Product Description

The Fractal Geometry ofNature is a mathematics text. But buried in the deltas and lambdasand integrals, even a layperson can pick out and appreciateMandelbrot's point: that somewhere in mathematics, there is anexplanation for nature. It is not a coincidence that fractal math isso good at generating images of cliffs and shorelines and capillarybeds. ... Read more

Customer Reviews (20)

A great insight into the world of fractals
This is a great book for someone looking to discover what all the hullabaloo is about fractals.It provides a wonderful insight into the mind of one of the great mathematical geniuses of our time, Benoit Mandelbrot.I think some people will find his writing style a bit too stodgy, almost arithmetical, but I find it interesting.I especially appreciate how marks off when he's going to be tangential with special brackets.Mandelbrot doesn't delve into too many rigorous mathematical proofs of the various topics he discusses.He broaches each subject in such a way that should be accessible to people from a wide array of sciences and disciplines.I don't recommend this book if you're trying to figure out how to create simple fractal programs.But I enthusiastically recommend it if you want to learn more about fractals, discover a new way to think about and understand nature, or are simply looking for a good bit of erudition.

A fractal is by definition
A set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.

The definition of a fractal pretty much sets the tone for the book. There are mostly definitions and monochrome diagrams to explain the more classical fractals. The book does shows some practical geometric uses for fractals but I would not let it get anywhere near my Koch Curve.

I am not being kind to this book as there is a color section in the center. That shows "The Great Wave" by Katsushika Hokusai (1760-12849.) And an extensive reference section.

The book its self could easily be used as a text book for school.

Fractals: Hunting the Hidden Dimension (Blu-ray) [Blu-ray]

An Eye For Fractals: A Graphic And Photographic Essay (Studies in Nonlinearity)

How We Mandel
This book is the latest version of a book the famous Benoit Mandelbrot wrote back in the mid-1970s, in which he coined the term "fractal". The subsequent version was released around 1980 and had, among other pictures, a black blotchy image on a white background which he called "the µ-map". Then some joker started calling it "The Mandelbrot Set" and he had to change the book again.

It is true that this is not the best-written book on fractal geometry. However, for a time it was the ONLY book on fractal geometry, and as such has incredible historic value. Imagine in ancient Greece where people had to borrow one of Euclid's latest scrolls to read about things found in no other work.

Personally, this book has taught me only a few things. I had already learned about fractals from articles in 1980s issues of Scientific American, and computer programs in Compute! magazine.

Many black-and-white images suffuse this tome, though there are some color plates which are by no means as complex as today's fractallographies, but will serve as an introduction to the genre.

The only Mandelbrot Set image is the blotchy one mentioned earlier. That's because Dr. Mandelbrot, though he discovered the set, wasn't the first to color the complement, and it was Heinz-Otto Peitgen's 1984 book "The Beauty of Fractals" that has the first color Mandelbrot pictures.

I wholeheartedly recommend this book for a glimpse into history, and the uninitiated may learn something as well; though I wouldn't demand that much of it.

A fractal is by definition
A set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.

The definition of a fractal pretty much sets the tone for the book. There are mostly definitions and monochrome diagrams to explain the more classical fractals. The book does shows some practical geometric uses for fractals but I would not let it get anywhere near my Koch Curve.

I am not being kind to this book as there is a color section in the center. That shows "The Great Wave" by Katsushika Hokusai (1760-12849.) And an extensive reference section.

The book its self could easily be used as a text book for school.

Fractals: Hunting the Hidden Dimension (Blu-ray) [Blu-ray]

An Eye For Fractals: A Graphic And Photographic Essay (Studies in Nonlinearity)

item as promised
it took a long time to get to me but it was delivered as descibed, great condition... good seller ... Read more

Product Description
In his latest book, former senior computer systems designer and bestselling author Gregg Braden merges these ancient and modern world views into a powerful new model of time. Marrying the modern laws of fractal patterns to the ancient concept of cycles, he demonstrates how everything from the war and peace between nations to our most joyous relationships and personal crises are the returning patterns of our past. As each pattern returns, it carries the same conditions of previous cycles—fractal patterns that can be known, measured and predicted!

        What makes this model so important today is that the returning cycles also carry a window of opportunity—a choice point—that allows us to choose a new outcome for the cycle. Braden suggests that if we can see time from this perspective, the patterns will show us what’s in store for the future, and perhaps how to avoid the mistakes of our past.

        After presenting the case histories that confirm the accuracy of fractal time calculations, the author crosses the traditional boundaries of science and spirituality to answer the question that must be asked: What does fractal time tell us about 2012, and beyond? Because the cycles repeat, the seed for 2012 has already happened and the pattern already exists! In a narrative format of easy-to-read science and true-life accounts, Fractal Time shows us what we can expect as we close the Great World Age described by the Mayan Calendar, and the secret to our moment in history.

Customer Reviews (74)

gregg braden should write science fiction, he is a fraud
again gregg braden is making a lot of money selling crap.
he is not a scientist, but rather a technologist. his theories are all scientifically wrong. scientists don't even bother with him .They work their butts off trying to get to the truth of things, and then this clown grasps a little tiny bit of what they say and goes on to write his fiction.
Russian scientists found out that the DNA molecule is magnetic, and this clown tells us that we can perform miracles becuase of that!!! like Jesus.yah! right It is like telling us that we can cure people holding a magnet in our hands. Gregg it it is high time that you stop the crap coming out of your head!!! and leave us alone

The Soul of a Woman's Vision
I have read many of Braden's books and seen him in person at workshops. This book reduces the fears that have been created regarding 2012. I loved his veiw point that the earth's location in space can trigger cycles of spiritual growth and what a window of opportunies opens up for us to enlarge our spiritual growth process.He promotes the power of personal choice better than any other author and proves mathematically how we can change the outcomes of our lives. We now have the power to change our future. The Soul of a WomanThank you Gregg!

Create your own windows of opportunity
Well-known bestselling author Gregg Braden merges his scientific background as a former computer scientist with his beliefs and research. Fractal Time explains his method of using events in the past to predict similar conditions in the future. The universe is not random, but operates in recognizable patterns. With the mystical December 21, 2012 date looming ever closer, Braden explains the Mayan calendar and that this upcoming date is merely the end of a cycle which is 5,125 years long. He presents traditions and predictions from indigenous cultures around the globe and across history, and they all point to this cycle end-date. Braden asserts that life on Earth presents windows of opportunities based on these fractal patterns. People can then exercise their free will and either go with the flow or consciously change the patterns.

-- Alice R. Berntson, New Connexion Journal
[...]

Interesting information about 2012
All Gregg Braden's books bear upon the larger scheme of things, esoteric messages encoded in ancient times, planetary themes such as the Shift, etc etc. This book is no exception.

It is divulged that we are rapidly approaching the end of a 5,125-year cycle of time, the end of a world age. This began in 3114 B.C. and will culminate in December 21, 2012.

Moreover, this cycle constitutes the last fifth of an even greater cycle (or world) describing "the precession of the equinoxes - our 25,625-year journey through the 12 constellations of the zodiac".

The date December 21, 2112 signifies the alignment of our Earth and the entire solar system with the "galaxy's equator" and "the heart of the Milky Way".

Braden defines "fractals" as "simple, self-similar, and repeating patterns" and it is his thesis that everything that happens in our universe occurs in repeating cycles.

This applies both to momentous world events and events in our personal lives. The appendices contain a simple method to calculate the reoccurrence of personal events or rather the reoccurrence of similar conditions in our lives to a particular seed event.

The momentous date in 2012 is not to be regarded as "the end of the world". What is occurring is not actually an event but an unfolding process. Just as life crises although seemingly horrendous actually present wonderful opportunities for growth, 2012 may prove to be the year where a global crisis forces mankind to transform to a new level of being, where we learn to allow heart-based decisions instead of those based on cold reason to take precedence.

Fractal Time
Greg Braden is an insightful person, his studies about the Universe and his ability to put his wisdom and knowledge into words that are easily assimilated by the reader is a masterful ability in itself. I am a fan of Braden's writing and thinking and am eager to learn as much as I can about the Mayan calendar and the cycles/patterns of the Universe. Braden has helps in putting the upcoming 2012 date in perspective. Fractal Time is one of the most important books for this time. ... Read more

Product Description
From the inventor/founder of fractal geometry, the award-winning book that turns modern financial theory on its head

Mathematical superstar and inventor of fractal geometry, Benoit Mandelbrot, has spent the past forty years studying the underlying mathematics of space and natural patterns. What many of his followers don't realize is that he has also been watching patterns of market change.

