Chaos and Fractals: New Frontiers of Science

by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe

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, cellular automata, L-systems, percolation and strange attractors. This new edition has been thoroughly revised throughout. The appendices of the original edition were taken out since more recent publications cover this material in more depth. Instead of the focused computer programs in BASIC, the authors provide 10 interactive JAVA-applets for this second edition.

Genres: Science, Mathematics, Nonfiction, Physics, Philosophy, Textbooks, Computer Science

901 pages, Kindle Edition

First published January 1, 1992

About the author

Peitgen studied mathematics, physics and economics from 1965 until 1971 in Bonn, later working for six years at the Institute for Applied Mathematics at the University of Bonn under Christian Fenske, where he received his PhD in 1973. His doctoral dissertation was Asymptotische Fixpunktsätze und Stabilität (en: Asymptotic fixed-point theorems and stability).

After receiving his habilitation in 1977, he first taught as private docent in Bonn before obtaining a professorship for mathematics at the University of Bremen.

In 1986 Peitgen and Peter Richter published their lavishly illustrated and very influential book The Beauty of Fractals, which was amongst the first books popularizing the concept of fractals to the general public. This book was followed up in 1988 by The Science of Fractal Images and in 1992 by a large and authoritative volume entitled Chaos and Fractals: New Frontiers of Science, written in collaboration with Hartmut Jürgens and Dietmar Saupe.

Peitgen is director of the Centre for Complex Systems and Visualization (Centrum für Complexe Systeme und Visualisierung - CeVis) at the University of Bremen. His research work emphasises dynamical systems, numerical analysis, image analysis, and data analysis, as well as the use of computers in image-based medical diagnostics.

Reviews

[Cara · Rating 4 out of 5]

Bruh.

I don't think I understood most of what I read, but I really just did that. I read that whole textbook.

[Paige McLoughlin · Rating 5 out of 5]

This is a book I read or skim or have varying levels of engagement with each time I pick it up. I always get something new from it. Fractals, Chaos, and Power laws started getting my attention in the late 80s in my deadhead days. The patterns created jived well with my psychedelic aesthetic at the time and I became intrigued by the subject and have had an eye out for books on it ever since.It seems nature quickly tired of the boring old geometry of Euclid (circles and triangles only take you so far) but an iteration of fleas being bitten by smaller fleas who in turn are bitten by smaller fleas and so on. The power of iteration seems to be a powerful form generator in nature and she uses it all the time. Using mixing as well another character that nature adores deterministic Chaos aka the butterfly effect where a system that starts with extremely close initial conditions soon diverges unpredictably. It is the reason why weather forecasting is pretty much useless extrapolating a week or two out into the future. And my favorite persona comes on stage the Mandelbrot another set of points that is created by a simple iteration process that generates infinite complexity and intricate beautiful patterns it is exhibit A for Plato's heavenly form. So little goes in to generate so much. Books like this get me to overcome my laziness and engage with the math. A well I draw upon often.

[Walter · Rating 5 out of 5]

Chaos theory is one of the most powerful and least understood paradigms to have emerged in natural science in the last 50 years. Developed in the 1980s and expanded in the 1990s, Chaos theory challenges the notion that complex processes that scientists see in such fields as physics, geology, biology and economics are not the results of random processes but are actually deterministic phenomena, the result of complicated conditions. The exciting part of this theory is that, unlike stochastic phenomena, processes that arise from deterministic conditions are, at least theoretically, able to be modeled, and thus predictable. This idea has breathed new life into scientific inquiry into a number of fields.

This book is a good and easy to read introduction to Chaos Theory and the Fractal Geometry that illustrates it. Peitgen explores the basic assumptions of Chaos Theory; the dependence on intial conditions, the non-linear dynamics that characterise it, and the effects of earlier observations of a phenomenon on its later direction. The mathematics of this theory are complicated, but Peitgen introduces the basic concepts of chaos theory in a way that the non-sophisticated mathematician can understand. He does this in a very readable and compelling way.

