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The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation
Gary William Flake develops in depth the simple idea that recurrent rules can produce rich and complicated behaviors. In this book Gary William Flake develops in depth the simple idea that recurrent rules can produce rich and complicated behaviors. Distinguishing "agents" (e.g., molecules, cells, animals, and species) from their interactions (e.g., chemical reactions, immune system responses, sexual reproduction, and evolution), Flake argues that it is the computational properties of interactions that account for much of what we think of as "beautiful" and "interesting." From this basic thesis, Flake explores what he considers to be today's four most interesting computational fractals, chaos, complex systems, and adaptation. Each of the book's parts can be read independently, enabling even the casual reader to understand and work with the basic equations and programs. Yet the parts are bound together by the theme of the computer as a laboratory and a metaphor for understanding the universe. The inspired reader will experiment further with the ideas presented to create fractal landscapes, chaotic systems, artificial life forms, genetic algorithms, and artificial neural networks.
514 pages, Paperback
First published July 10, 1998
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Displaying 1 - 19 of 19 reviews
August 27, 2012
I was somewhat disappointed by this book. Maybe 15 years ago the ideas presented here were radical and surprising, but now they're old hat. Much of the material is stuff that will be familiar to anyone who took intro level math and CS classes in college-- computability, lambda calculus, Godel's theorem, etc. The sections on fractals, cellular automata, neural nets, etc. cover well-trodden ground. There are better books if you want to delve deep into these subjects, and at a superficial level it's stuff you've already seen popularized elsewhere.
The biggest surprise to me was how little nature this book actually contains. There is certainly some-- fractal geometry in ferns, flocking of birds, emergent behavior in ants-- but much of the book is just theoretical discussion of the aforementioned math/CS topics.
I guess it's a sign of how much this whole area of inquiry succeeded, that it all seems rote today! We've played Sim-Ant and Sim City, fooled around with Mandelbrot sets in Javascript, seen machine learning practically applied all over the place. All par for the course.
If you're interested in fractals specifically, try Fractals Everywhere.
If you're interested in complex systems, criticality, and general applications of computation/math to nature, I highly recommend the Princeton Companion to Mathematics. It's a more modern take on these topics. If you're the kind of person who would read The Computational Beauty of Nature, you will definitely enjoy the Princeton Companion. Check it out!
The biggest surprise to me was how little nature this book actually contains. There is certainly some-- fractal geometry in ferns, flocking of birds, emergent behavior in ants-- but much of the book is just theoretical discussion of the aforementioned math/CS topics.
I guess it's a sign of how much this whole area of inquiry succeeded, that it all seems rote today! We've played Sim-Ant and Sim City, fooled around with Mandelbrot sets in Javascript, seen machine learning practically applied all over the place. All par for the course.
If you're interested in fractals specifically, try Fractals Everywhere.
If you're interested in complex systems, criticality, and general applications of computation/math to nature, I highly recommend the Princeton Companion to Mathematics. It's a more modern take on these topics. If you're the kind of person who would read The Computational Beauty of Nature, you will definitely enjoy the Princeton Companion. Check it out!
October 8, 2014
An introduction to several strands of mathematics and computer science which have parallels in nature and biology. The topics covered include:
+ Gödel's Incompleteness Theorem
+ Fractals
+ Chaos
Because of the breadth of topics, the subject matter is sometimes treated fairly lightly. Although this sort of introductory treatment was exactly what I wanted, I occasionaly found the explanations to be too hand-wavy, especially some of the philosophical asides.
Nevertheless, I enjoyed reading this book very much. It's probably the only book that combines all of these topics into one comprehensive volume.
The book has its own website with source code for the programs used to investigate the various subjects.
January 15, 2019
I very much enjoyed this book. While very little of the material was new to me, I enjoyed how it was presented, and more important, how it was connected. This is a somewhat dense book at times, and I had to re-read/re-parse some paragraphs multiple times. The sections "providing" mathematical background, however, are terse to the point of self-parody -- all they are good for is reminding someone about things they already knew, but might not have used recently.
