Rauzy fractal

In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution


 * $$s(1)=12,\ s(2)=13,\  s(3)=1 \,.$$

It was studied in 1981 by Gérard Rauzy, with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal set can be generalized to other maps over a 3-letter alphabet, generating other fractal sets with interesting properties, such as periodic tiling of the plane and self-similarity in three homothetic parts.

Tribonacci word
The infinite tribonacci word is a word constructed by iteratively applying the Tribonacci or Rauzy map : $$s(1)=12$$, $$s(2)=13$$, $$s(3)=1$$. It is an example of a morphic word. Starting from 1, the Tribonacci words are: We can show that, for $$n>2$$, $$t_n = t_{n-1}t_{n-2}t_{n-3}$$; hence the name "Tribonacci".
 * $$t_0 = 1$$
 * $$t_1 = 12$$
 * $$t_2 = 1213$$
 * $$t_3 = 1213121$$
 * $$t_4 = 1213121121312$$

Fractal construction


Consider, now, the space $$R^3$$ with cartesian coordinates (x,y,z). The Rauzy fractal is constructed this way:

1) Interpret the sequence of letters of the infinite Tribonacci word as a sequence of unitary vectors of the space, with the following rules (1 = direction x, 2 = direction y, 3 = direction z).

2) Then, build a "stair" by tracing the points reached by this sequence of vectors (see figure). For example, the first points are: etc...Every point can be colored according to the corresponding letter, to stress the self-similarity property.
 * $$1 \Rightarrow (1, 0, 0)$$
 * $$2 \Rightarrow (1, 1, 0)$$
 * $$1 \Rightarrow (2, 1, 0)$$
 * $$3 \Rightarrow (2, 1, 1)$$
 * $$1 \Rightarrow (3, 1, 1)$$

3) Then, project those points on the contracting plane (plane orthogonal to the main direction of propagation of the points, none of those projected points escape to infinity).

Properties

 * Can be tiled by three copies of itself, with area reduced by factors $$k$$, $$k^2$$ and $$k^3$$ with $$k$$ solution of $$k^3+k^2+k-1=0$$: $$\scriptstyle{k = \frac{1}{3}(-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}) = 0.54368901269207636}$$.
 * Stable under exchanging pieces. We can obtain the same set by exchanging the place of the pieces.
 * Connected and simply connected. Has no hole.
 * Tiles the plane periodically, by translation.
 * The matrix of the Tribonacci map has $$x^3 - x^2 - x -1$$ as its characteristic polynomial. Its eigenvalues are a real number $$\beta = 1.8392$$, called the Tribonacci constant, a Pisot number, and two complex conjugates $$\alpha$$ and $$\bar \alpha$$ with $$\alpha \bar \alpha=1/\beta$$.
 * Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of $$2|\alpha|^{3s}+|\alpha|^{4s}=1$$.

Variants and generalization
For any unimodular substitution of Pisot type, which verifies a coincidence condition (apparently always verified), one can construct a similar set called "Rauzy fractal of the map". They all display self-similarity and generate, for the examples below, a periodic tiling of the plane.