Rauzy fractal
# 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.

## Definitions

### 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:

* $t_{0}=1$
* $t_{1}=12$
* $t_{2}=1213$
* $t_{3}=1213121$
* $t_{4}=1213121121312$

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

### Fractal construction

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:

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

etc...Every point can be colored 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, 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.

* s(1)=12, s(2)=31, s(3)=1
* s(1)=12, s(2)=23, s(3)=312
* s(1)=123, s(2)=1, s(3)=31
* s(1)=123, s(2)=1, s(3)=1132