In The (Mis)Behavior of Markets, Mandelbrot joins with science journalist and former Wall Street Journal editor Richard L. Hudson to reveal what a fractal view of the world of finance looks like. The result is a revolutionary reevaluation of the standard tools and models of modern financial theory. Markets, we learn, are far riskier than we have wanted to believe. From the gyrations of IBM's stock price and the Dow, to cotton trading, and the dollar-Euro exchange rate--Mandelbrot shows that the world of finance can be understood in more accurate, and volatile, terms than the tired theories of yesteryear.The ability to simplify the complex has made Mandelbrot one of the century's most influential mathematicians. With The (Mis)Behavior of Markets, he puts the tools of higher mathematics into the hands of every person involved with markets, from financial analysts to economists to 401(k) holders. Markets will never be seen as "safe bets" again. ... Read more

Customer Reviews (70)

An opportunity for continued thinking
Benoit Mandelbrot's The (Mis) Behavior of Markets is a splendid read and very informative. As many reviewers have noted, Mandelbrot invented fractal geometry. He has also been on the cutting edge (some would say fringe, but he's thinking and questioning) in multiple disciplines, as his curiosity seem to know no bounds. Mandelbrot does a good job of describing the inadequacies of the efficient market hypothesis and CAPM and other sacrosanct theories in finance, and he offers for our consideration an alternative view. His view is based on his assertion of reality; namely that the world of finance is turbulent (as indeed, the world is turbulent), and linear tools relying on reliability and rational man will never tell the full story.

Mandelbrot, to his credit, warns the reader early on that his is not an investment guide. He simply offers his ideas, and admits that some don't offer as much insight currently as he wishes. He is, however, optimistic that his philosophies and his alternative (edge-type) thinking will prevail in some form.

Truly wish I read this before b-school, as he explains why I scratched my head through a good portion because the theories didn't look "right." As a couple of other reviewers have noted, the first part of the book is best, and sort of stumbles to find itself in the second half. All that said, this is a valuable contribution and highly recommended.

An unconventional view of financial markets
Benoit Mandelbrot presents a very sophisticated approach to describing the statistics of financial markets.The assumption of gaussian statistics is often the starting point for modeling many phenomena.Sophisticated modelers soon realize that gaussian statistics are inadequate in many cases; and so, it is not surprising that gaussian statistics fail in modeling financial markets.I can only hope that the organizations that analyze risk for retirement investments and the plans that these organizations recommend do not use gaussian statistics in predicting possible retirement financial scenarios.

Excellent Book
A must-read if you enjoyed Taleb's Black Swan and want more information on the ideas behind Taleb's postulations.

The Awesomeness of Fractals
(Note: first, I recommend that readers read Gaetan Lion's excellent review of this book.My review addresses narrower, somewhat technical aspects of the book.I apologize in advance for its great length.)

Second, some backstory to this book.Institutional investors such as insurance companies or brokerages usually do not focus on picking stocks per se; rather, they operate like an actuary, selecting the odds of particular outcomes and insuring against the bad outcomes in the cheapest way. For a large number of investors, the returns come from "cheaply-held risk," meaning that risky (but potentially higher-performing) assets are insured against massive losses in the least expensive way prudent.Naturally, a cheap way to hold risky assets is to not insure against risk, but of course this is extremely dangerous.


For a smaller number of investors, the returns come from providing the insurance: writing put options, exotic swaps, and so on. Limited liability partnerships that did this sort of thing used to be known as "hedge funds," since they were businesses that sold hedges against risk to other investors. More recently, portfolio insurance (in the forms of exotic derivatives) has become a major source of income for the entire FIRE sector. Even if one nets out the immense losses of 2008, it's been a very lucrative business worldwide.More about that in a minute.

As with other forms of insurance, the paramount consideration is pricing risk, which depends on the size and frequency of market fluctuations.In theory, these fluctuations not only ought to be random, but the frequency of each variation (ranked by size and direction) ought to map out a Normal (Gaussian) curve, as other random phenomena do.Instead, it maps out other curves.Since there MANY other curves that could potentially do the job--and softwares for applying them, too--this is not really a serious hurdle.But the problem doesn't really end there.


If anomalies like the 6 May 2010 1000-pt. drop in the Dow were randomly distributed THROUGH TIME, then adapting risk evaluation methods like the Black-Scholes formula (used to compute the value of derivatives) would be easy. Once we've replaced a Normal distribution with something that fits the data, we would still wrongly expect 6-May-2010 shocks to occur very far apart.But while volatility overall is not at unprecedented levels, individual anomalies are larger and more frequent than ever; and even if we focus on much more commonplace events, we find that they have very un-even distribution through time. In fact, such shifts come in consecutive (or almost-consecutive) bursts; this is known as "autocorrelation," and it makes a hash of any effort to actually price risk.


Prudence says that capital managers should keep value at risk below 20% of the value of the portfolio, which should be easy with a reasonable amount of hedging.But if one acknowledges the possibility both of very large numbers of anomalies, AND their autocorrelation (incidence close together in time), then hedging adequately becomes prohibitively expensive, eliminating gains from trading.


Mandelbrot argues that the patterns of price changes are fractal in nature; the (log) price history for the Dow, for example, looks rather the same if you show a 330-day chart or a 330-minute chart.Revealingly, he demonstrates how to use fractals to simulate a fictional trading chart that mimics the statistical behavior of the real stock.However, after explaining this backstory (and including the marginally relevant story of H.E. Hurst's study of the Nile), the authors have little to add.This is not the fault of fractals per se; indeed, several studies of multifractal models of asset returns (MMAR) compared to GARCH(1,1) suggest that fractals are indeed a stronger tool for capturing the true risk of asset returns.*


Part of the problem may be the difficulty of explaining the mathematics/nature nexus clearly. Mandelbrot is not merely a brilliant mathematician, he's a gifted writer; his 1997 paper introducing MMAR is remarkably easy to read, considering its historical significance.But for even broader diffusion, a prominent financial editor co-wrote the book, and probably decided the upper limit on how much could be explained to non-statisticians.


However, a more serious issue is the lack of rationale for the revised theory. What I mean by this is, we can always tweak the mathematical model to produce results that approximate reality very closely.As explained above, even a random (simulated) process is valuable to a trader, if the random process captures the statistical features of the real process--in this case, a time series of asset prices.But if the mathematical model is nothing more than yet another shrink-to-fit snapshot of history, it's nothing more than data mining writ large.It's as if I generated a Fourier series that precisely matched the movement of the Dow over many years; without an underlying narrative, it's nothing but an ingenious way to draw another chart of PAST price movements. The pattern and timing of anomalies could continue to change over time in a wholly unpredictable way, in the absence of an overriding explanatory narrative.
_______________________________________
* GARCH(1,1): generalized autoregressive conditional heteroskedasticity. The (1, 1) refers to the number of lags of the conditional variance and the number of lags of the squared innovations.An example of such a study would be Thomas Lux, "Multi-Fractal Processes as Models for Financial Returns" (1999).

Standard deviation, CAPM, Modern Portfolio Theory turned on their heads
If you are questioning the use of standard deviation, CAPM, and Modern Portfolio Theory as effective tools for money management supporting asset allocation portfolios at some of the largest and oldest institutions on Wall Street then this book will confirm for you the major flaws supporting these methologies and portfolios. ... Read more

Product Description
Since its original publication in 1990, Kenneth Falconer's Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general mathematical theory and applications of fractals in a way that is accessible to students from a wide range of disciplines. This new edition has been extensively revised and updated. It features much new material, many additional exercises, notes and references, and an extended bibliography that reflects the development of the subject since the first edition.
* Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals.
* Each topic is carefully explained and illustrated by examples and figures.
* Includes all necessary mathematical background material.
* Includes notes and references to enable the reader to pursue individual topics.
* Features a wide selection of exercises, enabling the reader to develop their understanding of the theory.
* Supported by a Web site featuring solutions to exercises, and additional material for students and lecturers.
Fractal Geometry: Mathematical Foundations and Applications is aimed at undergraduate and graduate students studying courses in fractal geometry. The book also provides an excellent source of reference for researchers who encounter fractals in mathematics, physics, engineering, and the applied sciences. ... Read more

Customer Reviews (6)

The Basic Text For Understanding Fractals
Fractal Geometry:Mathematical Foundations and Applications by Kenneth Falconer(second edition)is one of the most important mathematical books of this beginning of the 21-st century.It is a book of high mathematical level which can be very useful to non-mathematicians possessing a reasonable mathematical instruction and a logical mind.Mathematicians will find in this book deep and sophisticated notions and proofsand non-mathematicians will find all the concrete applications of the theory of fractals(see e.g.the new paragraph dedicated to fractals in finance).Prof.dr.Ion Chitescu
Faculty of Mathematics and Computer Science
University of Bucharest

A rare find
I agree with all that was said by the other reviews here but add one important point. The physical layout, (typeface, drawings, whitespace etc.) of this book is brilliantly done. This is often overlooked by the producers of technical works who do it "on the cheap", but it is vital if one is to use the book day after day, as I have had to.

