In some substitution tilings it is possible to decorate the prototiles with line segments such that they produce a grid of straight lines extending over the whole tiling. Here is an example for the case of the Penrose Rhomb tiling.

There are disputations among the experts how to define "aperiodic". One possibility is to use it synonymously with nonperiodic. This is somehow a waste of this term. Others refer to an "aperiodic tiling" as one, which is created by an aperiodic set of tiles. This is unsatisfactory since this is rather a property of the set of prototiles than the tiling itself. Another definition is: A tiling is aperiodic, if its hull contains no periodic tiling. Personally, I like the latter definition (DF). Then a sequence ...aaaaabaaaaaaa.... is not aperiodic (since its hull comtains the periodic sequence ....aaaaaaaaa....), but the Fibonacci sequence is aperiodic.

see patch.

Given a tile T in a tiling T, the 0-corona of T is just C₀(T)={T}. The n-corona of T (n>0) is C_n(T)={ S in T | S has nonemtpy intersection with some tile in C_n-1(T) }. Coronae can also be defined by starting with other objects in a tiling, like coronae of clusters, edges, vertices.... rather than tiles. The 1-corona of a vertex is also called vertex star.

A fundamental result in the theory of nonperiodic tilings was the discovery of the fact that some substitution tilings can be obtained by projecting certain points from higher dimensional point lattices. This was first carried out by deBruijn for the Penrose Rhomb tilings [de 81]. In the following years it was developed further by a lot of authors (we cannot list all of them; for a start, see the references in [Moo00] and [Fog02]). This work culminated in the development of the algebraic theory of model sets. Quickly it was realized that this theory is a reformulation of the work of Meyer [Lag96], [Moo97].

A point set S in R^d is called a Delone set, if it is uniformly discrete and relatively dense; i.e., if there are numbers R>r>0, such that each ball of radius r contains at most one point of S, and every ball of radius R contains at least one point of S.

One way to compute the window of a model set (resp. the Rauzy fractal of a tiling) is to consider the 'lifted' versions of the expanding maps of the substitution (which are automorphisms of the high dimensional lattice L) and their counterparts in the internal space. These are contracting maps yielding an iterated function system. The latter is known to have a unique compact nonempty solution. Multiplying the whole iterated function system by an appropriate factor yields the dual substitution. This was outlined in [Thu89], see also [Gel97], [Fre05].

A tiling has finite local complexity (flc) if it contains only finitely many types of patches with diameter less than some given R>0. 'Types of patches' is to be read either as congruence classes of patches, or as translation classes of patches. For instance, the pinwheel tiling is not flc w.r.t. translation classes, but it is flc w.r.t. congruence classes. In our classification, this qualifies the pinwheel tiling to be stored under infinite rotations, and there under finite local complexity.

The linear map that gives the scaling for a substitution rule, before the replacement by new tiles. Often, the linear map is just a scaling by a real number, or, in the plane case - where R² is identified with the complex plane - a multiplication by a complex number. Then this number is called the inflation factor.

The space where the window of a model set lives in. For details, see cut and project.

The equivalence class of tilings induced by the relation locally indistinguishable.

A nonperiodic tiling is called limitperiodic, if it is the union of countably many periodic patterns (up to a set of zero density). It is quite easy to see that this can only be the case if the inflation factor (or a power of it) is an integer number.

Two tilings are called locally indistinguishable, if a copy of each patch of one tiling occurs in the other tiling and vice versa. For instance, any two tilings arising from the same primitive substitution s are locally indistinguishable.

Many interesting substitution tilings can alternatively be generated by a matching rule. Examples are (again) the Penrose Rhomb tilings: Note the red arcs on each tile. The matching rule is given by the condition that tiles have to meet in a way such that the arcs of each tile are connected with arcs on the neighboured tiles.

Two tilings are called mld (mutually locally derivable), if one is obtained from the other in a unique way by local rules, and vice versa. For example, a tiling by Penrose Rhomb is obtained from a Robinson Triangle tiling easily: just delete the shortest and longest edges, keeping only the medium ones yields the Penrose Rhomb tiling; and vice versa: in a Penrose Rhomb tiling, add in each fat rhomb the long diagonal, and in each thin rhomb add the short diagonal. This gives again the Robinson Triangle tiling.

A model set is a discrete point set (more precisely, a Delone set) arising from a cut and project scheme. A theorem of Hof, generalized by Schlottmann, states that each model set is pure point diffractive [Hof95], [Sch00]. In connection with quasicrystals, it is of interest if a substitution tiling can serve as a model for a physical quasicrystal. Since physical quasicrystals are detected via their diffraction properties, this leads to the question whether a substitution gives rise to a model set. This is true for many well-known substitution tilings, in particular for the Penrose Rhomb tilings and the Ammann-Beenker tilings. These are mld with model sets with Euclidean internal spaces and with polytopal windows. Other model sets may arise from Euclidean internal spaces and fractally shaped windows, like the conch or the tribonacci tilings. There are also tilings corresponding to model sets with p-adic internal spaces - like the chair and the sphinx tilings - as well as tilings corresponding to model sets with mixed p-adic and Euclidean internal spaces, like the Watanabe Ito Soma 8-fold tilings.

Substitutions - in particular 1dim substitutions - with m prototiles can be formulated as a endomorphism of the free group with m generators. If the endomorphism is an automorphism, i.e., if it is invertible, then the substitution is also called invertible.

see mld

A tiling T is called nonperiodic, if from T + x = T it follows that x=0. In other words, if no translation (other than the trivial one) maps the tiling to itself. In the theory of nonperiodic tilings usually the repetitive ones are the objects to be investigated.

The internal space of a cut and project scheme is required to be a locally compact Abelian group. There are not too much locally Abelian groups out there. Besides the reals, there are the fields of p-adic numbers Q_p. Indeed, it turns out that some substitution tilings - like the chair and the sphinx - are model sets with p-adic internal spaces (In these two examples: H=Q₂ × Q₂). Others - like the Watanabe Ito Soma 8-fold tilings - are model sets with respect to products of Euclidean and p-adic internal spaces (here, H=R² × Q₂).