# Rauzy fractal

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

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

It was 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 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 : ${\displaystyle s(1)=12}$, ${\displaystyle s(2)=13}$, ${\displaystyle s(3)=1}$.[2][3] It is an example of a morphic word. Starting from 1, the Tribonacci words are:[4]

• ${\displaystyle t_{0}=1}$
• ${\displaystyle t_{1}=12}$
• ${\displaystyle t_{2}=1213}$
• ${\displaystyle t_{3}=1213121}$
• ${\displaystyle t_{4}=1213121121312}$

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

### Fractal construction

Consider, now, the space ${\displaystyle R^{3}}$ 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:

• ${\displaystyle 1\Rightarrow (1,0,0)}$
• ${\displaystyle 2\Rightarrow (1,1,0)}$
• ${\displaystyle 1\Rightarrow (2,1,0)}$
• ${\displaystyle 3\Rightarrow (2,1,1)}$
• ${\displaystyle 1\Rightarrow (3,1,1)}$

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 ${\displaystyle k}$, ${\displaystyle k^{2}}$ and ${\displaystyle k^{3}}$ with ${\displaystyle k}$ solution of ${\displaystyle k^{3}+k^{2}+k-1=0}$: ${\displaystyle \scriptstyle {k={\frac {1}{3}}(-1-{\frac {2}{\sqrt[{3}]{17+3{\sqrt {33}}}}}+{\sqrt[{3}]{17+3{\sqrt {33}}}})=0.54368901269207636}}$.
• 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 ${\displaystyle x^{3}-x^{2}-x-1}$ as its characteristic polynomial. Its eigenvalues are a real number ${\displaystyle \beta =1.8392}$, called the Tribonacci constant, a Pisot number, and two complex conjugates ${\displaystyle \alpha }$ and ${\displaystyle {\bar {\alpha }}}$ with ${\displaystyle \alpha {\bar {\alpha }}=1/\beta }$.
• Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of ${\displaystyle 2|\alpha |^{3s}+|\alpha |^{4s}=1}$.[6]

## 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.