While the subject matter is not easy, this is an excellent book to motivate one to get stuck into the underlying mathematics. The reward is a little insight into the often beatiful theorems and practical results found in this stimulating field of study.

What every student should know about fractals.
Fractals make headlines from time to time[--are they everywhere?], and and they make lovely color pictures; but they are also part of a substantial mathematical theory, one with an
exciting mathematical history. This very important book presents
the subject in a way that it can be taught to students, and it starts with the basics, systematically, step by step, building up the material. Or it can be used for selfstudy! It has great exercises too! In view of the many applications to geometric analysis, to PDE, and to statistics, it is likely that fractal geometry will soon be a standard math course taught in many (more) math departments. By now it is widely recognized that the selfsimilarity aspects of the wavelet algorithms are key to their sucess. The book came out in 1990, and the author has an equally attractive book on the subject from 1985[The geometry of fractal sets] with a slightly more potential theoretic bent.

Theoretical as well as practical insight
The first part of the book is essentially of a theoretical nature, with a thorough treatment of fractal geometry at a mathematical point of view. The second part on the other hand provides a flavour of the problems of fractal geometry in practice...so mathematicians as well as people interested in applications only should both find this book interesting. The maths are not easy but quite "understandable" for science undergrads...some notions of calculus or topology would help... but the introduction is excellent and allows anyone to follow the course of the book (but for understanding the proofs a good math background is required).

Excellent for understanding the geometrical properties of fractals.

Exposes fractal geometry as a real mathematical discipline.
I appreciate Falconer's books on fractal geometry because they show the topic as it really is: a whole mathematical discipline on its own right and not just a nice temporary fashion.

It begins introducing basictopological concepts and then proceeds to develop the theory for severalpossible definitions of fractal dimension, showing the relations betweenthem. Then it explores deeply the local geometry of different kinds offractal objects, and studies some other geometrical situations, like thepojection of fractals (ever thought of a DIGITAL sundial? Here it isdescribed!).

The book also includes a lot of applications to other areasof mathematics and physics, a great amount of graphics, and muchmore.

The text is suitable from third year undergraduate school and on.It is a larger but lighter version of "The Geometry of FractalSets". ... Read more

Product Description

Customer Reviews (12)

Excellent introduction to fractals and power laws
This book is an extraordinarily well written and presented introduction to fractals and power laws. It has a far deeper mathematical level and requires more time and effort to understand than the typical popularizations. I find it specifically well suited for the the mathematically inclined layman and scientists outside mathematics who want to delve deeper into this subject. Note, however, that if you are a hard core mathematician (of the "lemma - proof - corollary" type), you might probably find this book too superficial and light for your taste, so this book is not for you. If you are an Earth, Life or medical scientist, or just a mathematically curious guy, you will find this book interesting and rewarding. I wholeheartedly recommend it.

Collection of specific cases
This book explores many cases of self similar structures that give rise to fractals .
It is not mathematically oriented and the few mathematical arguments are easy .
It is full of examples of anecdotical character demonstrating power laws and self similarity (concert halls , music , image treatment etc) .
There are also some nice pictures .
However it is not by any account a book concerning the chaos theory .
As a physicist I have been disappointed .
It is too long to be a book on fractal esthetics and it is too short and too anecdotical to be a book about non linear dynamics .
The only description I can find would be : entertaining mathematical games on the concept of iteration and self similarity .

A Chaotic Heaven


What a head-trip! While the Pearly Gates of Paradise may be more than a few minutes away, you are almost certain to enjoy the journey with this book in hand. I purchased this book from Amazon back in 2002 and apart from the curling cellophane-coated front cover, I have nothing but praise for it. It simply gets better, every single time I read it - not unlike sipping some fine vintage even as it ages.

It must be difficult to write a book on a subject so intrinsically mathematical while retaining a healthy, comprehensible tone with a twist of the ridiculous. Schroeder has an enviable sense of comic timing in addition to his peculiarly personalized insight into the world of Number Theory. It is pretty amazing, considering the broad and variable scope of his exposition that the entire opus did not descend into an inexorable chaotic mess of formulae. He skillfully manages to avoid the quagmire of complexity by properly abbreviating lengthy explanations with diagrams, pretty color prints and even the occasional cartoon aside. This leaves him enough time for the most engaging (not to mention informative) anecdotes which allows him to bring the reader into certain obscure fields of research - bilingual poetry, cheating at roulette and on how to kill Germans with Gift(s) - so to speak.

Do not be fooled by the casual tone of the book because this is anything but a cursory tour. In fact, if this is your first encounter with Chaos and Fractals, it may be better to have more than one supplementary text at hand. (I suggest Peitgen, Jurgens and Saupe's Chaos and Fractals: New Frontiers of Science.) Schroeder's book is written for serious students, who want to see some practical (and sometimes not-so-practical) applications of what were once mere mathematical monstrosities. Neither Weierstrass nor Cantor could have predicted that their little monsters would turn out to dominate the physical world. This book gives you an insightful look at how far non-differentiable functions have come since those early pioneering days.

Go ahead and buy this book. It is what every scientific book should aspire to be - brilliant and funny (exactly in that order!)

Great Math Book
This is a one of the best semi-technical mathematics books I ever read. What I mean by "semi-technical" is, you need somewhat of a math interest and education to appreciate it, and if you have that, you can read it casually. You don't need pen and paper; it's not a textbook. However, occasionally you will want to grab the pen and paper to verify what the author writes.
For an ex-math person as myself, this book is an eye-opener as to how many areas of life are touched byfractals and chaos theory. Everything from nature, to economic markets, to music, to just plain theoretical stuff is mentioned here. And the writer delivers it in a well-organized, lucid, entertaining, and passionate fashion. And it is well-illustrated, which really helped me understand....
I'm on my 3rd reading of this book since 1992, and if I wear out the book, I'm buying another one! I rank this up there with "Prime Obsession" as the two best non-textbook math books I ever read.

For the uninitiated!.--Fun too!
For the uninitiated! --The author combines insight with story telling. He has a story to tell, and does it well! Not only does he know the theory inside out, he has the ability to get accross the central points so it (almost) seems easy, in any case entertaining, using pictures (including cartoons), humor, and equations when they are needed. He further make clear the many fascinating links between chaos theory, algorithms, technology, and areas of pure math, such as number theory. Highly recommended! ... Read more

Product Description
The first book to discuss fractals solely from the point of view of computer graphics, this work includes an introduction to the basic axioms of fractals and their applications in the natural sciences, a survey of random fractals together with many pseudocodes for selected algorithms, an introduction into fantastic fractals such as the Mandelbrot set and the Julia sets, together with a detailed discussion of algorithms and fractal modeling of real world objects. 142 illustrations in 277 parts. 39 color plates. ... Read more

Customer Reviews (4)

Fractals -- Applied Mathmatics and Computer Programming
From page 25" Fractals (a word coined by Mandelbrot in 1975) have blossomed tremendously in the past few years (written in 1988) and have helped reconnect pure mathematics research with both the natural sciences and computing."

This book has both Mathmatical equations and Computer Programs along with explanations and results (many graphs, plots, and color plate images).

If you have an interest in Fractals, Recursion, Computer programming, Image creation, this is a great book and filled with examples.

Great book on fractals and imaging
This old book is a timeless gem. It goes into the details of the mathematics of fractals and also shows well-commented C code for producing fractal imagery along with good color illustrations.
Chapter 1, "Fractals in Nature", uses computer generated images to build a visual intuition for fractal as opposed to Euclidian shapes. There is also a mathematical characterization with Brownian motion as the prototype.
In chapter 2, "Random Fractal Algorithms", randomness is introduced into the algorithms discussed in chapter one as a way of simulating natural phenomena. Ideas are extended to higher dimensions. C programs that produce mountain ranges using these ideas are presented, along with the resulting imagery.
Chapter 3, "Fractal Patterns Arising in Chaotic Dynamical Systems", turns to the topic of dynamical systems and is less mathematical than the first two chapters. There is some mathematics and some illustrations in 2D and black and white that should be familiar to any student of dynamical systems.
Chapter 4, "Fantastic Deterministic Fractals", demonstrates how genuine mathematical research experiments open a door to a new reservoir of fantastic shapes and images. Programs are shown that extend the ideas of chapter 3 into truly beautiful fractals. Ideas here stay mainly in 2D.
The final chapter, "Fractal Modelling of Real World Images", draws from the material of the previous chapters to present C programs that produce clouds, vegetation, smoke, and mountain ranges, all by altering a few of the parameters in the sample code presented by the authors.
This book is much better than more recent titles that bury their algorithms in complex high level languages or "toy books" on the subject that provide dumbed-down applications and in which the simplest possible explanation of fractals is given with no insight. I highly recommend this book to anyone interested in understanding fractal mathematics and in using that mathematics to produce stunning visual effects.

