Παίζει ο Θεός ζάρια: Η επιστήμη του χάους

by Ian Stewart, Κωνσταντίνος Σαμαράς (Translator)

Librarian's note: This is an Alternate Cover Edition for ISBN10: 9607122011 ISBN13: 9789607122018.

Οι μεγάλοι επιστήμονες του εικοστού αιώνα νόμισαν ότι είχαν ανακαλύψει τους αδήριτους νόμους ενός σύμπαντος που "δουλεύει" με την ακρίβεια ενός ωρολογιακού μηχανισμού. Όμως, έκαναν λάθος...

Από τα μικρότερα σωμάτια της κβαντικής φυσικής, τα "κενά" στη ζώνη των αστεροειδών, μέχρι τις μπάλες του μπιλιάρδου, τον καιρό και το χρηματιστήριο, τίποτε στο σύμπαν δεν συμπεριφέρεται με τον τρόπο που μπορεί ολοκληρωτικά να προβλεφθεί. Όμως, ερωτάται: αν ο πραγματικός κόσμος είναι τόσο τυχαίος, ποια χρησιμότητα έχουν οι απλές γραμμές, οι σφαίρες και οι κύκλοι των παραδοσιακών μαθηματικών;
-Πολύ λίγη.
Το υπογράφω υπεύθυνα, λέει ο Ian Stewart. Οι μαθηματικοί, για να αντιμετωπίσουν επιτυχώς τον "άτακτο" κόσμο, ανέπτυξαν την θεωρία του Χάους - την πιο παράξενη και επαναστατική σύλληψη των τελευταίων χρόνων. Η ιστορία, οι βασικές αρχές και πολλές πρακτικές εφαρμογές του Χάους, μας αποκαλύπτονται μέσα από το ελκυστικό αυτό βιβλίο.

Genres: Science, Mathematics, Nonfiction, Physics, Popular Science, Philosophy, Economics

358 pages, Paperback

First published January 1, 1989

About the author

Ian Nicholas Stewart is an Emeritus Professor and Digital Media Fellow in the Mathematics Department at Warwick University, with special responsibility for public awareness of mathematics and science. He is best known for his popular science writing on mathematical themes.
--from the author's website

Librarian Note: There is more than one author in the GoodReads database with this name. See other authors with similar names.

Reviews

[WarpDrive · Rating 4 out of 5]

This book is a solid, interesting and insightful introduction to Chaos theory (the relatively recent and fascinating branch of physics that deals with the study of nonlinear dynamical systems exhibiting extreme sensitivity to initial conditions, in which seemingly random complex behavior can derive from simple deterministic, innocuous-looking equations).

The material treated by the book is pretty standard for a good introduction to the subject: I think that it could actually be used as a supporting book for a non-mathematical undergraduate course in the subject. It would also be valuable reading for a course in the philosophy of science, as it contains, in a few places, fascinating discussions about the scientific method, about the contrast between the paradigm of modelling through partial differential equations and the methods of chaos theory, about the real meaning of complexity and of randomness and the challenges posed by chaotic behavior to the experimental verification of mathematical models, and other similarly interesting subjects.

Overall it is a quite enjoyable book, written with conceptual clarity, and one of the very few books about chaos theory that at least attempt to seriously get into the more subtle conceptual elements of this discipline.

However it must also be said that the subtitle “the new mathematics of chaos” is misleading - this is not a book about the mathematics of chaos, but it's more about the conceptual features of the phenomenon.
But, even if devoid of mathematics, it can be really fully appreciated only by readers who had some prior knowledge of basics of topology and of partial differential equations. From this perspective, it really leaves you wanting for a more mathematical, quantitative approach – and this is quite unsatisfactory – this book could so easily have been a real gem.
The lack of mathematical detail is occasionally frustrating (for example: Lorenz simplified mathematical model for atmospheric convection is shown, but there is no explanation of how these three differential equations are derived, nor any explanation of what the variables in the equations actually mean; another example: the concept of fractal dimension is introduced, but no mathematical detail is presented; even relatively simple examples like the driven oscillator or the double pendulum are not treated mathematically – something which the author could have at least done as a separate item in an appendix at the end of the book).

