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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.

Related Rauzy fractal constructions arise from other substitution systems. For example, A->AC, B->AB, C->B has dominant eigenvalue P^2, where P is the plastic constant, and gives three subtiles (Coulbois and Minervino 2017, pp. 6 and 12-13). The limiting fractal in the pentagonal construction for the plastic constant is similar to one of these subtiles in complex eigenvector coordinates.

RauzyFractalGjiniComparison

Another example is the central tile studied by Gjini (2001, pp. 305-306), associated with the supergolden ratio and digit strings in which each 1 is followed by at least two 0s. The figure above compares finite approximations of this central tile at left and the tent tile for the same parameter at right, scaled independently for display. Writing a for the root of x^3-x^2-1=0 with positive imaginary part, the central tile satisfies K=aK union (1+a^3K) (Gjini 2001, p. 306), while the tent tile satisfies F=aF union a^3(1-F) (Scheicher et al. 2026). The boundary of the central tile has Hausdorff dimension approximately 1.47131 (Gjini 2001, pp. 317-319). The boundary of the tent tile has Hausdorff dimension approximately 1.02952 (Scheicher et al. 2026), so the two attractors are not similar. This illustrates why an underlying algebraic number alone does not determine a Rauzy fractal.

See also

Fractal, Hokkaido Fractal, Iterated Function System, Nonperiodic Tiling, Numeral System, Pisot Number, Substitution System, Tent Tile, 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.Coulbois, T. and Minervino, M. "Tree Substitutions and Rauzy Fractals." 8 Nov 2017. https://arxiv.org/abs/1711.03535.Gjini, N. "A Self-Similar Tiling Generated by the Pisot Number Which Is the Root of the Equation x^3-x^2-1=0." Osaka J. Math. 38, 303-319, 2001. https://doi.org/10.18910/9669.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.Numberphile. "Tribonacci Numbers (and the Rauzy Fractal)." Jun. 3, 2019. https://www.youtube.com/watch?v=fMJflV_GUpU.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.Scheicher, K.; Sirvent, V. F.; and Surer, P. "Fractal Tiles Induced by Tent Maps." Disc. Contin. Dyn. Syst. 49, 159-188, 2026. https://doi.org/10.3934/dcds.2025160.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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