Biomorphs via modified iterations
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Authors
Krzysztof Gdawiec
- Institute of Computer Science, University of Silesia, Bedzinska 39, 41-200 Sosnowiec, Poland.
Wieslaw Kotarski
- Institute of Computer Science, University of Silesia, Bedzinska 39, 41-200 Sosnowiec, Poland.
Agnieszka Lisowska
- Institute of Computer Science, University of Silesia, Bedzinska 39, 41-200 Sosnowiec, Poland.
Abstract
The aim of this paper is to present some modifications of the biomorphs generation algorithm introduced
by Pickover in 1986. A biomorph stands for biological morphologies. It is obtained by a modified Julia set
generation algorithm. The biomorph algorithm can be used in the creation of diverse and complicated
forms resembling invertebrate organisms. In this paper the modifications of the biomorph algorithm in two
directions are proposed. The first one uses different types of iterations (Picard, Mann, Ishikawa). The second
one uses a sequence of parameters instead of one fixed parameter used in the original biomorph algorithm.
Biomorphs generated by the modified algorithm are essentially different in comparison to those obtained by
the standard biomorph algorithm, i.e., the algorithm with Picard iteration and one fixed constant.
Share and Cite
ISRP Style
Krzysztof Gdawiec, Wieslaw Kotarski, Agnieszka Lisowska, Biomorphs via modified iterations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 5, 2305--2315
AMA Style
Gdawiec Krzysztof, Kotarski Wieslaw, Lisowska Agnieszka, Biomorphs via modified iterations. J. Nonlinear Sci. Appl. (2016); 9(5):2305--2315
Chicago/Turabian Style
Gdawiec, Krzysztof, Kotarski, Wieslaw, Lisowska, Agnieszka. "Biomorphs via modified iterations." Journal of Nonlinear Sciences and Applications, 9, no. 5 (2016): 2305--2315
Keywords
- Biomorph
- escape time algorithm
- Mann iteration
- Ishikawa iteration.
MSC
References
-
[1]
D. Ashlock, J. A. Brown, Fitness Functions for Searching the Mandelbrot Set, in: Proc. of the 2011 IEEE Congress on Evolutionary Computation, (2011), 1108-1115.
-
[2]
D. Ashlock, B. Jamieson, Evolutionary Exploration of Generalized Julia Sets, in: Proc. of the IEEE Symposium on Computational Intelligence in Image and Signal Processing, (2007), 163-170.
-
[3]
M. Barnsley, Fractals Everywhere, Academic Press, Boston (1988)
-
[4]
M. Barnsley , Superfractals , Cambridge University Press, Cambridge (2006)
-
[5]
V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin (2007)
-
[6]
V. Drakopoulos, Comparing Rendering Methods for Julia Sets, J. WSCG, 10 (2002), 155-161.
-
[7]
V. Drakopoulos, N. Mimikou, T. Theoharis , An Overview of Parallel Visualization Methods for Mandelbrot and Julia Sets, Comput. Graph., 27 (2003), 635-646.
-
[8]
K. Gdawiec, Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns, Comput. Graph. Forum, (in press)
-
[9]
K. Gdawiec, W. Kotarski, A. Lisowska, Automatic Generation of Aesthetic Patterns with the Use of Dynamical Systems, in: Advances in Visual Computing, part 2, in: Lecture Notes in Computer Science, 6939 (2011), 691-700.
-
[10]
K. Gdawiec, W. Kotarski, A. Lisowska, Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods, Abstr. Appl. Anal., 2015 (2015), 19 pages.
-
[11]
S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
-
[12]
V. G. Ivancevic, T. T. Ivancevic , Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals, Springer, Berlin (2008)
-
[13]
S. M. Kang, H. H. Alsulami, A. Rafiq, A. A. Shahid , S-iteration Scheme and Polynomiography, J. Nonlinear Sci. Appl., 8 (2015), 617-627.
-
[14]
W. Kotarski, K. Gdawiec, A. Lisowska, Polynomiography via Ishikawa and Mann Iterations, in: Advances in Visual Computing, part 1, in: Lecture Notes in Computer Science, 7431 (2012), 305-313.
-
[15]
M. Levin, Morphogenetic Fields in Embryogenesis, Regeneration, and Cancer: Non-local Control of Complex Patterning, BioSystems, 109 (2012), 243-261.
-
[16]
J. Leys, Biomorphic Art: an Artist's Statement, Comput. Graph., 26 (2002), 977-979.
-
[17]
B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, New York (1983)
-
[18]
W. R. Mann, Mean Value Methods in Iteration , Proc. Amer. Math. Soc., 4 (1953), 506-510.
-
[19]
N. S. Mojica, J. Navarro, P. C. Marijuán, R. Lahoz-Beltra , Cellular ''bauplants'': Evolving Unicellular Forms by Means of Julia Sets and Pickover Biomorphs, BioSystems, 98 (2009), 19-30.
-
[20]
A. Negi, M. Rani , A New Approach to Dynamic Noise on Superior Mandelbrot Set , Chaos Solitons Fractals, 36 (2008), 1089-1096.
-
[21]
A. Ortega, M. de la Cruz, M. Alfonseca, Parametric 2-Dimensional L-systems and Recursive Fractal Images: Mandelbrot Set, Julia Sets and Biomorphs, Comput. Graph., 26 (2002), 143-149.
-
[22]
H. O. Peitgen, D. Saupe, The Science of Fractal Images, Springer-Verlag, New York (1988)
-
[23]
C. A. Pickover , Biomorphs: Computer Displays of Biological Forms Generated from Mathematical Feedback Loops, Comput. Graph. Forum, 5 (1986), 313-316.
-
[24]
C. A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World, Dover Publications, Mineola (2001)
-
[25]
P. Prusinkiewicz, A. Lindenmayer , The Algorithmic Beauty of Plants, Springer-Verlag, New York (1990)
-
[26]
M. Rani, R. Agarwal, Effect of Stochastic Noise on Superior Julia Sets , J. Math. Imaging Vis., 36 (2010), 63-68.
-
[27]
M. Rani, V. Kumar , Superior Julia Set , J. Korea Soc. Math. Educ. Ser. D Res. Math. Educ., 8 (2004), 261-277.
-
[28]
M. Rani, V. Kumar, Superior Mandelbrot Set, J. Korea Soc. Math. Educ. Ser. D Res. Math. Educ., 8 (2004), 279-291.
-
[29]
D. Saupe, Efficient Computation of Julia Sets and Their Fractal Dimension, Phys. D: Nonlinear Phenomena, 28 (1987), 358-370.
-
[30]
S. L. Singh, S. Jain, S. N. Mishra , A New Approach to Superfractals, Chaos Solitons Fractals, 42 (2009), 3110-3120.
-
[31]
J. A. Sprott , Strange Attractors: Creating Patterns in Chaos, M&T Books, New York (1993)