January 12, 2020
This was my first introduction to Complex Systems theory, and it got me hooked on the subject, playing a huge role in my research interests throughout college. Wasn’t the most technical, but for someone new to the ideas it was perfect. Definitely recommend.
July 27, 2013
As Amar Pai notes in his review, most of this is old hat to anyone who's paid attention to their undergraduate computer science curriculum and the pop math of the past 20 years. That said, it's exquisite writing, a wonderful intro for anyone new to computability, and has absolutely lovely digressions (often clearly offset as such). I found the material on computing via chaos via method of linear transforms new and fascinating. As Amar also notes, the title is kind of misleading; I was expecting a good bit more "nature" in this book. Still, well worth reading (you'll tear through it if you've got a sufficiently technical background; I read two-thirds of it waiting to meet my probation officer), and a charmingly deceptive book to leave out on the coffee table--softies will pick it up and actually learn something.
April 21, 2008
Best book of its kind I've read. Shows how mathematical algorithms shape the natural world and incites being gained with computer technology. The math isn't hard for someone who has studied the subject in college.
October 29, 2008
Amazing book about chaos theory, combinatorics, and fractals.
August 17, 2016
I used this for a textbook in my favorite college class: Biologically inspired computation.
October 21, 2021
"The beauty of physics is that simple differential equations can produce astonishingly complicated behavior. It was Newton 's most profound discovery that the forces that describe objects in motion are much simpler than the motions themselves." Gary William Flake
"The alphabet of calculus, being symbolic, is equivalent to the alphabet of the natural numbers. Hence, there are many things that can occur in continuous mathematics that cannot be described by symbolic mathematics precisely because there are more objects in the continuum than there are objects in the symbolic world. In the end, we simply do not have the language to pull it off." Gary William Flake
Ottimo libro che tratta diverse tematiche relative alla programmazione al fine di simulare fenomeni naturali.
Elenco quelle che ho trovato più interessanti con qualche estratto dal libro:
- Frattali: interessante il concetto di dimensionalità "fractals have a fractional dimension, as opposed to an integer dimension that idealized objects have. This characteristic means that a fractal with a dimension of 1.5 is in some way more than a line but less than a plane."
- Sistema Lindenmayer (L-system)
- M-set and the Julia sets: viene descritto lo pseudo-codice
- Caos: si spiega la differenza con comportamento casuale e si introducono gli attrattori.
- La mappa logistica: "is a simple population growth model that is defined by the iterative equation: Xt+l = 4*r*xt*(l - Xt)."
- Equazioni di Lotka-Volterra (Predator-Prey Systems): "consists of two differential equations, one for each species. In its simplest form the two equations are usually presented as where F represents the small -fish population and S represents the shark population."
- Sistemi complessi: "Whereas chaos is the science of how simple things produce complex behavior, complexity is the study of how complex collections of simple units produce a wide variety of behavior."
- Automa cellulare (Cellular Automata) tra cui Il Gioco della vita di Conway.
- Agenti autonomi e auto organizzazione: “At the lowest level, the most primitive agents perform almost ridiculously simple tasks that no one would refer to as intelligent. But as one progress es to higher levels, the emergent composite behavior appears more and more complex, eventually hitting on something that is " intelligent." But this idea is not reductionism. Reductionism claims that the whole can be understood as the sum of the parts. In every case that we have seen, the whole has revealed itself to be a surprise in that it is decidedly more than the sum of the parts. At all levels of nature, recursion and multiplicity of agents promote emergence and self-organization to yield an almost unexplainable form of complexity.”
- Competizione e cooperazione tra cui la teoria dei giochi e il dilemma del prigioniero.
“The key to cooperation being a winning strategy is iteration. If there is no chance that two parties will ever meet again, then defection is a better action from a game-theoretic point of view. Even biological systems obey this principle to a certain extent, in that some organisms are specially adapted to detect when future interactions are not likely to occur. Even your own gut bacteria will start to attack you if your stomach lining is perforated.”
