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 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

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

An one-dimensional substitution rule that uses an infinite number of prototiles. The inflation factor is $2.5$. The substitution rules are given by: $T_{0}\rightarrow T_{0},T_{1}$ $T_{1}\rightarrow T_{0},T_{0},T_{2}$ $T_{2}\rightarrow T_{0},T_{1},T_{3}$ $T_{k}\rightarrow T_{0},T_{k-1},T_{k+1}$ $T_{\infty}\rightarrow T_{0},T_{\infty},T_{\infty}$ The corresponding substitution matrix can be written as: $1 2 1 1 1 1 1 1 1 ...$ $1 0 1 0 0 0 0 0 0 ...$ $0 1 0 1 0 0 0 0 0 ...$

One Dimensional 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 property extended to the higher dimensional case. Unfortunately, as this example shows, this is not the case. A second example, with just two components is 2-component Rauzy fractal.

One Dimensional Euclidean Windowed Tiling Self Similar Substitution Polytopal Tiles

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 mirror image, $a \rightarrow ba, b \rightarrow bc, c \rightarrow a$. The scaling factor $\lambda \approx$ 1.7549 is the largest root of $x^3-2x^2+x-1=0$.

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 $\lambda \approx $ 2.20557 is the largest root of $x^3-2x^2-1=0$. This substitution has a simple dual, with three mildly fractal tiles, which are all similar to each other.

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. As the reversed substitution generates the same hull, it is mirror symmetric. 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 mildly fractal tiles, which are all similar to each other. The dual substitution scales by about 1.485, and rotates clockwise by about 81.22°.

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. 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 mildly fractal tiles, which are all similar to each other. The dual substitution scales by about 1.485, and rotates clockwise by about 81.22°.

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 substitution $a \rightarrow aca, b \rightarrow a, c \rightarrow b$. As a consequence, the Kolakoski-(3,1) substitution is MLD to its mirror image, even though it is not mirror symmetric itself.

Euclidean Windowed Tiling One Dimensional 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

One Dimensional Self Similar Substitution Polytopal Tiles

In some sense, the simplest cut and project tiling. It arises from the symbolic substitution a -> ab, b -> aa. Its internal space are the 2-adic integers.

P Adic Windowed Tiling One Dimensional Rep Tiles Self Similar Substitution

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

A classic. A lot of detail can be found in [JS99].

One Dimensional Self Similar Substitution

An one-dimensional substitution rule that uses an infinite number of proto tiles and yields a transcendental inflation multiplier. The inflation factor is approximately $2.7899$. The substitution rules are given by: $T_{0}\rightarrow T_{0},T_{1}$ $T_{1}\rightarrow 3T_{0},T_{2}$ $T_{2}\rightarrow 2T_{0},T_{1},T_{3}$ $T_{3}\rightarrow T_{0},T_{2},T_{4}$ $T_{4}\rightarrow 2T_{0},T_{3},T_{5}$ $T_{5}\rightarrow T_{0},T_{4},T_{6}$ $T_{6}\rightarrow T_{0},T_{5},T_{7}$ $T_{k}\rightarrow (1+f\left(k\right))T_{0},T_{k-1},T_{k+1}$ with $f\left(k\right)$ as the Thue-Morse sequence. The corresponding substitution matrix can be written as: $1 3 2 1 2 1 1 2 2 ...$

One Dimensional Self Similar Substitution

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