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 : , , . [2] [3] It is an example of a morphic word. Starting from 1, the Tribonacci words are: [4]
We can show that, for , ; hence the name " Tribonacci".
Consider, now, the space 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:
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).
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.
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 : , , . [2] [3] It is an example of a morphic word. Starting from 1, the Tribonacci words are: [4]
We can show that, for , ; hence the name " Tribonacci".
Consider, now, the space 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:
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).
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.