One of the best (if no the best) in the feild
You cant go past this book,

This book reads at any level, Great introduction to the field as well as an indespencible reference. Shows easy to implement code examples, and has lots of pictures showing what can beacheived.

This has been a main reference for a theisis I am currentlyworking on. The question is, why is it out of print. If you can find itit's worth it's wheight in gold.

A must
In my opinion, the best work ever written in the category not-for-beginner-but-available-to-non-specialist (such as Beauty of Fractals, by the same authors). An easy answer to question "How can Igenerate a fractal image with my PC?", from brownian motion to Juliasets. A must for reader interested in fractals (a bit out-of-fashion butstill very interesting field). ... Read more

Product Description

Fractals are unique patterns left behind by the unpredictable movements -- the chaos -- of the world at work. The branching patterns of trees, the veins in a hand, water twisting out of a running tap -- all of these are fractals. Learn to recognize them and you will never again see things in quite the same way.

Fractals permeate our lives, appearing in places as tiny as the surface of a virus and as majestic as the Grand Canyon. From ancient tribal peoples to modern painters to the animators of Star Wars, artists have been captivated by fractals and have utilized them in their work. Computer buffs are wild about fractals as well, for they can be generated on ordinary home computers.

In Fractals: The Patterns of Chaos, science writer John Briggs uses over 170 illustrations to clearly explain the significance -- and more importantly, the beauty -- of fractals. He describes how fractals were discovered, how they are formed, and the unique properties different fractals share. Fractals is a breathtaking guided tour of a brand new aesthetic of art, science, and nature. It will revolutionize the way you see the world and your place within it.

* Contains a special bibliography listing fractal generating software for desktop computers ... Read more

Customer Reviews (13)

The best introductory guide to fractals and chaos
Though this book is now out of print it has not yet been surpassed (or even replaced) as an inhtroduction to the worelds of fractal illustarations. The photographs are astouding - and this leads the reader to read the print to find out more about the patterns that this book describes. Though the book is about mathematical ideas there is adearth of complicated mathematical formulae. It is easy reading for anyone who can add, subtract and multiply (no division!) andwho understand the very simplest algebra. I have used this book in a course I teach at our local Senior College and all the students love it.I am anxiously awaiting a second updated edition.

staggeringly beautiful
Technically this book is good.The images, however, are staggeringly beautiful.I think Dr. Briggs teaches esthetics, which is the study of beauty.Of any book on fractals I have read, this is by far the most beautiful.I really liked this book, so much I donated it to the library of my alma mater.

Patterns to Inspire - A Captivating Look into Fractals
This book brings a comprehensive and visually intriguing approach to the study of fractal geometry and the chaos theory. Through thought provoking imagery and discernible explanations & comparisons, John Briggs has sparked my curiosity where I now look more closely at the world around me. I believe this book is intended to captivate those with the ability to visualize and appreciate the aesthetics and interconnectedness of the arts, sciences and the natural phenomena that surrounds us. An insightful & visually stimulating read!

Don't buy this book
This book says absolutely nothing.It has a few good pictures (the best one is one the cover by the way), but the text is utterly worthless and uninformative.My favorite quote from the book is "Nonlinear means not linear."Really?Don't waste your money.Now I understand why I found it at the used bookstore.

Great Photos, Poor Content
This is a fantastic source of images on the subject of fractals, but not a great source of learning. Most books on math and science are difficult for the general reader; few authors (like Isaac Asimov) can make complex things easily understood. But the author of this book is, in my opinion, doing the public a disservice by oversimplifying the subject. The explanations underestimate the public's ability to think, and even include a number of things which are either dead wrong or made-up! The subject of fractals is still new, and there are recently more books available to explain fractals to the general public. Again, this is a great source of images, if that's what you're looking for, but look for another source if you want to undersatnd and appreciate this incredible and important topic. ... Read more

Product Description
From Zeno to Mandelbrot: explore this new language with which you can describe the shape of cloud as precisely as an architect can describe a house. ... Read more

Product Description
A new treasure trove of stories that make mathematical ideas come to life with an unusual cast of characters. This book explores mathematical concepts and topics such as real numbers, exponents, dimensions, and geometry in both serious and humorous ways. 50 line drawings. ... Read more

Customer Reviews (10)

Math tales
I'm not sure the intended audience of this book.The pictures are too minimal for early elementary, with content far above them.The grades for whom the content is appropriate would probably find the stories unappealing.They didn't catch my interest either.To its credit, the book has some interesting sidebars and activities for teacher use.My resources would not have been lacking if I'd passed on this purchase.

Fractals and Googols. . . and Penrose
Theoni Pappas has written several books of mathematical 'tales' including the well-know Penrose books.
"Fractals, Googols and other Mathematical Tales" includes a total of 22 topical tales.

The table of contents helpfully includes information as to the topic discussed in that tale, as the chapter titles, 'Dr. Spacemath', for example, do not always indicate what the topic is!
Unfortunately, if you already have the Penrose books, you will find that at least 9 of the chapters in "Fractals" are repeated from the Penrose books.Several of these have been condensed and it seems as though the author's commentary has been added to some in a purple side-bar in keeping with the style of the rest of the book.

While the Penrose books are clearly written for and to those new to the world of mathematics, "Fractals" includes small-print, purple side-bars that seem to be written to a parent/teacher.
As in the Penrose books, there is a section in the very back that includes solutions to puzzles and challenges posted in the book.

We are big fans of the Penrose books and Theoni Pappas, and while 'Fractals' does indicate topics in the table of contents, I wish that all these books had a good index to make it easier to find tales and activities that are relevant to a student's current topic of study.
I was also disappointed that so many of the tales in "Fractals" are repeats from the Penrose books.

However, if you do not already have Penrose, this would be a great book to introduce you to Pappas' unique approach to mathematical tales.

Let's clear this up
C=pi*d.That means the diameter has to be **multiplied** by "three and a little bit" in order to be equivalent to, or wrap around, the circumference of the circle.It takes "three and a little bit" diameters to get around the circle one time, not one diameter to get around the circle three and a little bit times.The engineer was right!Other than that, it was still a pretty good book, though.If you're a teacher with an older class, you can see if they can spot the error.

Could I help clarify?
I believe the quote about the circumference and diameter is worded incorrectly.We all know that the size of the diameter and circumference are proportional (hence pi), but the diameter would never wrap around a circumference 3 and a little times!This is obviously false because the circumference is larger than the diameter.The circumference would wrap around the diameter 3 and a little times.Please check your books to determine if the quote is correct in R. Krapf's review - if so, then the book is WRONG!

Revieing the reviewer
I am not sure what R. Krapf "Engineer" (below) was thinking when he/she wrote his/her review...

The circumference of a circle (C) is calculated as 2 * pi * r (or pi * 2 * r)

Since r is the radius and 1/2 the diameter (d), that means C = pi * d

Since pi is about 3.14, that means the book is correct.The diameter of a circle would wrap around (the circumference of) the circle "3 and a little bit" ... Read more

Product Description
Fractals are characterized by the repetition of similar patterns at ever-diminishing scales. Fractal geometry has emerged as one of the most exciting frontiers on the border between mathematics and information technology and can be seen in many of the swirling patterns produced by computer graphics.It has become a new tool for modeling in biology, geology, and other natural sciences.

Anthropologists have observed that the patterns produced in differentcultures can be characterized by specific design themes.In Europeand America, we often see cities laid out in a grid pattern ofstraight streets and right-angle corners.In contrast, traditionalAfrican settlements tend to use fractal structure--circles of circlesof circular dwellings, rectangular walls enclosing ever-smallerrectangles, and streets in which broad avenues branch down to tinyfootpaths with striking geometric repetition.These indigenousfractals are not limited to architecture; their recursive patternsecho throughout many disparate African designs and knowledge systems.

Drawing on interviews with African designers, artists, and scientists,Ron Eglash investigates fractals in African architecture, traditionalhairstyling, textiles, sculpture, painting, carving, metalwork,religion, games, practical craft, quantitative technologies, andsymbolic systems.He also examines the political and socialimplications of the existence of African fractal geometry.His bookmakes a unique contribution to the study of mathematics, Africanculture, anthropology, and computer simulations. ... Read more

Customer Reviews (7)

Former Student
I had Ron Eglash as a professor at Rensselaer Polytechnic Institute. Discussing and analyzing aspects of this book, including self-organization in general, was very interesting and valuable to say the least. The book makes no assumptions in knowledge and will cleanly bring in the topic of fractals in african culture. I had read the book the winter break before taking the course and had no difficulty understanding the material even as a freshman. The concept is quite intriguing and shatters many of the held perceptions of "the hierarchy of mathematics." Ron Eglash is a great man and I know he loves talking with people that share similar interests in mathematics or cybernetics.

