Rauzy fractal

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

The infinite tribonacci word is a word constructed by iteratively applying the Tribonacci or Rauzy map :

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 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.