AUTHORS: Zhiyong Zhu, Enmei Dong
Download as PDF
ABSTRACT: Study on properties of Sierpinski-type fractals, including dimension, measure, Lipschitz equivalence, etc is very interesting. It is well know that studying fractal theory relies on in-depth observation and analysis to topological structures of fractals and their geometric constructions. But most works of simulating fractals are for graphical goal and often done by non-mathematical researchers. This makes them difficult for most mathematical researchers to understand and application. In [22], the authors simulated a class of Sierpinski-type fractals and their geometric constructions in Matlab environment base on iterative algorithm for the purpose of mathematical research. In this paper, we continue such investigation by adding certain rotation structure. Our results may be used for any graphical goal, not only for mathematical reasons.
KEYWORDS: Sierpinski-type square, Sierpinski-type triangle, IFS, deterministic algorithm, random iterated algorithm, Matlab
REFERENCES:
[1] A. Garg, A. Negi, A. Agrawal and B. Latwal, Geometric modeling of complex objects using iterated function system, International Journal of Scientific & Technology Research, Volume 3, Issue 6, 2014(6), pp. 1-8.
[2] M.-F. Barnsley, S.-G. Demko, Iterated Function Systems and the Global Construction of Fractals, Proc. Roy. Soc. London, Ser.A 399, 1985, pp. 243-275.
[3] G. David and S.Semmes, Fractured fractals and broken dreams: Self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications, vol. 7. The Clarendon Press Oxford University Press, New York (1997).
[4] F. Deng and F.-L. Xi, An Application of Lsystem and IFS in 3D Fractal Simulation, WSEAS Transactions on Systems. 4, 2008, pp. 352-361.
[5] K.-J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1997.
[6] K.-J. Falconer, Fractal Geometry - Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, England, 1990.
[7] S. Demko, L. Hodges and D. Naylor, Construction of fractal objects with iterated function systems, San Francisco, July, pp. 22-26.
[8] Y.-X. Gui and W.-X. Li, A generalized multifractal spectrum of the general sierpinsk carpets, J. Math. Anal. Appl. 348, 2008, pp. 180-192.
[9] J.-E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), pp. 713-747.
[10] B.-G. Jia, Bounds of Hausdorff measure of the Sierpinski gasket, J. Math. Anal. Appl. 330, 2007, pp. 1016-1024.
[11] K.-S. Lau, J.-J. Luo and H. Rao, Topological structure of fractal squares, Proceedings of the Edinburgh Mathematical Society. 155(1), 2013, pp. 73-86.
[12] Q.-H. Liu, L.-F. Xi and Y-.F. Zhao, Dimensions of intersections of the Sierpinski carpet with lines of rational slopes, Proceedings of the Edinburgh Mathematical Society. 50, 2007, pp. 411- 448.
[13] Mathworks, Matlab:The language of technical computing, version 6.5, 2002.
[14] Slawomir S.Nikiel, True-colour images and iterated function systems, Computer & Graphics, 5(22), 1998, pp. 635-640.
[15] T.-D. Taylor, Connectivity properties of Sierpinski relatives, Fractals, 19(4), 2011, pp. 481-506.
[16] H.-P. Wang, Research on dynamic simulation method of plants based on arithmetic of IFS, J.Changchun Inst.The.(Nat.Sci.Edi),6(2), 2005, pp. 49-52.
[17] Z.-X. Wen, Z.-Y. Zhu and G.-T. Deng, Lipschitz equivalence of a class of general Sierpinski carpets, J. Math. Anal. Appl., 385 (2012), pp. 16- 23.
[18] L.-F. Xi, and Y. Xiong, Self-similar sets with initial cubic patterns, C. R. Math. Acad. Sci. Paris, 348 (2010), pp.15-20.
[19] W.-Q. Zeng and X.-Y. Wang, Fractal theory and its computer simulation, Northeastern university press, Shenyang, China, 2001.
[20] Y. Fisher, E.W. Jacobs, R. D. Boss, Fractal image compression using iterated transformes, NOSC Techical Report, Naval Ocean System Center, San Diego, 1995.
[21] Z.-L. Zhou and F. Li, A new estimate of the Hausdorff measure of the Sierpinski gasket, Nonlinearity. 13(3), 2000, pp. 479-491.
[22] Z.-Y. Zhu and E.-M. Dong, Simulation of Sierpinski-type fractals and their geometric constructions in Matlab environment, Wseas Transactions on Mathematics, 12(10), 2013, pp. 2224- 2880.
[23] Z.-Y. Zhu and E.-M. Dong, Lipschitz equivalence of fractal triangles, J. Math.Anal.Appl., 433(2016), pp. 1157-1176.
[24] Z.-Y. Zhu, Lipschitz equivalence of totally disconnected general Sierpinski triangles, Fractals, Vol. 23, No. 2(2015), pp. 1550013 (14 pages).
[25] Z.-Y.Zhu, Y. Xiong and L.-F. Xi, Lipschitz equivalence of self-similar sets with triangular pattern, Sci. China. Ser. A, 54(2011), pp. 1019- 1026.