In mathematics, theRauzy fractal is afractal set associated with the Tribonaccisubstitution

${\displaystyle s(1)=12,s(2)=13,s(3)=1.}$

It was studied in 1981 by Gérard Rauzy,^[1] with the idea of generalizing the dynamic properties of theFibonacci morphism.That fractal set can be generalized to other maps over a 3-letter alphabet, generating other fractal sets with interesting properties, such as periodictiling of the plane andself-similarity in threehomothetic parts.

Theinfinite tribonacci word is aword constructed by iteratively applying theTribonacci orRauzy map :${\displaystyle s(1)=12}$,${\displaystyle s(2)=13}$,${\displaystyle s(3)=1}$.^[2][3] It is an example of amorphic word.Starting from 1, the Tribonacci words are:^[4]

• ${\displaystyle t_0=1}$
• ${\displaystyle t_1=12}$
• ${\displaystyle t_2=1213}$
• ${\displaystyle t_3=1213121}$
• ${\displaystyle t_4=1213121121312}$

We can show that, for${\displaystyle n>2}$,${\displaystyle t_n=t_{n-1}{t}_{n-2}{t}_{n-3}}$; hence the name "Tribonacci".

Consider, now, the space${\displaystyle R^3}$ with cartesian coordinates (x,y,z). TheRauzy fractal is constructed this way:^[5]

1) Interpret the sequence of letters of the infinite Tribonacci word as a sequence of unitaryvectors 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:

• ${\displaystyle 1\Rightarrow (1,0,0)}$
• ${\displaystyle 2\Rightarrow (1,1,0)}$
• ${\displaystyle 1\Rightarrow (2,1,0)}$
• ${\displaystyle 3\Rightarrow (2,1,1)}$
• ${\displaystyle 1\Rightarrow (3,1,1)}$

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).

• Can betiled by three copies of itself, with area reduced by factors${\displaystyle k}$,${\displaystyle k^2}$ and${\displaystyle k^3}$ with${\displaystyle k}$ solution of${\displaystyle k^3+k^2+k-1=0}$:${\displaystyle k=\frac{1}{3}(-1-\frac{2}{\sqrt[3]{17+3\sqrt{33}}}+\sqrt[3]{17+3\sqrt{33}})=0.54368901269207636}$.
• 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${\displaystyle x^3-x^2-x-1}$ as itscharacteristic polynomial. Its eigenvalues are a real number${\displaystyle \beta =1.8392}$, called theTribonacci constant, aPisot number, and two complex conjugates${\displaystyle \alpha }$ and${\displaystyle \overline{\alpha }}$ with${\displaystyle \alpha \overline{\alpha }=1/\beta }$.
• Its boundary is fractal, and theHausdorff dimension of this boundary equals 1.0933, the solution of${\displaystyle 2|\alpha {|}^{3s}+|\alpha {|}^{4s}=1}$.^[6]

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 displayself-similarity and generate, for the examples below, a periodic tiling of the plane.

• 
  s(1)=12, s(2)=31, s(3)=1
• 
  s(1)=12, s(2)=23, s(3)=312
• 
  s(1)=123, s(2)=1, s(3)=31
• 
  s(1)=123, s(2)=1, s(3)=1132

• List of fractals

• Arnoux, Pierre;Harriss, Edmund (August 2014)."WHAT IS... a Rauzy Fractal?".Notices of the American Mathematical Society.61 (7):768–770.doi:10.1090/noti1144.
• Berthé, Valérie; Siegel, Anne; Thuswaldner, Jörg (2010). "Substitutions, Rauzy fractals and tilings". InBerthé, Valérie; Rigo, Michel (eds.).Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge:Cambridge University Press. pp. 248–323.ISBN 978-0-521-51597-9.Zbl 1247.37015.
• Lothaire, M. (2005).Applied combinatorics on words. Encyclopedia of Mathematics and its Applications. Vol. 105.Cambridge University Press.ISBN 978-0-521-84802-2.MR 2165687.Zbl 1133.68067.
• Pytheas Fogg, N. (2002).Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.).Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin:Springer-Verlag.ISBN 3-540-44141-7.Zbl 1014.11015.

Wikimedia Commons has media related toRauzy fractals.

• Topological properties of Rauzy fractals
• Substitutions, Rauzy fractals and tilings, Anne Siegel, 2009
• Rauzy fractals for free group automorphisms, 2006
• Pisot Substitutions and Rauzy fractals
• Rauzy fractals
• Numberphile video about Rauzy fractals and Tribonacci numbers

• Fractals

• CS1 French-language sources (fr)
• Commons category link is on Wikidata