Chaos and Fractals

Rating:
Summary: A good introduction
Review: Chaos as a physical theory began essentially in the 1970's, but as a mathematical field it has existed since the early 1900's. This book covers only the mathematical study of chaos, and is addressed to those readers who have a fairly strong background in undergraduate mathematics. A knowledge of dynamical systems and measure theory would help in the appreciation of the book, but are not absolutely necessary. The application of fractals and chaos to finance is now legendary, but other applications, such as to packet networks and surface physics are not so well-known. Current research in chaos is done predominantly in the context of information theory, wherein the goal is to understand the difference between chaos and noise, and develop mathematical tools to quantify this difference. The BASIC code in the book gives away its age, but can be easily translated to one of the symbolic computing languages available now, such as Maple or Mathematica.

This is a sizable book, and space prohibits a detailed review, but some of the more interesting discussions in it include: 1. The video feedback experiment, which can be done with only a video camera and a TV set. This is always a crowd pleaser, at whatever level of the audience it is presented to. 2. The comparison between doing iteration of a chaotic map on two different calculating machines: a CASIO and an HP. The difference is very dramatic, illustrating the effect of finite accuracy arithmetic. 3. The pictures illustrating the Chinese arithmetic triangle and Pascal's triangle as it appeared in Japan in 1781. 4. The space-filling curve and its relation to the problem of defining dimension from a topological standpoint. This discussion motivates the idea of covering dimension, which the authors overview with great clarity. They also give a rigorous definition of the Hausdorff dimension and discuss its differences with the box counting dimension. 5. The many excellent color plates in the book, especially the one illustrating a cast of the venous and arterial system of a child's kidney. 6. The difficulty in measuring power laws in practice. 7. Image encoding using iterated function systems, which has become very important recently in satellite image analysis. This leads into a discussion of the Hausdorff distance, which is of enormous importance not only in the study of fractals but also in general topology: the famous hyperspaces of closed sets in a metric space. 8. The relation between chaos and randomness, discussed by the authors in the context of the "chaos game." 9. L-systems, which are motivated with a model of cell division. 10. the number theory behind Pascal's triangle. 11. The simulation of Brownian motion. 12. The Lyapunov exponent for smooth transformations. 13. The property of ergodicity and mixing for transformations, the authors pointing out that true ergodic behavior cannot be obtained in a computer where only a a finite collection of numbers is representable. 13. The concept of topological conjugacy. 14. The existence of homoclinic points in a dynamical system. These are very important in physical applications of chaos. 15. The Rossler attractor and its pictorial representation. 16. How to calculate the dimensions of strange attractors. 17. How to calculate Lyapunov exponents from time series, which is of great interest in many different applications, especially finance. 18. The Julia set, which the authors relate eventually to potential theory.

Rating:
Summary: Simply a fantastic book
Review: I purchased this book when it first came out, during the
initial wave of popularity of fractals and chaos theory.
Although the fadishness of chaos and fractals has died
down, a number of solid applications for this theory have
appeared in areas like computer graphics, finance,
modeling computer network traffic and data compression.

I have purchased a number of books on fractals and chaos and
how these concepts can be applied in a number of areas. I
have yet to see a better introduction to the topic. This is
a core reference and I keep coming back to it again and again.

In the spectrum of popular science books, this is definitely
on the technical end. You do not need an advanced background
in mathematics as you do for some books on chaos and fractals,
but the authors do not shy away from equations. However, the
ideas are clearly presented. I have used this book as a
reference for developing software for fractal brownian motion
and Hurst exponent estimation.

"Chaos and Fractals" covers a great deal of material. On a few
occasions I found that the algorithms or explaination were
difficult to follow. In some cases, like the generation of
Gaussian random numbers, I found better, simpler algorithms.

When this book was written, fractals and chaos were fairly new.
It is difficult to avoid comparing this book to an even thicker
book, "A New Kind of Science" by Stephen Wolfram. Although
cellular automata, the core topic of "A New Kind of Science"
are not exactly new, Wolfram claims new and profound
perspectives. Many, including this reviewer, feel that Wolfram's
claims are overblown and egotistical (he has a bad habbit of
claiming credit for innovation, even as he cites other work).
The authors of "Chaos and Fractals" do not make exalted
claims for this work. Yet without any fanfare, this book
really does deliver profound ideas. This is simply a
fantastic book. I recommend it for anyone in the applied
sciences (e.g., computer science, quantitative finance,
geology, etc...). Even for the mathematically sophisticated it
will provide an valuable overview, which is difficult to obtain
anywhere else.

Rating:
Summary: Add some DEPTH to your "Recreational Mathematics"
Review: I spent quite a bit of time looking for a good "fractals" book. For me, this is it. It is not a book for everyone, though. I'll try to offer guidelines to help you decide if it is for you. In summary: (a) its not just a picture book, but extremely visual, (b) its not math-intense but asks for math-comfort and offers options and (c) its not only for computer jockeys, but offers repeated links to that approach.