At times the author crosses the line where mathematics is "found" inside situations when it is not there
This book can be placed in the category of ethnomathematics, where the emphasis is on the ethno rather than the mathematics. Fractals are by definition structures that are self-similar over a large number of iterations and scales of measure. If you accept that only a few iterations are sufficient to define a fractal, then the structures described in this book can be considered fractals. However, the author does the best job of summing up the content in the first two sentences of chapter 11.

"Parts I and II of this book emphasized the geometric, symbolic, and quantitative aspects of African fractals. Some cases were more speculative than others - a difference that I hope was clearly indicated - but even in the use of mythic narrative, I generally restrained conclusions to those that had geometric or quantitative counterparts."

I agree that the author generally stayed within the bounds of reasonableness in describing what are called fractal structures in African design, but only if you stay within the bounds of a few iterations of shrinking self-similarity being sufficient to have a structure be considered a fractal.
In the last four chapters, the author makes some points that are both revealing and questionable. On page 182, after pointing out that anthropologists need years of study to understand a culture the author states, "My thin description fieldwork lasted only a year and moved through Senegal, Mali, Burkina Faso, The Gambia, Cameroon, Benin and Ghana." By his own admission, Eglash is not qualified to speak to deep cultural issues, yet he proceeds to do so. The section that begins on page 209 with the title "Recursion and sex - a cross-cultural comparison" simply went way beyond or more precisely out of my understanding of fractals, recursion and sex. I did not understand either the statement "This section will focus on the relation between recursion in mathematics and sexuality in culture" or the purported explanation.
While I am a strong supporter of the concept of ethnomathematics, people writing in support of it must take great care not to avoid finding mathematical ideas or intent inside situations where none exist. In my opinion, Eglash crosses that line in this book.

Published in Journal of Recreational Mathematics, reprinted with permission,

a good introduction to African mathematics and fractal geometry
This book starts out with a presentation of fractal geometry which is very comprehensible and enjoyable. Next it covers specific aspects of fractal geometry and their relation to African society, architecture, fashion, art, divination and games. This part of the book is very fascinating. I learned a lot about how recursion works and how it is used in African buildings and fashions in the chapter on recursion. Other chapters in this section are Geometric algorithms, Scaling, Numeric systems, Infinity and Complexity. They are all very interesting. The final section is on the implications of the fact that Africans used this kind of mathematics. The author emphasizes the application of African fractal geometry to education especially the education of African Americans who sometimes feel alienated from math classes which focus on the achievements of European peoples. One thing that the author stresses is that the fractal designs of, say city planning, made by African peoples are not more "natural" than the Western approach of dividing cities into rectangles. He says this assumption dovetails into a preconception of African societies as being somehow closer to nature and therefore unsophisticated. The author points out that fractal mathematics is hardly simple and also not easily intuited either. I did not find myself making this assumption but apparently some people do fall into this trap. Anyway, I highly recommend this book to anyone wanting an introduction, with applications, to fractal geometry and its use in African societies. I also recommend this book to educators looking for a way to get their students, regardless of their background, to be more interested in mathematics.

Connecting Africans ancient and modern
This is an amazing book! It clearly shows how many of the common things that people of African descent do have may scientific connections.Hair styles that are worn today by people of African descent, have been worn as far back to the ancient indigenous Africans known as the ancient Egyptians.So it really no surprise that there is mathematical and scientific knowledge being found today by scientist and scholars.

This book should be in every school and home in this country.I take that back, this book should be in every school globally.

Another scientific book that would make a great set for any school or home is, The African Unconscious.Written by Edward Bruce Bynum.You can find it here on Amazon.com.

This book helps to render obsolete long-held myths.
Ron Eglash's brilliant work on Afrikan fractals helps to shatter long-held myths and misconceptions about Afrikans, the most pervasive and pernicious of which is the notion of Afrikans (both on the Motherland and in the Diaspora) as inactive agents in history. This work motivated me to complete mine on chaos theory and Afrikan fractals. My longer reviews of Eglash's book appear in the Nexus Network Journal (vol. 2, 2000:165-168) and the Journal of Third World Studies (vol. xviii, no. 1, 2001:237-239), each reflecting the publication's genre and disciplinary focus. Dr. Abdul Karim Bangura is a researcher-in-residence at the Center for Global Peace and a professor of International Relations in the School of International Service at American University, and the director of The African Institution in Washington, DC. He is the author of 21 books and more than 200 scholarly articles. ... Read more

Product Description

This volume is the second edition of the highly successful Fractals Everywhere.The Focus of this text is how fractal geometry can be used to model real objects in the physical world.

This edition of Fractals Everywhere is the most up-to-date fractal textbook available today.

Fractals Everywhere may be supplemented by Michael F. Barnsley's Desktop Fractal Design System (version 2.0) with IBM for Macintosh software.The Desktop Fractal Design System 2.0 is a tool for designing Iterated Function Systems codes and fractal images, and makes an excellent supplement to a course on fractal geometry

* A new chapter on recurrent iterated function systems, including vector recurrent iterated function systems.
* Problems and tools emphasizing fractal applciations.
* An all-new answer key to problems in the text, with solutions and hints. ... Read more

Customer Reviews (12)

Way over my head!
Man, this got way too deep, way too fast. I have some algebra, some statistical analysis, some calc... This just took off running and left me in the dust. I was "fooled" by the "first course" title?

In fractal geometry we are concerned with the structure of subsets of various very simple "geometrical" spaces. Such a space is denoted by X. It is the space on which we think of drawing our fractals; it is the place where fractals live. What is a fractal? For us, for now, it is just a subset of a space. Whereas the space is simple, the fractal subset may be geometrically complicated.

So, why is *this* a fractal, but, *That*'s just a recursive function? Idunno, and I guess that I won't until I find another book?

This is a beautiful book. LOTs of graphics, TONs of information, but not for me.

The most intelligent and usefull book on fractals
I really enjoyed the author's style. It is not a typical mathematical book.So if you are into strict formal mathematics this is not for you. The author has deep understanding of fractals and he is an expert in Iterated Function Systems (IFS). The shadow theorem is one of the biggest hits on this book.
What I particularly like from this book is the illustrations. They let you understand fractals intuitively. It is definitelyan ideal book for engineers. Try also his other more advanced book SuperFractals
The only part of the book that is not very well developed is chaos, although the "chaotic dance" presented on the book is good starting point

A below average text book

Barnsley's book has a shortcoming common to many math text books -- it's poorly written.Barnsely's writing style is superfluous and rambling.What I learned from this book was in spite of Barnsley's writing, not because of it.

Furthermore, the book's illustrations are substandard.There are over five different fonts used in illustrations (including hand written text).This leads to confusion when you're unsure if the text in Barnsley's illustrations is referring to Greek letters or the conventional alphabet.Another shortcoming is that Barnsley intermingles end of chapter exercises with new concepts.You may have problems 1.1 to 1.5 reviewing what you've already learned, and then problem 1.6 introduces completely new material.This is a problem throughout the book, as important concepts are introduced in exercises or otherwise illogical locations.

On the positive side, solutions to most exercises are presented at the end of the book.Overall the book was useful, but learning the material was unnecessarily difficult due to the book's shortcomings.

Sometimes annoying but instructive
Although instructive, this book is sometimes annoying to read. The author seems to be playing his cards very close to the vest and not telling us everything.

For instance, there is little or no instruction on how to implement the IFS attractors presented as a panacea for data compression. This seems to be proprietary to his company. It also seems that hands-on manipulation is crucial to the images' production, contrary to the author's claims.

If you can understand the mathematics you may find the book useful, as I did when writing my book Fractals in MUsic.

A bad book for 7 th graders like me
this is a bad and very confusing book for a young student in, say... 7th grade, like me.The language is incomprehensible and there are no visual aids. ... Read more

Product Description
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.Amazon.com Review
Fascinating and authoritative, Chaos and Fractals: NewFrontiers of Science is a truly remarkable book that documentsrecent discoveries in chaos theory with plenty of mathematical detail,but without alienating the general reader. In all, this text offers anextremely rich and engaging tour of this quite revolutionary branch ofmathematical research.

The most appealing aspect about Chaos andFractals has to be its hundreds of images and graphics (withdozens in full-color) used to illustrate key concepts. Even themath-averse reader should be able to follow the basic presentation ofchaos and fractals here. Since fractals often mimic natural shapessuch as mountains, plants, and other biological forms, they lendthemselves especially well to visual representation.

Early chaptershere document the mathematical oddities (or "monsters") such as theSierpinski Gasket and the Koch Curve, which laid the groundwork forlater discoveries in fractals. The book does a fine job of placingrecent discoveries about chaos into a tradition of earliermathematical research. Its description of the work of mathematicianslike Pascal, Kepler, Poincaré, Sierpinski, Koch, and Mandelbrotmakes for a fine read, a detective story that ends with the discoveryof order in chaos. (For programmers, the authors provide shortalgorithms and BASIC code, which lets you try out plotting variousfractals on your own.)

