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Fractal n-hedral tilings of ℝd

Part of: Classical measure theory Discrete geometry Designs and configurations

Published online by Cambridge University Press:  09 April 2009

You Xu
You Xu
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA e-mail: [email protected]
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Abstract

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An n-hedral tiling of ℝd is a tiling with each tile congruent to one of the n distinct sets. In this paper, we use the iterated function systems (IFS) to generate n-hedral tilings of ℝd. Each tile in the tiling is similar to the attractor of the IFS. These tiles are fractals and their boundaries have the Hausdorff dimension less than d. Our results generalize a result of Bandt.

Keywords

Tilingfractalself-similar setiterated function system

MSC classification

Secondary: 52C22: Tilings in $n$ dimensions 28A80: Fractals 05B45: Tessellation and tiling problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 66 , Issue 3 , June 1999 , pp. 403 - 417
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Bandt, C., ‘Self-similar sets 5. Integer matrices and fractal tilings of ℝ’ , Proc. Amer. Math. Soc. 112 (1991), 549–562.Google Scholar
[2]Bandt, C., ‘Self-similar tilings and patterns described by mapping’, in: The mathematics of longrange aperiodic order (Waterloo, ON, 1995) 45–83. NATO ASI Series C Math. Phys. Sci., 489 (ed. Moody, R. V.) (Kluwer Acad. Publ., Dordrecht, 1997).CrossRefGoogle Scholar
[3]Bandt, C. and Gelbrich, G., ‘Classification of self-affine lattice tilings’, J. London Math. Soc. 50 (1994), 581–593.CrossRefGoogle Scholar
[4]Barnsley, M. F., Fractals everywhere (Academic Press, New York, 1988).Google Scholar
[5]Croft, H. T., Falconer, K. J. and Guy, R. K., Unsolved problems in geometry (Springer, New York, 1991).CrossRefGoogle Scholar
[6]Dekking, F. M., ‘Recurrent sets’, Adv. in Math. 44 (1982), 78–104.CrossRefGoogle Scholar
[7]Falconer, K. J., Fractal geometry: Mathematical foundation and applications (Wiley, New York, 1990).Google Scholar
[8]Grünbaum, B. and Shephard, G. C., Tilings and patterns (Freeman, New York, 1987).Google Scholar
[9]Hutchinson, J. E., ‘Fractals and self-similarity’, Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[10]Ito, Sh. and Kimura, M., ‘On Rauzy fractal’, Japan J. Indust. Appl. Math. 8 (1991), 461–486.CrossRefGoogle Scholar
[11]Kenyon, R., ‘Self-replicating tilings’, in: Symbolic dynamics and its applications (ed. Walters, P.), Contemporary Mathematics 135 (American Mathematical Society, Providence, 1992).Google Scholar
[12]Kenyon, R., ‘The construction of self-similar tilings’, Geom. Funct. Anal. 6 (1996), 471–488.CrossRefGoogle Scholar
[13]Lagarias, J. C. and Wang, Y., ‘Self-affine tiles in ℝ’, Adv. in Math. 121 (1996), 21–49.CrossRefGoogle Scholar
[14]Lau, K. S. and Xu, Y., ‘On the boundary of attractor with nonvoid interior’, Proc. Amer. Math. Soc. (to appear).Google Scholar
[15]Rauzy, G., ‘Nombres Algebriques et Substitutions’, Bull. Soc. Math. France 110 (1982), 147–178.CrossRefGoogle Scholar
[16]Rudin, W., Functional analysis (McGraw-Hill, New York, 1991).Google Scholar
[17]Schief, A., ‘Separation properties for self-similar sets’, Proc. Amer. Math. Soc. 122 (1994), 111–115.CrossRefGoogle Scholar
[18]Sirvent, V. F., ‘Note on some dynamical subsets of the Rauzy fractal’, Theoret. Comput. Sci. 180 (1997), 363–370.CrossRefGoogle Scholar
[19]Strichartz, R., ‘Wavelets and self-affine tilings’, Constructive Approx. 9 (1993), 327–346.CrossRefGoogle Scholar
[20]Wang, Y., ‘Self-affine tiles’, in: Advances in wavelets (ed. Lau, K. S.) (Springer, New York, 1998), 261–285.Google Scholar