Another fascinating aspect of this work is Peitgen's exploration of Fractal Geometry. Fractals are geometric shapes that include complicated spurs, peaks and other irregularities that one does not see in the neat lines, angles and arcs of Euclidian geometry. Peitgen illustrates the geometry of some of the processes that he maps out with beautiful maps of Fractal shapes. His illustrations show us how these processes can result in the kind of complex but beautiful shapes that we see in crystals, plants, shorelines and other natural occurances. These shapes also lend credence to the theory that Chaos processes could have led to the generation of some of these natural phenomena.

Pietgen further illustrates his topics with classical theories from some of the greatest scientists and mathematicians in history. Men like Blaise Pascal, Raul Julia and Benoit Mandelbrot have advanced theories in their times that have contributed to Chaos theory, so the reader can understand that Chaos theory was not pulled out of some theoretical hat. Furthermore, after every chapter, Peitgen lays out a Basic program that generates some of the processes and shapes that Peitgen describes in the chapter. If the reader can plug these short programs into a Basic compiler, he can visually see the concepts that Peitgen is teaching him. This is a powerful learning tool.

This book is absolutely essential for anyone who wants to understand Chaos Theory and Fractal geometry. It is very highly recommended.

[Andrew Davis · Rating 4 out of 5]

A great introduction to fractals and chaos with amazingly simple mathematics, and accompanied by a number of great pictures. The authors put a lot of effort into ensuring a clear presentation, which in turn resulted in its size presenting some challenges. As both a text and reference source would make a great addition for anyone interested in those subjects.

[Ken Sharp · Rating 3 out of 5]

The authors of this book tried to take a branch of math and make it approachable for non-math people. Mostly they were successful in that, although it's a little hard to keep people interested through a very dense, 900+ page book.

Full disclosure: I didn't read the entire book word-for-word. I was mostly interested in the example programs at the end of each chapter. I took it upon myself to convert the BASIC code to javascript that can run in your browser. My results are here: http://sharpk60.blogspot.com/2015/11/...

[Charles Frode · Rating 5 out of 5]

Wow! Wanna understand chaos theory and fractals? Who doesn't? Seriously, this college level text is full of wonderfully succinct explanations about how life unfolds according (chaos) and the wonderful patterns or structures it leaves (fractals). Check out Feigenbaum's constant if you want to understand the evolution of everything. Seriously. Dense text, math, but it's worth the digging to get the jewels.

[Paige McLoughlin · Rating 5 out of 5]

This is a book I read or skim or have varying levels of engagement with each time I pick it up. I always get something new from it. Fractals, Chaos, and Power laws started getting my attention in the late 80s in my deadhead days. The patterns created jived well with my psychedelic aesthetic at the time and I became intrigued by the subject and have had an eye out for books on it ever since.It seems nature quickly tired of the boring old geometry of Euclid (circles and triangles only take you so far) but an iteration of fleas being bitten by smaller fleas who in turn are bitten by smaller fleas and so on. The power of iteration seems to be a powerful form generator in nature and she uses it all the time. Using mixing as well another character that nature adores deterministic Chaos aka the butterfly effect where a system that starts with extremely close initial conditions soon diverges unpredictably. It is the reason why weather forecasting is pretty much useless extrapolating a week or two out into the future. And my favorite persona comes on stage the Mandelbrot another set of points that is created by a simple iteration process that generates infinite complexity and intricate beautiful patterns it is exhibit A for Plato's heavenly form. So little goes in to generate so much. Books like this get me to overcome my laziness and engage with the math. A well I draw upon often.
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following reviews


