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.

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.

points originally in the chaos basin y2=a(3,:); y3=a(4,:); are now attracted to the desired steady state. As we x=2:0.1:12.; increase Ac, and thus increase the amount of change y=2:0.2:12.; to the underlying system, some points which were ll=length(t); fori=l:ll originally in the steady state basin are no longer in=(yl(i)-2.)/0.1+l; stable. We believe all of these points lead to chaos n=(y2(i)-2,)/0.1+l; based on numerous simulations. Note that the basin time(m.n)=t(i); end boundary between the

was interested in preparing arrays of optical waveguides that were perfect as possible. Analysis of certain recursive tilings led Cook and his colleagues to conclude that the edges of optically useftil tilings were fractal in nature. This led to the developmen t of assembl y techniques and fractal array structures that allowed the Galileo researcher s to prepare highly ordered fiber arrays. One patent has already been granted on these techniques , and Incom, Inc. of Charlton, Massachusetts , has

description language ( S D L ). /* NOTES: 1. The radius of the tube is stored in the global variable r. 2. HasLinks and HasColors are Boolean global variables set by the user, 3. This code shows minimal use of bounding objects. If faster rendering is needed , form a composite object from several successiv e segment s and use another bounding object for the entire composite object. (This strategy can be extended to multiple levels by forming a larger composite object - with its own bounding

(1993). 12. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed., Addison-Wesley , Reading, MA (1991). 13. C. A. Pickover, Picturing randomnes s on a graphics supercomputer . IBM J. Res. Develop. 35, 227-230 (1991). 14. C. A. Pickover, Computers and the Imagination, St. Martin's Press, New York (1991). PART III CELLULAR, AUTOMATA, GASKETS AND KOCH CURVES This Page Intentionally Left Blank Chaos and Fractals: A Computer Graphical Journey C.A. Pickover (Editor) © 1998

the valuation of the army. (This transaction also defines s, i.e., s^ + s = 1.) With this assignmen t of hypercube values, the sum of hypercube values over all hypercube s whose first componen t (of their centers) is less than or equal to —3^7 + 1 is 1, as demonstrate d by the following string of equalities (which uses the geometric series formula and the identity ^^ + 5= 1). 2 - m ,+ |m2l + - 2 mi = -3rt+l m2=-oo m3=-oo = 2 ^y'C 2 s"")"-' m=-oo m=3n-l 00 OO = {s'"-')({/(\ Fig. 4'. A