These days computer-generated fractal patterns are everywhere, from squiggly designs on computer art posters to illustrations in the most serious of physics journals. Interest continues to grow among scientists and, rather surprisingly, artists and designers. This book provides visual demonstrations of complicated and beautiful structures that can arise in systems, based on simple rules. It also presents papers on seemingly paradoxical combinations of randomness and structure in systems of mathematical, physical, biological, electrical, chemical, and artistic interest. Topics include: iteration, cellular automata, bifurcation maps, fractals, dynamical systems, patterns of nature created through simple rules, and aesthetic graphics drawn from the universe of mathematics and art.
Chaos and Fractals is divided into six parts: Geometry and Nature; Attractors; Cellular Automata, Gaskets, and Koch Curves; Mandelbrot, Julia and Other Complex Maps; Iterated Function Systems; and Computer Art.
Additionally, information on the latest practical applications of fractals and on the use of fractals in commercial products such as the antennas and reaction vessels is presented. In short, fractals are increasingly finding application in practical products where computer graphics and simulations are integral to the design process. Each of the six sections has an introduction by the editor including the latest research, references, and updates in the field. This book is enhanced with numerous color illustrations, a comprehensive index, and the many computer program examples encourage reader involvement.
Introduction. Part 1. Geometry and Nature. Chaos game visualization of sequences (H. J. Jeffrey). Tumor growth simulation (W. Düchting). Computer simulation of the morphology and development of several species of seaweed using Lindenmayer systems (J.D. Corbit, D.J. Garbary). Generating fractals from Voronoi diagrams (K.W. Shirriff). Circles with kiss: a note on osculatory packing (C.A. Pickover). Graphical identification of spatio-temporal chaos (A.V. Holden, A.V. Panfilov). Manifolds and control of chaotic systems (H. Qammari, A. Venkatesan). A vacation on Mars - an artist's journey in a computer graphics world (C.A. Pickover). Part II. Attractors. Automatic generation of strange attractors (J.C. Sprott). Attractors with dueling symmetry (C.A. Reiter). A new feature in Hénon's map (M. Michelitsch, O.E. Rössler). Lyapunov exponents of the logistics map with periodic forcing (M. Markus, B. Hess). Toward a better understanding of fractality in nature (M. Klein, O.E. Rössler, J. Parisi, J. Peinke, G. Baier, C. Khalert, J.L. Hudson). On the dynamics of real polynomials on the plane (A.O. Lopes). Phase portraits for parametrically excited pendula: an exercise in multidimensional data visualisation (D. Pottinger, S. Todd, I. Rodrigues, T. Mullin, A. Skeldon). Self-reference and paradox in two and three dimensions (P. Grim, G. Mar, M. Neiger, P. St. Denis). Visualizing the effects of filtering chaotic signals (M.T. Rosenstein, J.J. Collins). Oscillating iteration paths in neural networks learning (R. Rojas). The crying of fractal batrachion 1,489 (C.A. Pickover). Evaluating pseudo-random number generators (R.L. Bowman). Part III. Cellular Automata, Gaskets, and Koch Curves. Sensitivity in cellular automata: some examples (M. Frame). One tub, eight blocks, twelve blinkers and other views of life (J.E. Pulsifer, C.A. Reiter). Scouts in hyperspace (S. Shepard, A. Simoson). Sierpinski fractals and GCDs (C.A. Reiter). Complex patterns generated by next nearest neighbors cellular automata (W. Li). On the congruence of binary patterns generated by modular arithmetic on a parent array (A. Lakhtakia, D.E. Passoja). A simple gasket derived from prime numbers (A. Lakhtakia). Discrete approximation of the Koch curve (S.C. Hwang, H.S. Yang). Visualizing Cantor cheese construction (C.A. Pickover, K. McCarty). Notes on Pascal's pyramid for personal computer users (J. Nugent). Patterns generated by logical operators (M. Szyszkowicz). Part IV. Mandelbrot, Julia and Other Complex Maps. A tutorial on efficient computer graphics representations of the Mandelbrot set (R. Rojas). Julia sets in the quaternions (A. Norton). Self-similar sequences and chaos from Gauss sums (A. Lakhtakia, R. Messier). Color maps generated by "trigonometric iteration loops" (M. Michelitsch). A note on Halley's method (R. Reeves). A note on some internal structures of the Mandelbrot set (K. J. Hopper ). The method of secants (J.D. Jones). A generalized Mandelbrot set and the role of critical points (M. Frame, J. Robertson). A new scaling along the spike of the Mandelbrot set (M. Frame, A.G. Davis Philip, A. Robocci). Further insights into Halley's method (R. Reeves). Visualizing the dynamics of the Rayleigh quotient iteration (C.A. Reiter).
The "burning ship" and its quasi-Julia sets (M. Michelitsch, O. E. Rössler). Field lines in Mandelbrot set (K.W. Phillip). A tutorial on the visualization of forward orbits associated with Siegel disks in the quadratic Julia sets (G.T. Miller). Image generation by Blaschke products in the unit disk (H.S. Kim, H.O. Kim, S.Y. Shin). An investigation of fractals generated by z→1/z -n + c (K.W. Shirriff). Infinite-corner-point fractal image generation by Newton's method for solving exp[-a (&zgr; + z)(&zgr; - z)] -1 = 0 (Y.B. Kim, H.S. Kim, H.O. Kim, S.Y. Shin). Chaos and elliptic curves (S.D. Balkin, E.L. Golebiewski, C.A. Reiter). Newton's methods for multiple roots (W.J. Gilbert). Warped midgets in the Mandelbrot set (A.G. Davis Philip, M. Frame, A. Robucci). Automatic generation of general quadratic map basins (J.C. Sprott, C.A. Pickover). Part V. Iterated Function Systems. Some nonlinear iterated function systems (M. Frame, M. Angers). Balancing order and chaos in image generation (K. Culik II, S. Dube). Estimating the spatial extent of attractors of iterated function systems (D. Canright). Automatic generation of iterated function systems (J.C. Sprott). Modelling and rendering of nonlinear iterated function systems (E. Gröller). Part VI. Computer Art. Automatic parallel generation of aeolian fractals on the IBM power visualization system (C.A. Pickover). Julia set art and fractals in the complex plane (I.D. Entwistle). Methods of displaying the behaviour of the mapping z→z2 + &mgr; (I.D. Entwistle). AUTUMN - a recipe for artistic fractal images (J.E. Loyless). Biomorphic mitosis (D. Stuedell). Computer art representing the behavior of the Newton-Raphson method (D.J. Walter). Systemized serendipity for producing computer art (D. Walter). Computer art from Newton's, Secant, and Richardson's methods (D. Walter). Author index. Subject index.