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

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

Topicscovered: Overview of fractals and chaos theory, feedback andmultiple reduction copy machines (MRCMs), the Cantor Set, theSierpinski Gasket and Carpet, the Pascal Triangle, the Koch Curve,Julia Sets, similarity, measuring fractal curves, fractal dimensions,transformations and contraction mapping, image compression, chaosgames, fractals and nature, L-systems, cellular automata basics,attractors and strange attractors, Henon's Attractor, Rössler andLorenz Attractors, randomness in fractals, the Brownian motion,fractal landscapes, sensitivity and periodic points, complexarithmetic basics, the Mandelbrot Set, and multifractal measures. ... Read more

Customer Reviews (13)

This book is a dream come true.
This book is a dream come true.
No other publication comes close to such complete coverage of the subject.
It is highly readable even for a novice like myself.
It has been a great joy to me.
Many thanks to the authors for doing such a great job.

It's all true: Best single source on fractals-but get the 1st ed.
Thanks to S.J. Will for the tip: Get the FIRST edition (used), as I did and save more than half the price, even of a used copy of this newer edition. Can't compare the two (having not seen the new one) but I can say the color images are very sharp in the older book. As far as content, I too have looked at and bought several books trying to understand fractals. (I am not math-literate, beyond high school algebra.) I found this book most helpful, but NOT easy for the general reader, beyond the first few, introductory pages. As other reviewers have noted, most of it is WAYYYY over the head of anyone who's not a college math major, but skipping through the examples and exercises (some of which are very rewarding if you can stay with it), I found the general explanations, the excitement of the authors, the broader significance of fractals all to be well-worth the price. -- And hey: at over 900 pages ( ! ) and with FORTY color plates, this book is an astounding bargain. Strongly recommended, even for novices.

"The Colors of Infinity," based on the video documentary by Arthur C. Clarke is a good introduction to fractals. An enjoyable DVD is included of the original TV program, especially if you learn better by watching and listening. The accompanying animated fractals are fascinating, but frustratingly poor resolution. For a more philosophical approach to fractals, I highly recommend "Heaven's Fractal Net" by William Jackson.

Compare the editions
I found the 1992 edition of this book at my local public library, and was (like all the other reviewers here) very impressed at the quality. The book deals with a highly technical subject, but does it in a way that you can follow even if you don't have advanced math training. The numerous color plates were also very beautiful. And to top it all off, there were "do it yourself" exercises at the end of the chapters, showing you how to program your computer to run these figures! OK, they use the old BASIC language, but still the code is clear enough that you can follow it and see what's really going on with these equations.

So I was so pleased to see a copy of the updated edition at a bookstore. In particular, I was eager to see if they'd updated those "do it yourself" exercises for use with EXCEL. However, as I read through it I was disappointed to notice two changes from the previous edition: first, all of the programming examples had been eliminated; second, the print quality of the color plates was noticeably poorer. And I didn't see much new material added - in fact one of the reviews above observes that the text itself is virtually unchanged. Considering the steep price of this tome, these were significant points to consider. Used copies of the old edition cost under 20 bucks, and IMHO are a better deal (I ended up buying one). So if you're ready to buy, just do yourself (and your wallet) a favor and compare the two editions first.

Excellent tutorial on nonlinearity
At least 50% of this book can be well understood by any 1st year, exact science student. There are a couple of mathematical issues that are more senior-like, but never mind. With the appropriate teaching or guidance, a lot of practical, advanced tasks can be tackled down.I could use this book all along for giving examples for college (university), undergraduate students of almost every mathematical subject: numerical analysis, calculus, linear algebra, group theory, algorithm theory, visualization in 2 and 3 dimensions, topology...you name it, after reading this book. No fuzzy theory or wavelets or any other advanced statistical method for dynamical systems is formally mentioned, though. However the concept of measure is very well introduced and described with examples. For physics is not bad for dynamical systems theory. Although no Hamiltonian or Lagrangian formalism is mentioned, the description on how to obtain Lyapunov exponents out of a set of differential equations is very good. Engineers get their share too: useful examples are given about, e.g., feedback and control theory (mind you, it is not a book specialized in, say, robotic control using chaos theory, but it is a good start). For philosophers and the layman there are quite a few pages as well. The foreword from Mitchel Feigenbaum, just to give an example, tells us a kind of summary which "warms up" the reader and "exorcises away" the possible fantasies an unprepared reader could have regarding (or against or in favor of) the word "chaos". Nice color plates for those with artistic inclinations and the graphics are just so very well printed, you can practically "follow" their computation. Not a bad book at all for your personal (or institutional) library, I may say.

A good introduction
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. ... Read more

Product Description
Benoit Mandelbrot's pioneering research in fractal geometry has affected many areas of mathematics, physics, finance and other disciplines. The papers reprinted in this third volume of his Selected Works center on a detailed study of fractional Brownian functions, best known as the mathematical tools behind the celebrated fractal landscapes. Extensive introductory material preceding the reprints incorporates striking new observations and conjectures. This book explores the fractal themes of "self-affinity" and "globality." The ubiquity of "wild" temporal and spatial variability led Mandelbrot, in the early 1960's, to conclude that those phenomena lie beyond the usual statistical techniques and represent a new state of indeterminism. New mathematical tools are needed, and this book contributes to their development. ... Read more

Product Description
The authors present an unusual attempt to publicize the field of Complex Dynamics, an exciting mathematical discipline of respectable tradition that recently sprang into new life under the impact of modern computer graphics. Where previous generations of scientists had to develop their own inner eye to perceive the abstract aesthetics of their work, the astonding pictures assembled here invite the reader to share in a new mathematical experience, to revel in the charm of fractal frontiers. 184 illustrations in 211 parts, 88 in color. ... Read more

Customer Reviews (7)

A Beautiful Introduction To Fractals
Fascinating photos, although the instructional text is indeed "complex" as the book itself states.

Anna Marie Fritz, author of "Funny Feline Fotos" and other amazon books.

One of the first fractal books!
One of the first books on fractals that gave info on the mathematics to creating them and what parameters or sections of the set that yield some beautiful renderings. The math is for a math major but still interesting.

The Essence of Beauty
I spent all last evening reading snippets of The Beauty of Fractals (those few paragraphs that a layman could understand) and admiring the sheer beauty of the diagrams/maps.I had not realised there was an aesthetic component to mathematics, and I certainly did not know that aspects of what is generally thought of as a dry science can be so visually appealing, not to say stunning.

I cannot understand why some people would argue the intrinsic artistic merit of something computer-generated and 'unnatural', when the results speak for themselves.

Beauty, true, is perceived, and lies in the eye of the beholder. It can be very subjective. But there are certain aspects of visual appeal that go beyond that. One would think that a symmetry of form, the complementary use of colours, the balance of shape and form, light and shade, arcs and curves--all these combine to give an objective, irrefutable fact of beauty that transcends thought and emotions, if not the senses.

In a couple of the chapters, it was said, and here I paraphrase:

The two modes of analysis and intuition as human means of understanding the natual world--need they be considered at opposite poles? Do they not complement one another? Are the thinker and the dreamer not one?

I find that very intriguing, just as I find the idea of chaos and order existing together in natural, dynamic processes being actually TYPICAL of Nature.

The word 'Chaos' has such negative connotations, implying confusion and destruction, but if I were to replace it with the word 'Disorder', then things begin to fall into place.

There can be no Order if there were no Disorder, for how then would we know the difference? In fact, one of the writers go so far as to say that it is the very existence of Disorder within Order that confers the essence of beauty found in Nature.
That is so true. It is the very non-linear aspect of Nature, that which mathematics, up till Mandelbrot, have been unable to map, that is so appealing in the visual sense.

In Nature, which, apart from abhorring vacuums, also has no place for a straight line (oh, how the poor, innocent straight line is maligned in the preface), beauty is inarguable, irrefutable, and only after that does it have history and context, different to and for each beholder.

So both Chaos/Disorder and Order co-exist in Nature, hand in hand. Order alone, rigidly disciplined, artificially-imposed, seems to require Disorder to breathe life into it.

Taking this a step further, our perception of beauty in all things is affected by Nature.

In yet another chapter, someone quoted someone else and here I go
paraphrasing again.

Beauty in science is the same as beauty in other disciplines-art, music,literature, what have you.'A fog of events, and suddenly you see a connection. It expresses a complex of human concerns that goes deeply to you, that connects things that were always in you that were never put together before.'

The thinker and the dreamer co-exist within each person, just as the analytical and intuitive modes of thought co-exist, not at opposite poles,but complementing one another.

