Rauzy fractal

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Rauzy fractal

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

s(1)=12,\  s(2)=13,\  s(3)=1 \,.

It has been 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 on 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 applying iteratively the Tribonacci or Rauzy map : s(1)=12, s(2)=13, s(3)=1.[2][3] Starting from 1, the Tribonacci words are:[4]

  • t_0 = 1
  • t_1 = 12
  • t_2 = 1213
  • t_3 = 1213121
  • t_4 = 1213121121312

We can show that, for n>2, t_n = t_{n-1}t_{n-2}t_{n-3}, hence the name "Tribonacci".

Fractal construction[edit]


Let's consider, now, the space R^3 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:

  • 1 \Rightarrow (1, 0, 0)
  • 2 \Rightarrow (1, 1, 0)
  • 1 \Rightarrow (2, 1, 0)
  • 3 \Rightarrow (2, 1, 1)
  • 1 \Rightarrow (3, 1, 1)

etc...Every point can be coloured 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, reduced by factors k, k^2 and k^3 with k solution of k^3+k^2+k-1=0: \scriptstyle{k = \frac{1}{3}(-1-\frac{2}{\sqrt[3]{17+3 \sqrt{33}}}+\sqrt[3]{17+3 \sqrt{33}}) = 0.54368901269207636}.
  • Stable by 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 for characteristic polynomial x^3 - x^2 - x -1, its eigenvalues are a real number \beta = 1.8392, called Tribonacci constant, a Pisot number, and two complex conjugated numbers \alpha and \bar \alpha with \alpha \bar \alpha=1/\beta.
  • Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of 2|\alpha|^{3s}+|\alpha|^{4s}=1.[6]

Variants and generalization[edit]

For any unimodular substitution of Pisot type, which verifies a coïncidence 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. 

External links[edit]