“Yet, the most important point in all of this is that unilateral, mindless cooperation and defection are both bad strategies. The stability, robustness, and resistance of a strategy depend on a special mixture of cooperation and defection.”
- Adattamento: “Our knowledge of evolution is like a mountain viewed from a distance: We may not be able to discern every little peak, bump, or pebble, but the outline of the mountain is so clear and distinct that we have virtually no doubt as to the mountain 's existence.”
“The property of adaptation, from an evolutionary point of view, is often described by the equation adaptation = variation + heredity + selection.”
- Algoritmi genetici: “By simulating a few simple principles (crossover, mutation, reproduction, and selection), one can get a glimpse of how novel and creative innovations are made in nature.”
- Reti neurali tra cui il percettrone (perceptron).
Nel complesso un libro ben scritto e denso di informazioni. L'unica pecca è che è stato scritto nel 2000 e per un libro di (e non solo) programmazione è molto tempo fa. Se cercate qualcosa di più recente e più semplice ma sugli stessi argomenti consiglio l'ottimo The Nature of Code.
"The alphabet of calculus, being symbolic, is equivalent to the alphabet of the natural numbers. Hence, there are many things that can occur in continuous mathematics that cannot be described by symbolic mathematics precisely because there are more objects in the continuum than there are objects in the symbolic world. In the end, we simply do not have the language to pull it off." Gary William Flake
Ottimo libro che tratta diverse tematiche relative alla programmazione al fine di simulare fenomeni naturali.
Elenco quelle che ho trovato più interessanti con qualche estratto dal libro:
- Frattali: interessante il concetto di dimensionalità "fractals have a fractional dimension, as opposed to an integer dimension that idealized objects have. This characteristic means that a fractal with a dimension of 1.5 is in some way more than a line but less than a plane."
- Sistema Lindenmayer (L-system)
- M-set and the Julia sets: viene descritto lo pseudo-codice
- Caos: si spiega la differenza con comportamento casuale e si introducono gli attrattori.
- La mappa logistica: "is a simple population growth model that is defined by the iterative equation: Xt+l = 4*r*xt*(l - Xt)."
- Equazioni di Lotka-Volterra (Predator-Prey Systems): "consists of two differential equations, one for each species. In its simplest form the two equations are usually presented as where F represents the small -fish population and S represents the shark population."
- Sistemi complessi: "Whereas chaos is the science of how simple things produce complex behavior, complexity is the study of how complex collections of simple units produce a wide variety of behavior."
- Automa cellulare (Cellular Automata) tra cui Il Gioco della vita di Conway.
- Agenti autonomi e auto organizzazione: “At the lowest level, the most primitive agents perform almost ridiculously simple tasks that no one would refer to as intelligent. But as one progress es to higher levels, the emergent composite behavior appears more and more complex, eventually hitting on something that is " intelligent." But this idea is not reductionism. Reductionism claims that the whole can be understood as the sum of the parts. In every case that we have seen, the whole has revealed itself to be a surprise in that it is decidedly more than the sum of the parts. At all levels of nature, recursion and multiplicity of agents promote emergence and self-organization to yield an almost unexplainable form of complexity.”
- Competizione e cooperazione tra cui la teoria dei giochi e il dilemma del prigioniero.
“The key to cooperation being a winning strategy is iteration. If there is no chance that two parties will ever meet again, then defection is a better action from a game-theoretic point of view. Even biological systems obey this principle to a certain extent, in that some organisms are specially adapted to detect when future interactions are not likely to occur. Even your own gut bacteria will start to attack you if your stomach lining is perforated.”
“Yet, the most important point in all of this is that unilateral, mindless cooperation and defection are both bad strategies. The stability, robustness, and resistance of a strategy depend on a special mixture of cooperation and defection.”
- Adattamento: “Our knowledge of evolution is like a mountain viewed from a distance: We may not be able to discern every little peak, bump, or pebble, but the outline of the mountain is so clear and distinct that we have virtually no doubt as to the mountain 's existence.”
