiperiodic and aperiodic have been used in a wide variety of ways in a wide variety of fields, leading to considerable confusion. Moreover, the word "tiling" itself is quite problematic.
In the context of 'Aperiodic tiling', a non-periodic tiling is simply one with no period, as discussed above, and aperiodicity is a property of tiles: a set of tiles is aperiodic if and only if it admits only non-periodic tilings. There is no mathematical concept of aperiodic tiling per se. Quasiperiodic tilings, generally, mean those obtained by the cut-and-project method; however William Thurston's influential lecture notes  used the term to mean repetitive tilings. The Penrose tiles themselves are a source of much of the confusion, for the tilings they admit are quasiperiodic, in both senses, and non-periodic, and they themselves are aperiodic.
Moreover the terms aperiodic, non-periodic and quasiperiodic are widely used in other fields, such as dynamical systems, with altogether different meanings; and there is much literature on tilings in which, inappropriately, the distinction is not made. It is important to note however, that the core results of the field simply are not meaningful without this careful delineation.
The word "tiling" is problematic as well, despite its straightforward definition. There is no single Penrose tiling, for example: the Penrose rhombs admit infinitely many tilings (which cannot be distinguished locally) and even established figures in the field informally refer to "aperiodic tiling", knowing full well that this is not technically defined. A common solution is to try to use the terms carefully in technical writing, but recognize the widespread use of the informal terms.
- Actually, a set of aperiodic prototiles can always form uncountably many different tilings, even up to isometry, as proven by Nikolaï Dolbilin in his 1995 paper The Countability of a Tiling Family and the Periodicity of a Tiling
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