Euclidean Windowed Tiling
A one dimensional substitution rule with a two component Rauzy Fractal. For a second example and more details see infinite component Rauzy fractal.
One Dimensional Euclidean Windowed Tiling Self Similar Substitution
A classic simple substitution rule with Rauzy Fractal:
One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles
The dual tiling of the 1D tiling a->ab, b->c, c->a, resp. the version with polygonal tiles.
Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogramm Tiles
In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings. These were published as late as 1987 in [GS87] , where they were named Ammann A2 (our Ammann Chair), Ammann A3, Ammann A4 and Ammann A5 (better known as Ammann …
Without Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution
A self-similar version of the Ammann-Benker tiling. The colours of the triangles in the rule image indicate the orientation of the triangles: the orange triangle is just the ochre triangle reflected. Hence the rhomb supertile has two axes of mirror symmetry.
With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogramm Tiles Self Similar Substitution
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 …
Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling One Dimensional Polytopal Tiles Parallelogram Tiles Self Similar Substitution Mld Class Fibonacci
Part of an infinite series, where most tilings in this series are not flc, this one is the exception.
The reason is that the inflation factor is a - real - PV number.
By an argument in [PR] this forces flc.
Interestingly, the shape of the tiles can vary.
That is, there is one free parameter $l$ , …
Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution
In order to generate the golden triangle tilings by matching rules, L. Danzer and G. van Ophuysen found this substitution for coloured prototiles. The list of its vertex stars serves as matching rules. For more details, see golden triangle and the references there.
Without Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Self Similar Substitution
This tiling and Nautilus are dual tilings generated by non-PV morphisms. As such they are the first step in a generalisation of the work of G. Rauzy, P. Arnoux, S. Ito and others for PV substitution rules. The work that developed out of G. Rauzy’s seminal paper [Rau82] .
The inflation factor for …
Finite Rotations Euclidean Windowed Tiling
A volume hierarchic version of Conch.
Finite Rotations Euclidean Windowed Tiling 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 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 …
Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Rhomb Tiles Self Similar Substitution
An invertible substitution rule with a disconnected Rauzy Fractal. For two letter substitution rules the Rauzy fractal is connected if and only if the substitution is invertible. In fact as the window is one dimensional for these tilings it is an interval. It was hoped that the connectedness …
One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles
Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles
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
The substitution $a \rightarrow ab, b \rightarrow cb, c \rightarrow a$ is the composition of the one with the smallest PV scaling factor, $a \rightarrow bc, b \rightarrow a, c \rightarrow b$, and its mirror image, $a \rightarrow cb, b \rightarrow a, c \rightarrow b$. As such, it is MLD to its own …
Euclidean Windowed Tiling One Dimensional Self Similar Substitution
The substitution $a \rightarrow aca, b \rightarrow a, c \rightarrow b$ has palindromic and thus mirror symmetric variant of the Kolakoski-(3,1) substitution, which is in the same MLD class, along with the further variants A (mirror symmetric) and B (with its mirror image). The scaling factor …
Euclidean Windowed Tiling One Dimensional Self Similar Substitution
The substitution $a \rightarrow bcc, b \rightarrow ba, c \rightarrow bc$ is a member of the MLD class of the [Kolakoski-(3,1) sequence] (/substitution/kolakoski-3-1/). As the reversed substitution generates the same hull, it is mirror symmetric. The scaling factor $\lambda \approx $ 2.20557 is the …
Euclidean Windowed Tiling One Dimensional Self Similar Substitution
The substitution $a \rightarrow abcc, b \rightarrow a, c \rightarrow bc$ is a member of the MLD class of the [Kolakoski-(3,1) sequence] (/substitution/kolakoski-3-1/). The scaling factor $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$.
This substitution has a simple dual, with three …
Euclidean Windowed Tiling One Dimensional Self Similar Substitution
The substitution $a \rightarrow abc, b \rightarrow ab, c \rightarrow b$ is closely related to the Kolakoski-(3,1) sequence, and is one of the examples whose windows (dual tiles, Rauzy fractals) have been analysed in detail [BaS04] . It is MLD to the mirror symmetric variant given by the palindromic …
Euclidean Windowed Tiling One Dimensional Self Similar Substitution
A volume hierarchic version of Nautilus
Finite Rotations Euclidean Windowed Tiling Self Similar Substitution
A companion to infinite component Rauzy fractal. As mentioned for that rule, it was hoped that the result for two symbol substitution rules that the window is connected if and only if the rule is invertible. This substitution rules is not invertible and yet the Rauzy fractal is connected:
Euclidean Windowed Tiling One Dimensional Polytopal Tiles Self Simmilar Substitution
A substitution rule where the tiles are allowed to overlap. The image left indicates, that the yellow and the green tiles do overlap. It is unknown whether these tilings are mld to the Penrose Rhomb tilings.
Finite Rotations Euclidean Windowed Tiling Polytopal Tiles
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 by F. Gähler, in particular its cut and project scheme, the local matching rules and diffraction properties [Gah88]. The window of …
With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield and Socolar Matching Rules
Finite Rotations Euclidean Windowed Tiling Polytopal Tiles Polytopal Windowed Tiling Canonical Substitution Tiling Parallelogram Tiles Plastic Number
The three letter substitution rule whose scaling is the smallest PV number, the Plastic Number which is a root of the polynomial $x^3 - x - 1 = 0$.
Though it might not look it at first glance, the Rauzy fractal is connected.
This can be shown using the method of A. Siegel described in [Sie04].
The Rauzy fractal:
One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal 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 …
Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Parallelogram Tiles Canonical Substitution Tiling Mld Class Shield and Socolar Matching Rules
The three letter substitution rule analysed by G. Rauzy in [Rau82] . The Rauzy fractal for this tiling is the Rauzy fractal.
Euclidean Windowed Tiling One Dimensional Polytopal Tiles Self Similar Substitution
Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Canonical Substitution Tiling Polytopal Tiles Parallelogram Tiles Self Similar Substitution
There is a very simple rule to transform the wheel tiling into the shield tiling: Replace each edge in the tiling by an edge orthogonal to it, of equal length, such that the old and new edge intersect in their midpoints. Applying this rule to the wheel tiling yields the shield tiling and vice versa. This is a very simple example of tilings which are mld.
With Decoration Finite Rotations Euclidean Windowed Tiling Polytopal Windowed Tiling Polytopal Tiles Mld Class Shield and Socolar