Intuition and analysis complement, rather than confound (or they should, gods-willing).

The artist and the scientist complement each other, i.e. Art and Science are not the opposing polarites of disciplines as some would have us think.

The thinker and the dreamer ARE one.

And this book has shown that the essence of beauty lies in the marriage of Art and Science.

(Disclaimer: Mere thoughts from a layman.)

Time goes by... the pictures fade
Amazon dot com gives no credit for my earlier reviews.... I wrote the first review without thinking or spell checking, fast.There are still parts of this book I can't duplicate! And parts that are just written so bad no one can understand them, but most of it stands the test of time, like Mandelbrot's article. This will probably be a classic in the future! I think the free Fractint documentation is probably better for a beginner or Hans Lauwerier's " Fractals".

Swirly
Although one of the earliest titles to bring fractals into the mainstream, 'The Beauty of Fractals' isn't as visually exciting as the follow-up, 'Chaos and Fractals', and it's a very dry read - in 1986 complex dynamics were an esoteric field of mathematics that had yet to transfer to student posters and rave videos. At this price it's restricted to people who absolutely need it, although along with 'Godel, Escher, Bach' it's one of the seminal hackish coffee-table books. ... Read more

Product Description
Fractal geometry is the geometry of the natural world. It mirrors the uneven but real shapes of nature, the world as we actually experience it. Introducing Fractal Geometry traces the development of this revolutionary new discipline. ... Read more

Customer Reviews (10)

NEVER in Stock!!!!!
I ordered this book in July, specifically, along with many other curious items since I was ordering this anyway; I was extremely anxious to receive it.Unfortunately, I NEVER received it but DID just (middle of October) receive word I never would and they cancelled the order.I would not have placed this order if I had known they would never get this in stock - EVER!!I never shop bait-n-switch stores OR any affiliate.I'm sure the book is great, but I would not order from Amazon since they are not able to obtain it.Try another book store!

Interesting book, Illogical progression
I bought this book because I wanted something that would introduce me to the topic, fractal geometry, without overwhelming me with technical jargon, and this book does do that. However, the way in which the book is constructed is quite convoluted. It has illustrations on every page, some of which are left without any description, that may or may not be related to the topic discussed on the page.

The book doesn't progress in terms of chapters, rather sort of chronologically. However, in doing this it just comes off disjointed. The paragraphs are jumpy and disconnected from short telegraphic sentences.
While reading the first 1/3 of the book, I had convinced myself that the book must have been written for an audience much, much younger than I. However, I don't see how that can be logical given the subject material. The latter 2/3 of the book also convinced me otherwise.
So in general, yes, you will learn from this book, provided you are able to follow the progression and are able to correctly interpret the meanings of the diagrams.

Excellent introduction to fractals
This book was an excellent introduction to fractals with clear writing and numerous illustrations.I'm ready for the next level!

disappointing, irregular

1.0 out of 5 stars mixed up, February 10, 2009
By Andres Amador "pickletruck" (san francisco, ca) - See all my reviews
(REAL NAME)
i'm not new to fractals by any means but i picked up this book in a bookstore and appreciated the simple way the ideas were being conveyed and with lots of graphics. after getting into it though, i was dismayed by it combination of simplicity and then sometimes dense detail. I mean, sometimes he would bring up something tantalizing, something ripe for expansion and then let it drop. and then other times he would go into detail that had me wondering if even HE knew what he was saying.

for a much better read with lots of great images, i highly recommend 'turbulent mirror' its a necessity for those curious is fractals and chaos.

loved it
Want to learn what fractal geometry is?Get this book, you'll know enough to be able to join in conversations on the topic. ... Read more

Product Description
Now in paperback--the lively first novel featuring former Marine JAG and private eye Pepper Keane. When math professor Jayne Smyers discovers that three murder victims with apparently unrelated cases were each an expert in fractal geometry, she hires Pepper to find the killer. ... Read more

Customer Reviews (39)

mystery
I am not going into a lengthy review about this book I would just like to say that Mark Cohen is a fantastic writer with a sense of humor. I have read both of his books and wonder when or if he will write another.

Great story, unique subject
I love reading mysteries (essentially junk food for my mind) that increase my awareness of more esoteric subjects. I still know next to nothing about fractals, but at least I know they exist. And I had a great time reading the Fractal Murders while that happened.

Pepper Keane is a great character, a sort of Travis McGee of our time. Actually, a great deal about the Fractal Murders has very much a John D. MacDonald flavor to it. Keane is a protgonist who has rebelled against "the man" and lives by his own rules as an investigator. A woman brings him a case that is complex and full of intrigue. Happily, Mark Cohen is a little more light-handed with the sexism and the sex than MacDonald!.

The Fractal Murders has great background that I never found boring, Pepper Keane is a charming main character and his family and friends are also full of personality and fully envisioned. No one-dimensional characterizations. The mystery leads you most of the way and then has a twist I didn't see coming. I liked this book very much and would read more by this author. I hope there are more!

A pretty good mystery
Three fractal mathematician die and another one who was a close friend
of one of them thinks it isn't a coincidence. The FBI says that there
is no connection. So Pepper Keane gets the case...
No resemblance to any living fractal a mathematicians seems to exist,
and the premise of the conclusion of time transform doesn't seem likely,
but the novel is well researched. In some cases I think the explanations are better than some popularizations that I have read.
It turns out that there is an FBI connection to the gun that killed
one of the mathematicians in Washington state.
From there on the plot has some surprising twists!
I liked it and for a mystery it was well written.
The point seems to be that once Pepper was close,
the pile of files at his house would have been a target
more than himself. From Waco we know that the FBI can
have an agenda outside their legal one.
To think that mathematicians might be targets
because of an economic model is kind of 'romantic'
but not very realistic... a Levy Flight?

Smart
"The Fractal Murders" offers a terrific premise and the delivery lives up to it, too. The sleuthing is dogged, hard work. It's smart and well organized--the kind of diligent, relentless focus that is probably the norm out there for actual detectives. The Colorado settings are well rendered. I thought a few key things happened a bit too conveniently: the jet ride to Boston, the handy night vision goggles, access via a friend to key commercial airline passenger manifests. I also could have done without all the restaurant and eating descriptions; they just don't add up to much. For a guy who spends a lot of time running and working out, there is very little physical action. Most of the work is mental. Readers looking for a good brain exercise will enjoy how Pepper Keane approaches his work and the end packs a nifty double twist.

Great math mystery that's not too technical
University of Colorado math professor Jayne Smyers hires Pepper Keane, former Marine JAG turned PI, to investigate the deaths of three other math professors.The Feds have investigated and found no link between the deaths.Jayne is convinced there's a link.Pepper is skeptical at first but agrees to look into it.

There's plenty of bad blood between Pepper and FBI agent Polk who did some of the investigating.This history adds to Pepper's determination to investigate these deaths.

As Pepper digs deeper into the deaths, he begins to see some similar threads that continue to propel him forward.With romance in the air, Pepper worries that Jayne may be the next victim.

Can he decipher the pattern and unmask the killer before anyone else is killed?Can he protect Jayne as well?

I thoroughly enjoyed this refreshing mystery.Pepper is a fabulous character, even with his baggage.It is explained throughout the book, so we aren't left floundering.His interactions with Polk, Jayne, detectives where each mathematician was killed, his brother, his neighbors, and his best friend really help us to get to know him.

I found the math to be explained in plain English so that it was easy to understand. It also didn't detract from the investigation; it actually enhanced it.I am not a mathematician, but I really enjoyed this book.I hope he writes more in this series.I can't wait to read them.I highly recommend this book.
... Read more

Product Description
Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular ``chaotic'' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory--Cantor sets, Hausdorff dimension, box dimension--using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science--the FitzHugh-Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008. ... Read more

Product Description
Now in a greatly expanded second edition, this book relates fractals and chaos to a variety of geological and geophysical applications and introduces the fundamental concepts of fractal geometry and chaotic dynamics. In this new edition, Turcotte expands coverage of self-organized criticality and includes statistics and time series to provide a broad background for the reader. Topics include drainage networks and erosion, floods, earthquakes, mineral and petroleum resources, fragmentation, mantle convection, and magnetic field generation. The author introduces all concepts at the lowest possible level of mathematics consistent with their understanding, so that the reader requires only a background in basic physics and mathematics. He includes problems for the reader to solve. This book will appeal to a broad range of readers interested in complex natural phenomena. ... Read more

Customer Reviews (2)

