One of the tilings discovered R. Ammann in 1977, when he found several sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings. These were published much later, in 1987, in [GS87] , where they were named Ammann A2 (our Ammann Chair), Ammann A3, Ammann A4, and Ammann A5 (better known as Ammann Beenker). The A4 tilings are mld to the well-known Ammann Beenker tilings. Thus they share most properties with the latter.

With Decoration Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Self Similar Substitution Mld Class Ammann

The substitution rule a1->a1 b1, a2->b2 a2, b1->a2, b2->a1. The tilings generated become Fibonacci tilings under the projection a1,a2->a and b1,b2->b. Alternatively one can simply remove the colour labels on the tiles. The name comes from the projection structure of the tiling. The expansion predecessor of the tiling is itself a projection tiling with the window lying at the center of the window for the full tiling. For more information see [HL].

Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling One Dimensional Polytopal Tiles Parallelogram Tiles Self Similar Substitution Mld Class Fibonacci

A simple variant of the domino tilings (aka table tilings). C. Goodman-Strauss pointed out in [Goo98] the following. B. Solomyak proved in Sol98, that for each nonperiodic substitution tiling the substitution rule is invertible: One can tell from $\sigma(T)$ its predecessor $T$ uniquely. But this is true only if the prototiles have the same symmetry group as the first order supertiles. By using decorated tiles this can always be achieved. (And now Chaims remark:) Here we see a case where such a decoration is necessary.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Parallelogram Tiles Polyomio Tiling Rep Tiles Self Similar Substitution

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles

In his PhD thesis, E. Harriss classified all substitution tilings which are canonical projection tilings. Here one example is shown, derived from the cut and project scheme of the Ammann-Beenker tilings.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

The classical example to explain the cut and project method (see figure, lower part): In the standard square lattice $\mathbb{Z}^2$, choose a stripe with slope $\frac{1}{\tau}$ (where tau is the golden ratio $\frac{1+\sqrt{5}}{2}$ ) of a certain width $\cos(\arctan(\frac{1}{\tau})) + \sin(\arctan(\frac{1}{\tau})) = \frac{1+\tau}{\sqrt{2+\tau}}$. Then take all lattice points within the strip and project them orthogonally to a line parallel to the strip. This yields a sequence of points. There are two values of distances between neighboured points, say, $S$ (short) and $L$ (long).

Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling One Dimensional Parallelogram Tiles Self Similar Substitution Mld Class Fibonacci

The 2dim analogue of the famous Fibonacci tiling in one dimension. It is just the Cartesian product of two Fibonacci tilings $F_{1}$, $ F_{2} : \{ T_{1} \times T_{2}\ |\ T_{i}\ in\ F_{i}\}$. Obviously, it can be generated by a substitution with three prototiles. It shares a lot of nice features with the 1dim Fibonacci tiling: It is a model set (better: it’s mld with one), so it has pure point spectrum.

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution

A simple variant of Fibonacci times Fibonacci, the latter arising from the one-dimensional Fibonacci tiling.

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution

Whereas it is simple to generate rhomb tilings with n-fold symmetry by the cut and project method, it can be hard to find a substitution rule for such tilings. Here we see a rule for n=7. This one was later generalized by E. Harriss to arbitrary n.

Finite Rotations Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Harrisss Rhomb

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles

A polygonal version of Kenyon (1,2,1). The boundary is generated by the morphism $a \to b, b \to c, c \to c a' b' b'$ (where $x'$ is the inverse of $x$).

Finite Rotations Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Kenyons Construction

A polygonal version of Kenyon 2. The edges are generated by the morphism: a->b, b->c, c->d, d-> b’a’ (where x’ is the inverse of x).

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Parallelogram Tiles Kenyon'S Construction

A simple substitution, yielding a tiling which is not of finite local complexity (flc). The substitution maps the single prototile, a unit square, to three columns of three squares each, where the third column is shifted by an irrational amount t. In higher iterates of the substitution, there are neighboured squares shifted against each other by t, 3t+t, 9t+3t+t,… mod 1. Since t is irrational, these sequence contains infinitely many values (mod 1), thus there are infinitely many pairwise incongruent pairs of tiles.

Polytopal Tiles Parallelogram Tiles Rhomb Tiles

A substitution tiling with inflation factor sqrt(3), using a single prototile, namely a 60º rhomb. The substitution sends one rhomb to seven rhombs (instead of three, as one would expect from the inflation factor), thus the tiles in higher iterations do overlap. But the substitution is chosen in a way such that tiles do either overlap completely, or not at all. So overlapping tiles can be identified, and the substitution yields a proper tiling.

Finite Rotations P Adic Windowed Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles

A plane substitution tiling which does not possess flc. It arises from the 1-dimensional substitution a -> abbb, b -> a, which inflation factor is not a PV-number. In the last sentence, ‘arises’ is to be understood as follows: Whenever one has a 1-dimensional substitution, it defines a d-dimensional substitution just by taking the Cartesian product. For an example, see Fibonacci times Fibonacci. Then, a 1-dimensional cut through each such d-dimensional tiling along a direction of some edge is the 1-dimensional tiling itself.

Polytopal Tiles Parallelogram Tiles Self Similar Substitution

This one is mld to the semi-detached house tiling. A view at the latter (hopefully) explains the name. This version was realized in order to prove (or disprove) that the semi detached house tiling is a cut and project tiling with p-adic internal space. This is not the case, as was shown in [FS].

Finite Rotations Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution

Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Plastic Number

In connection with physical quasicrystals, the most interesting 2dim tilings are based on 5-, 8-, 10- and 12-fold rotational symmetry. This 12-fold tiling was studied thoroughly in [Soc89], where J. Socolar described the generating substitution as well as the local matching rules and the cut and project scheme, As well as the Penrose Rhomb tilings (5- resp. 10-fold) and the Ammann-Beenker tilings (8-fold), it allows a decoration by Ammann bars (see [GS87]).

Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Canonical Substitution Tiling Mld Class Shield And Socolar Matching Rules

MLD to the more popular chair tiling, this version allows a simple translation into a coloured lattice: Replace each square of type i (1,2,3, or 4) with its midpoint, and assign to it colour i. Then each set of all points of colour i is a model set with internal p-adic space with p=2. This was first shown in [BMS98], a general framework is given in [LMS03].

With Decoration Finite Rotations P Adic Windowed Tiling Polytopal Tiles Self Similar Substitution Parallelogram Tiles Rhomb Tiles Mld Class Chair

Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Self Similar Substitution