It has been 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 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.
We can show that, for , , hence the name Tribonacci".
Let's consider, now, the space 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:
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 , and with solution of : .
- 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 , its eigenvalues are a real number , called Tribonacci constant, a Pisot number, and two complex conjugated numbers and with .
- Its boundary is fractal, and the Hausdorff dimension of this boundary equals 1.0933, the solution of .
Variants and generalization
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.
- Rauzy, Gérard (1982). "Nombres algébriques et substitutions". Bull. Soc. Math. Fr. (in French) 110: 147–178. Zbl 0522.10032.
- Lothaire (2005) p.525
- Pytheas Fogg (2002) p.232
- Lothaire (2005) p.546
- Pytheas Fogg (2002) p.233
- Messaoudi, Ali (2000). "Frontière du fractal de Rauzy et système de numération complexe. (Boundary of the Rauzy fractal and complex numeration system)". Acta Arith. (in French) 95 (3): 195–224. Zbl 0968.28005.
- 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". In Berthé, Valérie; Rigo, Michel. Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications 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 105. Cambridge University Press. ISBN 978-0-521-84802-2. MR 2165687. Zbl 1133.68067.
- Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
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