靜宜非線性及相關問題研討會

1. 靜宜非線性及相關問題研討會 An algorithm to generate fractal reptiles Peng-Jen Lai ( 賴鵬仁 ) Department of mathematics, National Kaohsiung Normal University 高雄師範大學數學系 20120829

2. Contents • Introduction: 1. Tilings 2. Reptiles 3. Fractals 4. Fractal tilings 、Fractal reptiles • Motivation • Survey of methods to generate Fractal reptiles • A geometric algorithm to generate Fractal reptiles • Computer simulation with Maple • Future directions • Reference

3. Tilings with tiles as regular polygons

4. The definition of tilings • Definition: [Grünbaum et al. 1987] A plane tiling is a countable family of closed topological disk which cover the plane without gaps or overlaps. More explicitly, the union of the sets ( which are known as the tiles of ) is to be the whole plane, and the interior of the sets are to be pairwise disjoint.

5. Tiling with non-polygonal tiles

6. Tiling in non-Euclidean spaces Escher 作 品 欣 賞

7. The definition of reptiles(自我複製磁磚) • The idea reptilewas invented by S. W. Golomb in 1962. More information about the k-rep tile is presented in [Darst et al 1998, Grunbaum]. • Definition: A k-rep tile is defined as any tile T that can be dissected into k congruent parts each of which is similar to T.

8. The following figure shows a famous example of a 4-rep tile called Sphinx. Remark: Every reptile can tile the plane!

9. 4-rep tiles

10. Wiki Fractal figures

11. 碎形天線陣列

12. Space-filling curves: the Sierpinski curve drawn with Maple [Rovenski 2000]

13. Definition of fractals: Using iterated function system (IFS) [Barnsley]

14. Definition of fractal tiles • Def: [Feder 1998] A prefractalis an intermidate shape of generating a fractal using IFS method. • , is a prefractal. • Def: [Lai 2009]A (kth) prefractal tile is a tile with (kth) prefractal boundaries. Now we can define when a fractal tile can tile the plane. • Def: [Lai 2009]A fractal tile can tile the plane if its every kth prefractal tile can tile the plane for . • From now on, when we say a fractal tile, it means a disk-like set with fractal boundary and being able to tile the plane. Figure:Fractal tiling with tiles as Fudgeflakes

15. Def: A fractal reptile is a fractal tile which is also a reptile. • The aim of this talk: how to design a fratcal reptile?

16. Motivation • In the internet we see many fractals which are claimed to be able to tile the plane. Because of the complicated boundaries of the fractals, it is not easy to understand how they can tile the plane. • It is more difficult to understand why some well-known fractal is also a reptile.

17. Wiki

18. Gosper island is a 7-rep tile

19. Terdragon is a 3-rep tile

20. Escher style rules Parallel translation Glide-reflection

21. To check if a fractal is a tile by Escher style rules [Lai]Example: The quadratic Koch curve [Feder]

22. How to generate a fractal reptile? • Survey of methods: • [Bandt 1991] Suppose that M represents an expansive map ( e.g., ), {y_1,,,y_m} is a complete residue system for M, and . Then the attractor set is the union of m tiles that have disjoint interior and satisfy A_j=f_j(A). Such tiles are called m-rep tiles. ( 此處之 m-rep tile 是較廣義的，因為 M 不見得是相似變換。) • [Bandt et al 2001] They give conditions to guarantee that a self-affine tile in R^2 is homeomorphic to a disk.

23. 3. The tiles generated by the IFS method (wavelet) are not always good-looking. In [Flaherty and Wang 1999] They stated as follows:

24. The classification of 2-reptiles by Ngai et al. [Ngai 2000] • Twindragon, • Levy dragon, • Heighway dragon, • triangle, • rectangle (e.g. A4 paper), • tame twindragon Graphs in Wiki

25. A geometric algorithm to generate fractal reptile [Lai] Investigation on the Gosper island Remark: This view of point is stated in Wiki, but not generalized for general case.

26. Criterion 1:

27. Another example: Greek cross

28. Computer simulation with Maple • We can use Maple to draw the prefractal of Greek cross fractal reptile up to several generation easily. 4th generation of Greek cross Fractal reptile

29. Future directions • Wavelet 、quasi-crystal and fractal tiles. • Find some conditions to ensure that a fractal reptile is a topological disk. • How many well-known examples can be checked by this algorithm?

30. References • M. F. Barnsley, Fractals Everywhere, AP Professional, 2ed, 1993. • C. Bandt, self-similar sets 5. Integer matrices and fractal tilings of R^n, Proceedings of AMS 112 (1991), 549-562. • C. Bandt, Y. Wang, Disk-like self-affine tiles in R^2, DiscreteComput. Geom. 26 (2001), 591-601. • T. Flaherty, Y. Wang, Haar-type multiwavelet bases and self-affine multi-tiles, Asian J. Math. Vol.3 No. 2 (1999) 387-400. • R. Darst, J. Palagallo, T. Price, Fractal Tilings in the Plane, Math. Magazine, Vol. 71, No.1.(1998), p.12-23. • J. Feder, Fractal, Plenum Press,1988. • K. Falconer, Fractal geometry mathematical foundations and applications, John Wiley and Sons, 1990. • B. Grünbaum and G. C. Shephard, Tilings and patterns, W.H.Freeman and Company, 1987. • Ngai, Sirvent, Veerman, Wang, On 2-reptiles in the plane, Geom. Ded. 82 (2000) 325-344. • Peng-Jen Lai, How to make fractal tilings and fractal reptiles, Fractals, 2009. • V. Rovenski, Geometry of Curves and Surfaces with Maple, Birkhauser 2000. • 賴鵬仁編著，幾何學講義第二部鑲嵌圖形之幾何結構與碎形幾何學以電腦軟體輔助探索Learning tilings and fractals with the aid of maple GSP and CorelDraw，白象文化，ISBN: 978-986-6453-09-0 ( in Chinese). • 上課自編教材: 1.Maple快速入門. 2.Maple programing. 還有螢幕操作之錄影檔，有提供給思渤科技(參考思渤科技網頁). • 洪維恩著, 數學魔法師Maple6, 基峰出版. • Maple10 Programming Guide, WaterlooMaple.

31. Thanks for your attention! 北大武日出、雲海、鐵杉，清大登山社朋友潘庭、之中、鳴君等攝2007

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39. If we use the view point of initiator and generator of boundary. Then it’s hard to understand why it is a 7-reptile