This book is doubtless great for a high-school or college course in fractals. But I think it is also a worthy buy, albeit a pricey one, for a certain type of layperson with a fascination for mathematics presented in some depth. If you enjoy math but find some of the "popularizations" a bit too shallow, then the realm of fractals and chaos is a great place to explore in depth. This is a fine guidebook for that exploration.

"Chaos and Fractals" is not a book for the reader who is primarily fascinated with the visual representations of fractals. BUT it i!s chock-full of b/w illustrations (686 by the authors count) and nicely sprinkled with gorgeous color plates. The visual element is not central, but is very strongly represented and I found that almost every important concept was enhanced by the addition of a diagram or illustration.

This is definitely a book that delves into the mathematics of fractals. It does so in a well-crafted dual-track form. The core of the book should be comfortable and enjoyable mathematical reading for anyone with a sound and fairly current familiarity with high school math (Not that such "currency" suggests its only for youngsters! This old-timer preserves essentially that level of math by regular exposure to recreational math and the like). On the second track, the book provides mathematically in-depth views of selected topics. This is really nice if you like to stretch your mathematical horizons since you can use the core to steady your foundation understanding of a topic and then dive int!o the advanced mathematical topics at will; mustering strategic retreat when necessary, without loss of face, but sometimes learning how more advanced mathematics can be used.

Finally, the book makes an effort to scaffold some computer exploration of fractal concepts that succeeded for me but might not for you. For every chapter the authors provide a "Program of the Chapter" which allows exploration of one or more of the fractal forms and concepts explored therein. These are usually quite short and are written in Microsoft BASIC. This latter might be a problem for some. Nowadays, users with more advanced operating systems might not know where to find their version of BASIC (and it might not even be supplied), much less how to fire it up.

I would not belabor the BASIC program element too much except that experimenting with such code is an excellent way for anyone to better understand an algorithmic process. A program is, after all, such a process - a sequence of !discrete steps. I'd urge you to search your Windows disk for something like an "oldmsdos" folder and dig out the Qbasic files found there and fire them up. Even if you've never written a program, this kind of applied-use is a fine way to learn!

For the right sort of reader, this is unquestionably a 5-star book.

Rating:
Summary: Excellent for intermediate knowledge of chaos
Review: This book is a great entertainer for anyone who wants to spend many evenings "playing with chaos". The code in the book is a little dated (BASIC), but you won't have problems to use it as a good reference. The book will guide you through the understanding of the exciting realm of chaos and its hidden monsters.

Chaos and fractals are subjects that sound modern, interesting and eye-catching in the most of the cases. However, the applications and implications of chaos in the real world constitute the great achievement of human knowledge that the concept represents.

The lecture of this book doesn't require an extensive knowledge of math (but it would be helpful), it requires many will and passion for rediscovering your conception of the universe instead.

Before reading this book I'd recommend "Chaos: the Making of a New Science" by James Gleick and for those who are looking for a more compact but challenging material "Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise" by Manfred Schroeder will be just fine.

Rating:
Summary: Well worth the cost
Review: This is possibly the best and most thorough of all books on fractals. The discussion is excellent, the illustrations superb. After all, these are the guys who developed the computer art exhibits that toured Europe and parts of the US in the 1980s.

The mathematics is somewhat advanced, but not so advanced that most persons with a thorough background in high school mathematics cannot understand it. After all, I used it as a primary reference for my book Fractals in Music!

Rating:
Summary: Excellent for a Math/ Computer Science student
Review: While Chaos and Fractals isn't really a book for the layman, I highly recommend it for those of you out there who want a deep and comprehensive look at these things. I've read several fractal books, some simple (FractalVision: Making Fractals Work For You), some highly mathematical (Fractal Image Compression, Science of Fractals), but this is easily the best of the lot, not only for in-depth but understandable reading, but also for separation. If you only want to learn about bifurcation in repeated iteration, or only about strange attractors, just pick the appropriate chapter. If you don't want to know about the more complex proofs, skip them; they're in small print and set off with lines to mark them as optional.

I do recommend some mathematical education and an interest in (not necessarily a talent for) proofs to get the most from this book. They cite a lot of stuff that you probably haven't seen before if you haven't had some college calculus, at least the basics. And you won't understand the more complex stuff (basic topology, mainly) unless you've had some kind of proofs-based calc course. However, even without that, it's a _really_ neat book. There's a lot here that even the layman can understand, it's just that he'll be intimidated by the set-off parts that prove the results he's only skimming.

I highly recommend this to anyone who is serious about fractals, or thinks they might try to be so in the future. It will take quite some time for even a dedicated fractal enthusiast to become bored with the book, even if it's the only one you own.