READING PROGRESS
December 24, 2020 – Started Reading
December 24, 2020 – Shelved
December 26, 2020 – page 14
1.62% "found the other big doorstop of a book I enjoy so much. Scribble scribble,"
December 26, 2020 – page 61
7.06% "intro examples in chapter one the Sierpinski gasket, The multiple copier machine copy, shrink, twist (repeat) nature does that a lot. The logistic iteration and deterministic chaos and butterfly effect and fun with Fibonacci's rabbit sequence and how it tends to the golden ratio. All the hits are in the first chapter."
December 26, 2020 – page 104
12.04% "This is a perfect book for a voraciously curious but very undisciplined mind. Good stuff."
December 26, 2020 – page 128
14.81% "This is cool I get new insights (incites) every time I read it,"
December 26, 2020 – page 163
18.87% "Math and nature love to iterate. We are all constructivists now."
December 26, 2020 – page 192
22.22% "Hausdorf measures and measuring fractional dimensional freaks that seem to populate actual existing reality we have here."
December 27, 2020 – page 192
22.22% "an empirical method of figuring out fractional dimensions in box-counting methods that work for objects in the real world presented to the observer."
December 27, 2020 – page 214
24.77%
December 27, 2020 – page 276
31.94% "going into the gory details of actually building fractals via specific iterative methods."
December 27, 2020 – page 329
38.08%
December 27, 2020 – page 329
38.08%
December 27, 2020 – page 376
43.52% "space-filling curves"
December 27, 2020 – page 433
50.12% "I love the Mandelbrot set so easy to generate simply points on the border between contained and escaping to infinity of complex numbers added and squared over and over again. The equation for people into that is C(n+1)= (C(n) + K)^2 over and over again. I will drop a video here. https://www.youtube.com/watch?v=NGMRB...
https://www.youtube.com/watch?v=FFftm..."
December 27, 2020 – page 433
50.12% "Randomizing fractals and power spectra with Brown, White, Pink, and Black noise. Fun stuff I am fascinated with the fact that there are different flavors of randomness that obey different distributions of statistical noisiness. I love when something that seems like one thing from far away turns out to be many things up close."
December 27, 2020 – page 433
50.12% "Deterministic Chaos (Butterfly effect) and the mixing or folding analogy for how it comes about mathematically"
December 27, 2020 – page 541
62.62%
December 27, 2020 – page 541
62.62% "Period doubling as a route to chaos and Feigenbaum number that expresses the proportions to that doubling. Behold another video.

https://www.youtube.com/watch?v=ovJcs..."
December 27, 2020 – page 705
81.6% "the strange attractor which has chaotic orbits that neither repeat nor are predictable but stay in a certain range. The Lorentz attractor for example can be constructed from a handful of partial differential equations that can't be cleanly solved but makes just such a strange attractor that has chaotic orbits but in a certain well definite range in 3-dimensional space."
December 27, 2020 – page 839
97.11% "Mandelbrot sets and their relation to Julia sets. They are connected and generated by very similar methods."
December 27, 2020 – Shelved as: early-twenty-first-century
December 27, 2020 – Shelved as: european-history
December 27, 2020 – Shelved as: general-science
December 27, 2020 – Shelved as: mathematics
December 27, 2020 – Shelved as: philosophy
December 27, 2020 – Shelved as: physics
December 27, 2020 – Finished Reading
January 17, 2021 – Shelved as: chemistry
January 17, 2021 – Shelved as: college-textbook
January 17, 2021 – Shelved as: computer-science
January 17, 2021 – Shelved as: engineering
January 17, 2021 – Shelved as: late-twentieth-century

[Krishna · Rating 4 out of 5]

This was my first introduction to the theory of chaos and non linear dynamics. Provides a comprehensive approach to understanding chaos in many natural phenonomena happening around us, while keeping the reader interested

[Oren]

science hurts.

[Asgar · Rating 5 out of 5]

Very hard read. To be honest, I'm only two-thirds a way through it.
I'm keeping more as a reference book, which is what it's meant to be for in the first place...

[Ron Banister · Rating 4 out of 5]

Tough to read but worth the effort!!

[Tex Kaplan · Rating 5 out of 5]

Beautifully written and illustrated with tons of cool stuff. Discusses many topics which can be explored independently.