Biomorphs via Modified Iterations

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. c ©2016 All rights reserved.

[1]  Daniel A. Ashlock,et al.  Fitness functions for searching the Mandelbrot set , 2011, 2011 IEEE Congress of Evolutionary Computation (CEC).

[2]  Agnieszka Lisowska,et al.  Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods , 2015 .

[3]  T. Ivancevic,et al.  Complex Nonlinearity: Chaos, Phase Transitions, Topology Change and Path Integrals , 2008 .

[4]  Mamta Rani,et al.  Effect of Stochastic Noise on Superior Julia Sets , 2009, Journal of Mathematical Imaging and Vision.

[5]  Vasileios Drakopoulos,et al.  An overview of parallel visualisation methods for Mandelbrot and Julia sets , 2003, Comput. Graph..

[6]  Przemyslaw Prusinkiewicz,et al.  The Algorithmic Beauty of Plants , 1990, The Virtual Laboratory.

[7]  V. Berinde.  Iterative Approximation of Fixed Points , 2007 .

[8]  Nelly Selem Mojica,et al.  Cellular "bauplans": Evolving unicellular forms by means of Julia sets and Pickover biomorphs , 2009, Biosyst..

[9]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[10]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[11]  Agnieszka Lisowska,et al.  Polynomiography via Ishikawa and Mann Iterations , 2012, ISVC.

[12]  D. Ashlock,et al.  Evolutionary Exploration of Generalized Julia Sets , 2007, 2007 IEEE Symposium on Computational Intelligence in Image and Signal Processing.

[13]  Mamta Rani,et al.  Superior Mandelbrot Set , 2004 .

[14]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[15]  Dietmar Saupe,et al.  Efficient computation of Julia sets and their fractal dimension , 1987 .

[16]  Clifford A. Pickover.  Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , 2001 .

[17]  Vasileios Drakopoulos.  Comparing Rendering Methods for Julia Sets , 2002, WSCG.

[18]  J. Sprott.  Strange Attractors: Creating Patterns in Chaos , 1993 .

[19]  Michael Levin,et al.  Morphogenetic fields in embryogenesis, regeneration, and cancer: Non-local control of complex patterning , 2012, Biosyst..

[20]  Krzysztof Gdawiec,et al.  Inversion Fractals and Iteration Processes in the Generation of Aesthetic Patterns , 2017, Comput. Graph. Forum.

[21]  Agnieszka Lisowska,et al.  Automatic Generation of Aesthetic Patterns with the Use of Dynamical Systems , 2011, ISVC.

[22]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[23]  Sarika Jain,et al.  A new approach to superfractals , 2009 .

[24]  Alfonso Ortega,et al.  Parametric 2-dimensional L systems and recursive fractal images: Mandelbrot set, Julia sets and biomorphs , 2002, Comput. Graph..

[25]  Michael Cory.  Computers, pattern, chaos and beauty, graphics from an unseen world , 1991 .

[26]  C A Pickover,et al.  Biom orphs: Computer Displays of Biological Forms Generated from Mathematical Feedback Loops , 1986, Comput. Graph. Forum.

[27]  Ashish Negi,et al.  A new approach to dynamic noise on superior Mandelbrot set , 2008 .

[28]  S. Ishikawa.  Fixed points by a new iteration method , 1974 .

[29]  Shin Min Kang,et al.  S-iteration scheme and polynomiography , 2015 .