“The property of adaptation, from an evolutionary point of view, is often described by the equation adaptation = variation + heredity + selection.”
- Algoritmi genetici: “By simulating a few simple principles (crossover, mutation, reproduction, and selection), one can get a glimpse of how novel and creative innovations are made in nature.”
- Reti neurali tra cui il percettrone (perceptron).
Nel complesso un libro ben scritto e denso di informazioni. L'unica pecca è che è stato scritto nel 2000 e per un libro di (e non solo) programmazione è molto tempo fa. Se cercate qualcosa di più recente e più semplice ma sugli stessi argomenti consiglio l'ottimo The Nature of Code.
May 26, 2021
I read this book years ago. Coming back to it recently I noticed chapters on Neural Nets and gradient descent methods of Machine Learning (this book came out in 2000) I am surprised that this stuff was talked about then. I thought the rudiments of Neural Nets were newer, but they seem to have been around in the 1980s and 1990s although the computational muscle wasn't nearly as strong as it is today when it has been paying off.
There are chapters covering Chaos theory, complexity, game theory especially iterated games, and maps of iterated game players using different strategies and tracing their fortunes in interactive maps over time. It is a hodgepodge of cool ideas in a recreational math format. Good stuff quite fun for the mathematically inclined.
There are chapters covering Chaos theory, complexity, game theory especially iterated games, and maps of iterated game players using different strategies and tracing their fortunes in interactive maps over time. It is a hodgepodge of cool ideas in a recreational math format. Good stuff quite fun for the mathematically inclined.
September 24, 2020
I loved this book when it came out, but I was re-reading part of it recently and just surprised by how little is really covered. Maybe it's the oversaturation of these ideas (see Amar Pai's review). But looking at it now, the best way to interact with this book is through its accompanying code, and to expect it to be a starting point, not a reference.
November 2, 2022
No book has made me appreciate nature through looking at computation as much as this book. It being a mathematics book, I didn't expect reading it cover to cover, but I ended up doing so. It's fascinating how it knits together topics like computation, chaos, fractals, cellular automata, adaptation and complex systems
February 11, 2020
Fantastic introduction to many fields within mathematics, systems, and computation. Each chapter presents a concise exposition of different topics
Want to read
October 4, 2020recommendation: Memetics: Hacking Belief Systems
February 8, 2011
I feel a strange draw towards two poles. I love the highly technial man-made achievements and I love the completely unspoiled "nature of nature". This book sythesizes the two extremes beautifully. As our computers push the envelope of mathematics, we are better able to sythesize and understand the structure and appearance of natural things. This book added to my appreciation of nature by showing me the level of computation required to simulate it. At the same time, it added to my appreciation of science by tying it to the emotional response that I feel when I see beauty in nature.
Very little of this is really share-able in a high school classroom. The kids love the fractals and this book helps explain the practical applications of fractals. But fractals weren't even in the CA teaching standards in the first place, so it's all kindof extra-curricular.
But, it's a beautiful book. Beautiful in the way that "Eternal Golden Braid" was beautiful. And much more concise.
Very little of this is really share-able in a high school classroom. The kids love the fractals and this book helps explain the practical applications of fractals. But fractals weren't even in the CA teaching standards in the first place, so it's all kindof extra-curricular.
But, it's a beautiful book. Beautiful in the way that "Eternal Golden Braid" was beautiful. And much more concise.
May 3, 2012
Like my experience with Manfred Schroeder's "Chaos, Fractals, and Power Laws", but extremely well organized, and clear. The example pseudo-code and included examples really got me excited to write some code and play with each topic myself.
June 11, 2012
Very interesting book. The subjects are well selected and complementary to each other. Even if it is more about computing than nature, it is interesting to see the connections that are made between some natural phenomena and math/CS theories.
February 9, 2014
Emergence of complexity from chaos ... How the blind forces of physics and chemistry at the lower level can and does produce a mind boggling complexity as perceived in the natural world.
April 12, 2015
Best science book every written.
Displaying 1 - 19 of 19 reviews