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Rauzy Fractal

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The Rauzy fractal is a planar fractal associated with the tribonacci constant and the substitution system 0->01, 1->02, 2->0 (Rauzy 1982, pp. 148-150; Messaoudi 1998, p. 136). It connects numeral systems, tilings, and dynamical systems.

Let alpha be the nonreal root of

x^3-x^2-x-1=0
(1)

with positive imaginary part. The real root is the tribonacci constant beta>1, and |alpha|=beta^(-1/2)<1 (Rauzy 1982, p. 150). One normalization of the Rauzy fractal is

K={sum_(k=0)^inftyepsilon_kalpha^k:epsilon_k in {0,1}, epsilon_kepsilon_(k+1)epsilon_(k+2)=0 for all k>=0}.
(2)

Thus the binary digits may not contain three consecutive 1s. Messaoudi (1998, p. 136) uses the scaled normalization E=alpha^3K.

RauzyFractalPieces

The three pieces are illustrated above using finite sums. Grouping the admissible digit strings by their initial blocks 0, 10, and 110 gives the three affine pieces

K=alphaK union (1+alpha^2K) union (1+alpha+alpha^3K).
(3)

The corresponding contraction mappings form an iterated function system.

RauzyFractalTiling

The scaled set E=alpha^3K tiles the complex plane by translations from the point lattice Z+Zalpha (Messaoudi 1998, p. 138). The illustration shows finite approximations of E and its six neighboring translates by +/-1, +/-alpha, and +/-(1+alpha). Its three pieces also give a nonperiodic tiling (Messaoudi 1998, p. 136).

The boundary of K is a Jordan curve (Messaoudi 2000, p. 208). Its Hausdorff dimension is

dim_H(partialK)=(lnrho)/(ln|alpha|)=1.0933641...,
(4)

where rho is the largest real root of x^4+2x^3-1=0 (Messaoudi 2000, pp. 210-211). The scaling between K and E does not change this Hausdorff dimension.

See also

Fractal, Iterated Function System, Nonperiodic Tiling, Numeral System, Pisot Number, Substitution System, Tribonacci Constant, Tribonacci Sequence

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References

Akiyama, S. "On the Boundary of Self Affine Tilings Generated by Pisot Numbers." J. Math. Soc. Japan 54, 283-308, 2002. https://doi.org/10.2969/jmsj/05420283.Arnoux, P.; Berthé, V.; Ei, H.; and Ito, S. "Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions." In Proc. Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) (Ed. R. Cori, J. Mazoyer, M. Morvan, and R. Mosseri). Paris, France: Maison de l'Informatique et des Mathématiques Discrètes, pp. 59-78, 2001. https://doi.org/10.46298/dmtcs.2291.Holton, C. and Zamboni, L. Q. "Geometric Realizations of Substitutions." Bull. Soc. Math. France 126, 149-179, 1998. https://doi.org/10.24033/bsmf.2324.Ito, S. and Kimura, M. "On Rauzy Fractal." Japan J. Indust. Appl. Math. 8, 461-486, 1991. https://doi.org/10.1007/BF03167147.Ito, S. and Sano, Y. "Substitutions, Atomic Surfaces, and Periodic Beta Expansions." In Analytic Number Theory (Ed. C. Jia and K. Matsumoto). Dordrecht, Netherlands: Kluwer, pp. 183-194, 2002. https://doi.org/10.1007/978-1-4757-3621-2_12.Messaoudi, A. "Propriétés arithmétiques et dynamiques du fractal de Rauzy." J. Théor. Nombres Bordeaux 10, 135-162, 1998. https://www.numdam.org/item/JTNB_1998__10_1_135_0/.Messaoudi, A. "Frontière du fractal de Rauzy et système de numération complexe." Acta Arith. 95, 195-224, 2000. https://doi.org/10.4064/aa-95-3-195-224.Messaoudi, A. "Tribonacci Multiplication." Appl. Math. Lett. 15, 981-985, 2002. https://doi.org/10.1016/S0893-9659(02)00073-3.Messaoudi, A. "Fractals de Rauzy." C. R. Acad. Sci. Paris, Ser. I 341, 175-178, 2005. https://doi.org/10.1016/j.crma.2005.06.019.Messaoudi, A. "Propriétés arithmétiques et topologiques d'une classe d'ensembles fractales." Acta Arith. 121, 341-366, 2006. https://doi.org/10.4064/aa121-4-5.Rauzy, G. "Nombres algébriques et substitutions." Bull. Soc. Math. France 110, 147-178, 1982. https://doi.org/10.24033/bsmf.1957.Robinson, E. A., Jr. "Symbolic Dynamics and Tilings of R^d." In Symbolic Dynamics and Its Applications (Ed. S. G. Williams). Providence, RI: Amer. Math. Soc., pp. 81-119, 2004. https://doi.org/10.1090/psapm/060.Siegel, A. "Pure Discrete Spectrum Dynamical System and Periodic Tiling Associated with a Substitution." Ann. Inst. Fourier 54, 341-381, 2004. https://www.numdam.org/article/AIF_2004__54_2_341_0.pdf.Sirvent, V. F. "On Some Dynamical Subsets of the Rauzy Fractal." Theor. Comput. Sci. 180, 363-370, 1997. https://doi.org/10.1016/S0304-3975(96)00331-3.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. §1.1.1. https://www.mathematicaguidebooks.org/.Vince, A. "Digit Tiling of Euclidean Space." In Directions in Mathematical Quasicrystals (Ed. M. Baake and R. V. Moody). Providence, RI: Amer. Math. Soc., pp. 329-370, 2000.

Cite this as:

Weisstein, Eric W. "Rauzy Fractal." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RauzyFractal.html

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