Un libre excelente, an excellent book!
El Profesor Donald Turcotte desde principios de los 80 ha venido trabajando en el modelamiento de problemas geológicos con fractales y caos, y su libro es una de las mejores introducciones al tema.Desde su Capítulo1 se ve claramente el estilo concreto y práctico que lo caracteriza, y lamanera como introduce el concepto de la dimensión fractal por medio deejemplos sencillos es sin duda invaluable.El capítulo de fragmentaciónpresenta algunos modelos interesantes de este complejo fenómeno, y losejercicios son muy ilustrativos al respecto.El capítulo de sismicidad ytectónica presenta algunos de los más importantes resultados en lacaracterización de fallas y terremotos, aunque se queda un poco corto enlos avances recientes a nivel de evolución de sistemas de fallamiento.Elcapítulo de relación entre el tenor y el tonelaje para yacimientosminerales aún tiene su vigencia dado lo poco que se ha realizado en estecampo.Su relación con los procesos magmáticos de diferenciación es muyinteresante y se puede realizar mucha investigación al respecto.Sucapítulo de Geomorfología presente avances interesante como el concepto deordenamiento de tokunaga, pero también se queda un poco corto en losrecientes avances en este campo.El capítulo de fractales auto-afines esuna de las mejores introducciones a esta temática.El autor explica demanera clara algunos conceptos importantes como el exponente de Hurst,movimiento fraccionado browniano y cálculo de dimensiones fractales contransformada de fourier, además de introducir algunos conceptos de procesosestocásticos.El profesor Turcotte dedica algunos capítulos para hablaracerca de la teoría de sistemas dinámicos no lineales y sus posiblesaplicaciones en ciencias de la Tierra.Esta vez son las inversiones delcampo geomagnético y la convección en el manto los fenómenos que presentaneste tipo de comportamiento y sus implicaciones en la geología y lageofísica son profundas.Su capítulo de criticalidad autoorganizadapresenta claramente los conceptos fundamentales de lo que hoy en día hallegado a ser uno de los grandes paradigmas científicos.Este capítulopermite entender la manera como se aplican este tipo de ideas alentendimiento de los terremotos, y en un futuro se extenderá a muchos otroscampos de las ciencias de la tierra.Enhorabuena que el Profesor Turcottehaya publicado una obra como esta, que se convertirá en obra de referenciapara la futura generación de investigadores en ciencias de la Tierra.

Superb application and explanation of chaos and fractals
Dr. Turcotte explains and applies fractals to a wide variety of geological and geophysical phenomena.The writing is accessible to any educated reader who is willing to follow the mathematics.I have found it veryuseful in applying fractals and chaos to my own fields of study.Thesecond edition is a great improvement over the first:it is much expandedand clearer and begins at a more basic level.The book conveys a lapidaryconcision of thought that is impressive even in a pure theoreticalscientist.Dr. Turcotte is to be congratulated in this importantcontribution, which is so very useful to scientists in other fields. ... Read more

Product Description
A leading pioneer in the field offers practical applications of this innovative science. Peters describes complex concepts in an easy-to-follow manner for the non-mathematician. He uses fractals, rescaled range analysis and nonlinear dynamical models to explain behavior and understand price movements. These are specific tools employed by chaos scientists to map and measure physical and now, economic phenomena. ... Read more

Customer Reviews (9)

Gets you up and running with chaos theory for time series
This book includes a very detailed description of how to apply some chaos theory techniques - primarily R/S analysis - to time series data.With this technique, one can gauge whether a time series is completely random, completely predictive, or a mixture of these.

This book glosses over some conceptual topics such as Efficient Market Theory and the Fractal Market Hypothesis in favor of details to perform a rigorous statistical analysis.These conceptual topics are better covered in Peters' earlier work "Chaos and Order in the Capital Markets".

For the analytically oriented reader, there can be much frustration as equations are often initially presented in sloppy and unusable forms with undefined parameters (hence 4 of 5 stars).However, these are subsequently broken down and presented in a step-by-step manner that will allow most readers to implement his techniques.

Overall, this is an excellent introductory book for the practitioner or economist, not so great for the non-technical reader.

This book is a disappointment
As a market analyst for an oil company, I spend considerable effort in trying to find new ways and theories to "decode" the markets and overcome uncertainty. I had hoped Mr. Peters book would offer a model of analysis to test and hopefully use.

Unfortunately, the deeper you get into the "meet and potatoes" of this text, the more disappointing it gets. This book offers nothing. Readers less skilled in the subject matter might attribute this to their shortcomings or lesser math skills. Readers well-versed in this subject matter will easily determine that this text is a waste of time...

Good for newcomers to FMA
I have found Mr. Peters book excellent. I am no mathematician and still I found it easy to read. I was interested in learning about fractal market analysis and I found what I needed in this book.

Almost useless
In this book, Peters explains general capital market in an interesting way; but when he gets into the Fractals - the main theme of the book, he seems to just pick things up from previous research papers and condensesthem. At the end of the day you will be wondering if it is worthwhile topurchase and read the book after all.

On the editing side, I think Wileycould have done better. The math in the book is typeset in a programminglanguage style and you have to give extra effort in order to readthem.

The three stars is for being one of the very few works available inthis field.

simple as possible intro to fractals and markets
The title only indicates part of the true subject matter of this book.The book teaches about fractal analysis of any data set, and uses financial markets as special cases to illustrate the concepts involved in fractalanalysis.He begins with a brief, but facinating history of fractals, andyou learn the concepts you will need to form your own trading strategies.Mr. Peters demonstrates an easy familiarity with fractals, and this servesto keep the book interesting through its most difficult mathmaticalpassages. ... Read more

Product Description

Mandelbrot is world famous for his creation of the new mathematics of fractal geometry. Yet few people know that his original field of applied research was in econometrics and financial models, applying ideas of scaling and self-similarity to arrays of data generated by financial analyses. This book brings together his original papers as well as many original chapters specifically written for this book.

Customer Reviews (4)

Typical Mandlebrot
The good side of Mandlebrot is that he is always entertaining, and this is no exception. The downside is that this entire book is based on a complete false premise. Even the most cursory glance at the literature of mathematical finance shows that the mainstream of financial mathematics is perfectly aware of the fact that distributions of returns are non-normal, but Mandlebrot pretends that this is some sort of great insight on his part and proceeds to attack the nonexistent misconception that things are otherwise.

Secondly, on the question of self-similarity and scaling: tick data at various level of scale do not look the same. At the scale of seconds or sub seconds the data look dramatically different to longer scales and it seems silly to suggest that this is truly fractal.

A clear warning to all those financial analysts using N(0,1)
This book deserves to receive 6 stars.Mandelbrot serves up overwhelmingempirical,statistical,and historical evidence that financial decision makers are dead wrong in assuming,contrary to the available evidence, that a normal probability distribution describes the outcomes accurately in financial markets .In fact,the Cauchy distribution is substantially more relevant than the normal distribution.Mandelbrot's work simply means that the standard theoretical models taught in all colleges and universities,the CAPITAL ASSET PRICING MODEL(CAPM) and the BLACK-SCHOLES equation,give correct answers if and only if the relevant probability distributions about the movement of prices in financial markets over time are all normal.However, the evidence shows that they are NOT normal.Mandelbrot confirms ,by massive data analysis, Keynes's original 1921 objections to the misuse in application of (by merely assuming the applicability of such a distribution without examining the actual data)the normal probability distribution made in chapters 29 and 30 of the A Treatise on Probability(1921).Unfortunately,it appears that little,if any ,of Mandelbrot's scientific approach and analysis is being integrated into economics and finance.

scientific way of evaluating price movement
in this book, Mandelbrot is trying to prove that first, the price movement's distribution is scaling invariant, meaning a security's log price-change's distribution is same as with its 5-min's or with its daily's(or even monthly); second, price movement is not purely random/normaldistribution/brownian/random walk on street(they are all same description), meaning if u use normal distribution as one of ur bases for ur model, u will not only be theoretically wrong, but also be punished in real-life trading, such as the case of long-term capital. third, price movement does have cycle, but it length can not be determined in trading time, meaning u will not be able to decide when those cycles are going to start or end; fourth, changes of price movements do concentrate, meaning big moves will happen continouesly, or very closely to each other. the major implication to me is that many current financial theories are wrong, specially, those using normal distribution(such as option model) as basic assumption for security price movement. it also may prove that some of current price-based models(such as some trend following system) have some merit. but manay systems based on channel(such as bollinger bands)will not work in long-run. with those in mind and many available mathematical tools, one should be able to build a good financial model.

A book to make you think differently about the markets
To read this book you need truly to understand math and the markets. There is no questions that Mandelbrot is one of the greatest figures of our time. What he claimed based on his studies on cotton trading in the early 60s might not be close to the reality of today, but the way he approached it makes you think twice about the markets. Cotton trading is so different from stock market trading because it is either spoting trading or futures trading, and it is based on margins. The market usually has poor liquidity and with few players in it. The conclusions the book made could poorly extend to the general markets. The hard-to-follow math notations kept distracting me from following the main subjects. Anyway, this book will teach you something new, but you have to understand math and the markets, deeply. ... Read more