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.

• 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]

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

• List of fractals