SuperFractalsSuperFractals is the long-awaited successor to Fractals Everywhere, in which the power and beauty of Iterated Function Systems were introduced and applied to producing startling and original images that reflect complex structures found for example in nature. This provoked the question of whether there is a deeper connection between topology, geometry, IFS and codes on the one hand and biology, DNA and protein development on the other. Now, 20 years later, Barnsley brings the story up to date by explaining how IFS have developed in order to address this issue. New ideas such as fractal tops and superIFS are introduced, and the classical deterministic approach is combined with probabilistic ideas to produce new mathematics and algorithms that open a whole theory that could have applications in computer graphics, bioinformatics, economics, signal processing and beyond. For the first time these ideas are explained in book form, and illustrated with breathtaking pictures. |
Contents
Codes metrics and topologies | 10 |
Transformations of points sets pictures and measures | 89 |
Semigroups on sets measures and pictures | 190 |
Hyperbolic IFSs attractors and fractal tops | 313 |
Superfractals | 385 |
| 471 | |
Common terms and phrases
1-variable acting addresses affine applied approximations associated belonging boundary called chaos game Chapter circle closed code space colour compact complete compute consists constructed contains continuous contractive converges corresponding defined Definition denote described directed discussed distance diversity domain dynamical system elements Equation euclidean example Exercise exists Figure fixed point follows fractal fractal sets function geometry given Hausdorff homeomorphism hyperbolic IFSs illustrated illustrated in Figure invariant invertible limit linear look M¨obius transformations mapping mathematical means measure metric space natural Notice objects obtained orbital picture pair panels pixel plane points probability produced projective transformations Proof properties Prove random represented respect result segment semigroup sequence set attractor Show shown structure subsets superfractal Suppose symbolic Table Theorem tiling topology tops tree underlying union unique