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FRACTAL TILINGS FROM SUBSTITUTION TILINGS

XINCHANG WANG

School of Mathematics and Physics, Jinggangshan University, Jiangxi, P. R. China

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PEICHANG OUYANG

School of Mathematics and Physics, Jinggangshan University, Jiangxi, P. R. China

E-mail Address: g_fcayang@163.com

Corresponding author.

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KWOKWAI CHUNG

Department of Mathematics, City University of Hong Kong, Tat Chee Avenue Kowloon Tong, Hong Kong

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XIAOGEN ZHAN

School of Mathematics and Physics, Jinggangshan University, Jiangxi, P. R. China

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HUA YI

School of Mathematics and Physics, Jinggangshan University, Jiangxi, P. R. China

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, and

XIAOSONG TANG

School of Mathematics and Physics, Jinggangshan University, Jiangxi, P. R. China

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https://doi.org/10.1142/S0218348X19500099Cited by:2 (Source: Crossref)

Abstract

A fractal tiling or $f$ $f$ -tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of $f$ $f$ -tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

Keywords:

References

1. D. Schattschneider, M.C. Escher: Visions of Symmetry (Harry N. Abrams, 2004). Google Scholar
2. R. W. Fathauer, Fractal tilings based on kite- and dart-shaped prototile, Comput. Graph. 25(2) (2001) 323–331. Web of Science, Google Scholar
3. R. W. Fathauer, Self-similar tilings based on prototiles constructed from segments of regular polygons, in Proc. 2000 Bridges Conf., ed. R. Sarhangi (Central Plain Book Manufacturing, Winfield, 2000), pp. 285-292. Google Scholar
4. R. W. Fathauer, Fractal tilings based on V-shaped prototiles, Comput. Graph. 26(4) (2002) 635–643. Web of Science, Google Scholar
5. R. W. Fathauer, Fractal tilings based on dissections of polyhexes, in Proc. 2005 Bridges Conf. (Central Plain Book Manufacturing, Winfield, 2005), pp. 427–434. Google Scholar
6. R. W. Fathauer, Fractal tilings based on dissections of polyominoes, in Proc. 2006 Bridges Conference (Tarquin Publications, 2006), pp. 293-300. Google Scholar
7. B. Grünbaum and G. C. Shephard, Tilings and Patterns (W. H. Freeman, 1987). Google Scholar
8. M. F. Barnsley, Fractals Everywhere (Academic Press, New York, 1988). Google Scholar
9. C. B. Bandt, Self-similar sets 5: Integer matrices and fractal tilings of $R^{n}$ $R^{n}$ , Proc. Amer. Math. Soc. 112(2) (1991) 549–562. Web of Science, Google Scholar
10. K. Grochenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl. 1(2) (1994) 131–170. Google Scholar
11. H. N. Huynh, and L. Y. Chew, Arc-fractal and the dynamics of coastal morphology, Fractals 19(2) (2011) 141–162. Link, Web of Science, Google Scholar
12. P. J. Lai, Concise methods to generate fractal tilings, Int. J. Nonlinear Sci. Numer. Simul. 10(5) (2009) 585–596. Web of Science, Google Scholar
13. J. Palagallo and M. Salcedo, Symmetries of fractal tilings, Fractals 16(1) (2008) 69–78. Link, Web of Science, Google Scholar
14. N. P. Frank, S. B. G. Webster and M. F. Whittaker, Fractal dual substitution tilings, J. Fractal Geom. 3(3) (2014) 265–317. Google Scholar
15. P. C. Ouyang, H. Yi, Z. Y. Deng and X. Huang, Boundary dimensions of fractal tilings, Fractals 23(4) (2015) 1550035. Link, Web of Science, Google Scholar
16. P. C. Ouyang and R. W. Fathauer, Beautiful math, part 2: Aesthetic patterns based on fractal tilings, IEEE Comput. Graph. Appl. 34(1) (2014) 68–76. Web of Science, Google Scholar
17. K. W. Chung and H. M. Ma, Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems, Chaos Solitons Fractals 24(4) (2005) 1145–1158. Web of Science, Google Scholar
18. E. O. Harriss, On canonical substitution tilings, Ph.D. thesis, Department of Mathematics, Imperial College London (2004). Google Scholar
19. D. Frettlöh, Duality of model sets generated by substitutions, Rev. Roum. Math. Pures Appl. 50(5–6) (2012) 619–639. Google Scholar
20. C. Goodman-Strauss, Matching rules and substitution tilings, Ann. Math. 147(1) (1998) 181–223. Web of Science, Google Scholar
21. D. Frettlöh and E. Harriss, Substitutions (2018), http://tilings.math.uni-bielefeld.de/substitution/. Google Scholar
22. L. M. Surhone, M. T. Tennoe and S. F. Henssonow, Penrose Tiling (Betascript Publishing, 2013). Google Scholar
23. R. W. Fathauer, Extending Escher’s recognizable-motif tilings to multiple-solution tilings and fractal tilings, M. C. Escher’s Legacy (Springer, 2003), pp. 154–165. Google Scholar
24. P. Webster, Fractal Islamic geometric patterns based on arrangements of $n / 2$ $n / 2$ stars, in Proc. Bridges 2013: Mathematics, Music, Art, Architecture, Culture, Enschede, The Netherlands (2013), pp. 87–94. Google Scholar
25. D. Frettlöh and E. Harriss, Equithirds (2018), http://tilings.math.uni-bielefeld.de/substitution/equithirds/. Google Scholar
26. D. Frettlöh and E. Harriss, T2000 (2018), http://tilings.math.uni-bielefeld.de/substitution/t2000/. Google Scholar