This monograph introduces the reader to the world of superfractals, fractal tops and Iterated Function Systems (IFS). It contains all the necessary groundwork in topology, metric spaces, semigroups, measures, and transformational geometry, but a solid background in any of these topics will make the reading of this book much easier.
The book is well structured and an enjoyable read. The chapters zero through three serve as an introduction and to lay the mathematical foundation. "Superfractals" is very well illustrated but the reader should not just look at the pretty pictures at the surface. The book contains some very interesting, very deep and even enthralling mathematics. The author's ability to clearly describe complicated phenomena paired with a pleasant writing style make this book a must read for all people interested in the mathematics of fractals.
Superfractals are at the confluence of deterministic and completely random fractal structures. Barnsley compares superfractals to leaves which are classified by the type of tree they come from but still exhibit randomness amongst the leaves of the same kind. Superfractals are attractors of IFS's on cross products of Hausdorff spaces.
I found Barnsley's explanation of the process of colour stealing particularly fascinating, because it creates truly remarkable fractals. Colour stealing is when you color the fractals with colors from separate pictures. There are many wonderful, colorful pictures that spark one's imagination. Examples spread liberally throughout the book together with relevant exercises help illuminate this new look at "patterns in nature".
Kai Brunkalla (firstname.lastname@example.org) is an assistant professor of mathematics at Walsh University. His PhD is in operator theory and his recent interests include fractal portraits of DNA sequences.
Part I. Geometries and Transformations: 1. Codes, metrics and topologies; 2. Transformations of points, sets, pictures and measures; 3. Semigroups on sets, measures and pictures; Part II. Iterated Function Systems: 4. IFS acting on measures; 5. More on IFS; Part III. Applications to Graphics: 6. Digital content production; 7. Image compression; 8. Super IFS.