In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution
s ( 1 ) = 12 , s ( 2 ) = 13 , s ( 3 ) = 1 . {\displaystyle 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 {\displaystyle s(1)=12} , s ( 2 ) = 13 {\displaystyle s(2)=13} , s ( 3 ) = 1 {\displaystyle s(3)=1} . It is an example of a morphic word. Starting from 1, the Tribonacci words are:
t 0 = 1 {\displaystyle t_{0}=1}
t 1 = 12 {\displaystyle t_{1}=12}
t 2 = 1213 {\displaystyle t_{2}=1213}
t 3 = 1213121 {\displaystyle t_{3}=1213121}
t 4 = 1213121121312 {\displaystyle t_{4}=1213121121312}
We can show that, for n > 2 {\displaystyle n>2} , t n = t n − 1 t n − 2 t n − 3 {\displaystyle t_{n}=t_{n-1}t_{n-2}t_{n-3}} ; hence the name "Tribonacci".
Fractal construction
Consider, now, the space R 3 {\displaystyle 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 ⇒ ( 1 , 0 , 0 ) {\displaystyle 1\Rightarrow (1,0,0)}
2 ⇒ ( 1 , 1 , 0 ) {\displaystyle 2\Rightarrow (1,1,0)}
1 ⇒ ( 2 , 1 , 0 ) {\displaystyle 1\Rightarrow (2,1,0)}
3 ⇒ ( 2 , 1 , 1 ) {\displaystyle 3\Rightarrow (2,1,1)}
1 ⇒ ( 3 , 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).
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