Rauzy fractal

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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.


Tribonacci word[edit]

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".

Fractal construction[edit]


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).


  • Can be tiled by three copies of itself, with area reduced by factors , and with solution of : .
  • 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 as its characteristic polynomial. Its eigenvalues are a real number , called the Tribonacci constant, a Pisot number, and two complex conjugates and with .
  • Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of .[6]

Variants and generalization[edit]

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.

See also[edit]


  1. ^ Rauzy, Gérard (1982). "Nombres algébriques et substitutions" (PDF). Bull. Soc. Math. Fr. (in French). 110: 147–178. Zbl 0522.10032.
  2. ^ Lothaire (2005) p.525
  3. ^ Pytheas Fogg (2002) p.232
  4. ^ Lothaire (2005) p.546
  5. ^ Pytheas Fogg (2002) p.233
  6. ^ Messaoudi, Ali (2000). "Frontière du fractal de Rauzy et système de numération complexe. (Boundary of the Rauzy fractal and complex numeration system)" (PDF). Acta Arith. (in French). 95 (3): 195–224. Zbl 0968.28005.

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