# Rauzy fractal

Rauzy fractal

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

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

It has been studied in 1981 by Gérard Rauzy,[1] with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal set can be generalized to other maps on 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.

## Definitions

### Tribonacci word

The infinite tribonacci word is a word constructed by applying iteratively the Tribonacci or Rauzy map : $s(1)=12$, $s(2)=13$, $s(3)=1$.[2][3] Starting from 1, the Tribonacci words are:[4]

• $t_0 = 1$
• $t_1 = 12$
• $t_2 = 1213$
• $t_3 = 1213121$
• $t_4 = 1213121121312$

We can show that, for $n>2$, $t_n = t_{n-1}t_{n-2}t_{n-3}$, hence the name Tribonacci".

### Fractal construction

Construction

Let's consider, now, the space $R^3$ with cartesian coordinates (x,y,z). The Rauzy fractal is constructed this way:[5]

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:

• $1 \Rightarrow (1, 0, 0)$
• $2 \Rightarrow (1, 1, 0)$
• $1 \Rightarrow (2, 1, 0)$
• $3 \Rightarrow (2, 1, 1)$
• $1 \Rightarrow (3, 1, 1)$

etc...Every point can be coloured according to the corresponding letter, to stress the self-similarity property.

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, 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 by 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 for characteristic polynomial $x^3 - x^2 - x -1$, its eigenvalues are a real number $\beta = 1,8392$, called Tribonacci constant, a Pisot number, and two complex conjugated numbers $\alpha$ 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 $2|\alpha|^{3s}+|\alpha|^{4s}=1$.[6]

## Variants and generalization

For any unimodular substitution of Pisot type, which verifies a coïncidence 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.