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.

A set of prototiles is called an aperiodic set, if the prototiles allow only nonperiodic tilings. See also matching rules.

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.

A plane tiling generates a dynamical system (X,R²), where X is the closure (w.r.t. a certain topology) of the orbit of the tiling under the actions of R². This X is called the hull of the tiling.

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.

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).

Dealing with tilings, it is useful to consider finite parts of a tiling. These are called patches (or clusters). The definition is simply: a patch is a finite subset of a tiling. Sometimes one requires, in addition, that the support of a patch should be connected, or homeomorphic to a ball. The latter leads to problems in higher dimensions. There are tilings where no patch with more than one tile is homeomorphic to a ball (although the prototiles are), see Fig. 3 in [Fre02].

A non-negative Matrix M is called primitive, if there is a power of M which has strictly positive entries. A substitution is called primitive, if the corresponding substitution matrix is primitive.

The tiles, which serve as building blocks for tilings. For more details, see tiling.

An algebraic integer q is a PV number (for Pisot-Vijayaraghavan number), if all its algebraic conjugates (except its complex conjugate, if q is complex) are less than one in modulus. A profound theorem by Meyer states essentially, that a substitution tiling can only be a model set, if the inflation factor is a PV number [Mey95].

Essentially, a Rauzy fractal is just the window set of a cut and project scheme. But usually this term - or the term 'generalized Rauzy fractal' - is only used for fractally shaped Euclidean windows in dimension 2, arising from one-dimensional substitutions. To be precise, originally it was coined for the particular window arising from the tribonacci substitution, which dynamical properties were studied by G. Rauzy in [Rau82].

A tiling T is called repetitive, if for every r>0 there is R>0, such that a copy of every r-patch in T is contained in every R-patch in T. In plain words, this means that each local part of the tiling occurs 'everywhere' in the tiling. In even plainer words: If you stand on a repetitive tiling, then your local surrounding do not tell you the in the slightest way where you are, even if you have a map of the whole tiling:

A substitution (at least here) consists of rules how to enlarge a tile and replace the enlarged tile with other tiles. If the union of the latter ones is similar to the original tile, then the substitution is called self-similar substitution. For example, the substitution for the Penrose Rhombs is not self-similar, but the substitution for the Robinson Triangles is.

In other words, a substitution is a self-similar substitution, if σ(T)=T. A substitution tiling is called self-similar, if it can be generated by a self-similar substitution. It is known that any - sufficiently nice, i.e., repetitive and flc wrt translations - tile substitution in the plane can be made self-similar, by using fractal boundaries.

A weaker version is described by the term 'self-affine' tiling [LW96], [BG94]. The definition of this reads exactly as above if one replaces 'similar to' with 'affine image of'.

In general, even restricted to mathematics, the term substitution can have several meanings. In connection with tilings, it describes a simple but powerful method to produce tilings with many interesting properties. The main idea is to use a finite set of building blocks {T₁, T₂ ... T_m} (the prototiles), an expanding linear map Q (the inflation factor) and a rule, how to dissect each scaled tile QT_i into copies of the original prototiles T₁, T₂...T_m. This information can be encoded in terms of parametrized tiles and affine maps (see example below), or, more appealing, in a diagram. In our encyclopedia, we use the latter method. Essentially, from such a diagram one can extract all needed information about the substitution. An important object in this context is the substitution matrix, which contains quite a lot information about the corresponding tilings.

To a substitution s with prototiles T₁, ... T_m we assign the substitution matrix M_s = (m_ij)_{i,j = 1,...m}, where m_ij is the number of copies of T_i in s(T_j).

The n-th iterate sⁿ(T) of a prototile under a substitution s.

This is not a mathematical term in this context. By taxonomy we mean the classification of the tilings with respect to the following main categories:

For our purposes, a tile in R^d is defined as a nonempty compact subset of R^d which is the closure of its interior. Sometimes a tile is required to be connected, or to be homeomorphic to a closed ball.

A tiling T in R^d is a countable set of tiles, which is a covering as well as a packing of R^d. I.e., the union of all tiles in T is R^d, and the intersection of the interior of two different tiles in T is empty.

For any tiling T, the translation module is the Z-span of all translations t, such that there is a tile T in T, and T+t is also in T.

One can start a long discussion about the definition of a vertex in a tiling. We leave that to others and define a vertex in a plane tiling as a point in R² which is contained in more than two tiles of the tiling.

A vertex star is the 0-corona of a vertex of a tiling. In other words, the vertex star of a vertex x in a tiling T is the set of all tiles in T which have nonempty intersection with {x}.

The compact set in the internal space of a model set which tells which lattice points are projected. For details, see cut and project.