Rauzy fractal

Definitions

Tribonacci word

The infinite tribonacci word is a word constructed by iteratively applying the Tribonacci or Rauzy map : s(1)=12, s(2)=13, s(3)=1. It is an example of a morphic word. Starting from 1, the Tribonacci words are:

• t_0 = 1
• t_1 = 12
• t_2 = 1213
• t_3 = 1213121
• t_4 = 1213121121312 We can show that, for n2, t_n = t_{n-1}t_{n-2}t_{n-3}; hence the name "Tribonacci".

Fractal construction

Consider, now, the space 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 \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 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).

Properties

• Can be tiled by three copies of itself, with area 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 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 x^3 - x^2 - x -1 as its characteristic polynomial. Its eigenvalues are a real number \beta = 1.8392, called the Tribonacci constant, a Pisot number, and two complex conjugates \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.

Variants and generalization

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. Image:Rauzy1.png|s(1)=12, s(2)=31, s(3)=1 Image:Rauzy2.png|s(1)=12, s(2)=23, s(3)=312 Image:Rauzy3.png|s(1)=123, s(2)=1, s(3)=31 Image:Rauzy4.png|s(1)=123, s(2)=1, s(3)=1132

References

References

1. Rauzy, Gérard. (1982). "Nombres algébriques et substitutions". Bull. Soc. Math. Fr..
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)". Acta Arith..