Chaos and Fractals: New Frontiers of Science - Hardcover

Peitgen, Heinz-Otto; Jurgens, Hartmut; Saupe, Dietmar

Hardcover

ISBN 10: 0387979034 ISBN 13: 9780387979038

Publisher: Springer-Verlag, 1992

Synopsis

For almost 15 years chaos and fractals have been riding a wave that has enveloped many areas of mathematics and the natural sciences in its power, creativity and expanse. Traveling far beyond the traditional bounds of mathematics and science to the distant shores of popular culture, this wave captures the attention and enthusiasm of a worldwide audience. The fourteen chapters of this book cover the central ideas and concepts of chaos and fractals as well as many related topics including: the Mandelbrot Set, Julia Sets, Cellulair Automata, L- systems, Percolation and Strange Attractors. Each chapter is closed by a "Program of the Chapter" which provides computer code for a central experiment. Two appendices complement the book. The first, by Yuval Fisher, discusses the details and ideas of fractal images and compression; the second, by Carl J.G. Evertsz and Benoit Mandelbrot, introduces the foundations and implications of multifractals.

About this title

Review:

Fascinating and authoritative, Chaos and Fractals: New Frontiers of Science is a truly remarkable book that documents recent discoveries in chaos theory with plenty of mathematical detail, but without alienating the general reader. In all, this text offers an extremely rich and engaging tour of this quite revolutionary branch of mathematical research.

The most appealing aspect about Chaos and Fractals has to be its hundreds of images and graphics (with dozens in full-color) used to illustrate key concepts. Even the math-averse reader should be able to follow the basic presentation of chaos and fractals here. Since fractals often mimic natural shapes such as mountains, plants, and other biological forms, they lend themselves especially well to visual representation.

Early chapters here document the mathematical oddities (or "monsters") such as the Sierpinski Gasket and the Koch Curve, which laid the groundwork for later discoveries in fractals. The book does a fine job of placing recent discoveries about chaos into a tradition of earlier mathematical research. Its description of the work of mathematicians like Pascal, Kepler, Poincaré, Sierpinski, Koch, and Mandelbrot makes for a fine read, a detective story that ends with the discovery of order in chaos. (For programmers, the authors provide short algorithms and BASIC code, which lets you try out plotting various fractals on your own.)

This is not, however, only a book of pretty pictures. For the reader who needs the mathematics behind chaos theory, the authors in no way dumb down the details. (But because the richer mathematical material is set off from the main text, the general reader can still make headway without getting lost.)

There have been advances in the field since this book's publication in 1992, but Chaos and Fractals remains an authoritative general reference on chaos theory and fractals. A must for math students (and math enthusiasts), Chaos and Fractals also deserves a place on the bookshelf of any general reader or programmer who wants to understand how today's mathematicians and scientists make sense of our world using chaos theory. --Richard Dragan

Topics covered: Overview of fractals and chaos theory, feedback and multiple reduction copy machines (MRCMs), the Cantor Set, the Sierpinski Gasket and Carpet, the Pascal Triangle, the Koch Curve, Julia Sets, similarity, measuring fractal curves, fractal dimensions, transformations and contraction mapping, image compression, chaos games, fractals and nature, L-systems, cellular automata basics, attractors and strange attractors, Henon's Attractor, Rössler and Lorenz Attractors, randomness in fractals, the Brownian motion, fractal landscapes, sensitivity and periodic points, complex arithmetic basics, the Mandelbrot Set, and multifractal measures.

Review:

“It is relatively discursive and easy to read, with each chapter telling a coherent story, and it highlights the key concepts and ideas, examining a few models in detail and using worked numerical examples as well as visualisations and illustrations ... . makes an excellent entry to the broader mathematics of fractals and chaos, especially for students who are curious about the details as well as the core concepts but don’t want to get bogged down in formal mathematics.” (Danny Yee, Danny Yee’s Book reviews, February, 2016)

"It is one of the best introductions to chaos and fractals around. ... Unlike some other books on fractals, it can be read by non-specialists ... . The book is beautifully produced and well illustrated so it is a pleasure to read." (Hugh Williams, The Mathematical Gazette, Vol. 90 (5l9), 2006)

"The first edition of this vast introduction to chaos and fractals appeared in 1992. This new edition is virtually identical to the original except for some material ... . the book is ... a wonderful tour of a fascinating area of mathematics, and now the reader can take this tour while carrying around a slimmer (but still hefty) volume. ... The authors have a friendly conversational style ... . This is a great book ... ." (Raymond N. Greenwell, MathDL, May, 2005)

"Chaos and Fractals: New Frontiers of Science is an amazing introduction to the ideas of fractal geometry and chaotic dynamics ... . The authors have done a tremendous job in explaining quite difficult concepts in an elegant and simple way ... . I enjoyed this book tremendously – the authors have put in a tremendous amount of work in making a vast and interesting subject accessible ... . I wholeheartedly recommend this book to anyone with even a passing interest in the subject matter." (Dr. S. Virmani, Contemporary Physics, Vol. 46 (6), 2005)

"There appeared many books in the 1980’s and early 1990’s that ... required only a limited mathematical background to understand. They made the fractals, chaos and the Mandelbrot and Julia sets quite popular ... . The ... book that is under review here is one of these popular books. ... The book will remain what it has been so far: an outstanding book that contains all you ever wanted to know about fractals and chaos accessible to all levels of mathematically skilled." (Bulletin of the Belgian Mathematical Society, Vol. 12 (3), 2005)

"The book is written for everyone who wants to learn details of chaos theory and fractal geometry, also for readers who have not much knowledge of technical mathematics. In the fourteen chapters the central ideas and concepts of chaos and fractals are developed ... ." (F. Haslinger, Monatshefte für Mathematik, Vol. 144 (4), 2005)

"This is the second ... edition of what has been a bestseller since its first publication in 1992. ... All the laudatory comments heard twelve years ago about this fascinating book remain entirely valid. No one has succeeded in better presenting ... . the presentation has not aged at all – the comprehensiveness of the underlying mathematics and the illustrative power of the figures has never been surpassed. Twelve years after its first edition this book remains a must buy." (André Hautot, Physicalia, Vol. 57 (3), 2005)

"Numerous books have appeared in recent years that either explore the beauty of fractal art, describe techniques for its creation, or investigate some aspect of the related field of chaotic behavior. The present work attempts to accomplish all three goals in one huge volume...the authors should be applauded for their ambitions undertaking." Mathematical Reviews

"This book ... contains all one ever wanted to know about fractals, and more. Written by–next to Mandelbrot–the greatest popularizer of the concept of fractal geometry ... It contains a wealth of information on nearly every angle of the topic... I enjoyed reading the book for its lucid approach, its attempt at completeness, and especially, for the large number of illustrative figures and pictures." Zentralblatt Mathematik

About this edition

• Publisher: Springer-Verlag
• Publication date: 1992
• ISBN 10: 0387979034
• ISBN 13: 9780387979038
• Binding: Hardcover
• Edition number: 2
• Number of pages: 984