Thus the binary digits may not contain three consecutive 1s. Messaoudi (1998, p. 136) uses the scaled normalization .
The three pieces are illustrated above using finitesums. Grouping the admissible digit
strings by their initial blocks 0, 10, and 110 gives the three affine pieces
where
is the largest realroot
of (Messaoudi 2000, pp. 210-211).
The scaling between
and
does not change this Hausdorff dimension.
Related Rauzy fractal constructions arise from other substitution systems. For example, ,, has dominant eigenvalue, where 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.
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 for the root of with positiveimaginary
part, the central tile satisfies (Gjini 2001, p. 306), while the tent tile satisfies (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.
Akiyama, S. "On the Boundary of Self Affine Tilings Generated by Pisot Numbers." J. Math. Soc. Japan54, 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 ."
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. France126, 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 Bordeaux10, 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. I341,
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. France110,
147-178, 1982. https://doi.org/10.24033/bsmf.1957.Robinson,
E. A., Jr. "Symbolic Dynamics and Tilings of ." 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. Fourier54, 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
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Amer. Math. Soc., pp. 329-370, 2000.
, 
, 
(Rauzy 1982, pp. 148-150; Messaoudi 1998, p. 136).
It connects numeral systems, tilings,
and dynamical systems.
be the nonreal root
of
, and 
(Rauzy 1982, p. 150). One normalization
of the Rauzy fractal is
.
tiles the complex plane
by translations from the point
lattice 
(Messaoudi 1998, p. 138). The illustration shows finite
approximations of 
and its six neighboring translates by 
, 
, and 
. Its three pieces also give a nonperiodic
tiling (Messaoudi 1998, p. 136).
is a Jordan curve (Messaoudi
2000, p. 208). Its Hausdorff dimension
is
is the largest real root
of 
(Messaoudi 2000, pp. 210-211).
The scaling between 
and 
does not change this Hausdorff dimension.
, 
, 
has dominant eigenvalue 
, where 
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.
for the root of 
with positive imaginary
part, the central tile satisfies 
(Gjini 2001, p. 306), while the tent tile satisfies 
(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.
."
Osaka J. Math. 38, 303-319, 2001. 
." In Symbolic Dynamics and Its Applications
(Ed. S. G. Williams). Providence, RI: Amer. Math. Soc., pp. 81-119,
2004.