On the other hand, it is not an over-simplistic book: many fascinating features of chaos theory are addressed in a pretty rigorous, occasionally deep, but always approachable manner:
- there is an excellent introduction to the relationship between the topological features of the phase space, and the overall behavioral pattern of the dynamics of the associated system (in particular I enjoyed the part about Poincare sections and how they relate to phase portraits and attractors)
- the logistic mapping is treated beautifully, and the introduction to the concept of strange attractors is quite enjoyable, possibly one of the best I have seen
- the same applies to the concept of self-similarity and how the the enormously varied range of possible mappings gets lumped together into universality classes, where within each class the scaling ratio is always the same (for example: the famous 4.669 for the class of mappings structurally similar to the logistic mapping)
- the frustratingly complex phenomenon of turbulence is treated really well
- the relationship between strange attractors and their fractal dimension is very interesting

Overall, it is a very good introduction to chaos theory and nonlinear dynamics, recommended to readers with no prior exposure to this fascinating discipline who are interested in a serious but non-mathematical treatment of the subject.

[Charbel · Rating 4 out of 5]

Before we start with the review, let's take a moment to appreciate how good of a science communicator Ian Stewart is.

Now on with the nitty gritty.

When faced with accepting Quantum Mechanics, Einstein famously said: "God does not play dice with the universe", to which Stephen Hawking wittily replied: "Not only does God play, but he sometimes throws them where they cannot be seen". Quantum Mechanics, you see, cannot be handled with simple every day linear mathematics. Instead we attempt to explain it using probability. The reason behind this is chaos.

Chaos may not be apparent in the overview of things, but when the smaller details add up, chaos becomes the main force behind them. It is the reason why so many behaviours seem unpredictable, or even random. To understand chaos, we cannot rely on classical linear mathematics, in fact just to glimpse it mathematicians had to become absorbed in the world of topology; a world of saddles, sinks, sources, and attractors; where a hole is not the lack of something but is in itself something (I love that!). I admit that before reading this book I underestimated topology, thinking that it could not rival calculus, statistics, and probability. But once I was asked to visualise an object in four dimensions, let alone fix or six, I understood my own ignorance. In the end, what we end up with is that chaos is not only unpredictable, but also stable; making it one of the most dazzling paradoxes around.

Now enough about chaos and topology, let's talk about the book; after all that's what a review is for. Now, can you read this book without an advanced background in mathematics? Yes, I did. Will it be easy? Not particularly. The larger part of Does God Play Dice is conceptual. You have to put in an effort. If I had to compare this book to something, I'd say that it's close to an introductory course on chaos. It explains a whole lot, but it leaves you with so many questions. The best aspect of this book is that some of the most difficult things to understand are explained clearly with Ian Stewart's subtle sense of humour. And so even when I had my eyes closed trying to visualise something, like attractors, or writing down notes on the Butterfly Effect ( which is pretty useful to me), it was still fun. Challenging, but fun!

Best of all, the book prepares you to read more about chaos. Because let's face it, when you finally finish this book, you're going to have one of two reactions: either "Wow, I'm so glad I read this! I need to learn more about chaos!" or "I don't even want to hear the C word again! Now where's the aspirin?".
Fortunately for me, I had the former reaction. Some chapter were fantastic (chapter 16 comes to mind), others, like the pendulum chapter, could have used more "bling".

So would I recommend this book? Yes, definitely. But a word of advice: take your time with it. Let the new concepts sink in first. Don't rush through it; read the sentence (or the chapter) multiple times if you have to, until you get it. Because once you do, it's worth it.
If you do decide to pick it up, I hope you enjoy it. Have fun, and sorry about the long review.

[Natalia · Rating 4 out of 5]

What a great introduction to chaos theory! This book is not only well-written, but it's also incredibly interesting.

[Eleanore · Rating 4 out of 5]

An extremely accessible history of the emergence of chaos theory and description of its fundamental elements and dynamics. Written with an eye for humor, the book is a real triumph of conceptual clarity for the non-mathematically inclined and reflects an important extension to the basic qualitative understanding of science, the ramifications of which are still working themselves out even in the hard scientific disciplines. I am, however, thoroughly looking forward to the eventual impact this new field has on the social sciences! A salutary warning to the whole range of intro-quantitative methods courses: "Linearity is a trap. The behavior of linear equations - like that of choirboys - is far from typical."

[Karel Baloun · Rating 5 out of 5]

It's stunning and intriguing review of nonlinear systems (chaos), from countless real world perspectives. Stewart's humorous and engaging writing style makes the book a pleasure. He starts from simple mathematical equations and simple physical systems such as pendulums and turbulent water, and routinely takes the idea out to cutting edge research or engineering possibilities.

Now I know what math textbooks and areas of study to proceed to, and Stewart has given mea geometric ability to visualize, necessary for success

[Kerem · Rating 4 out of 5]

At times a bit technical, this is indeed an intriguing book, a passionate account of Stewart on chaos theory and how endless applications and uses it has. I really enjoyed it, and recommend it to anyone to challenge your perspective on how things happen in world.

[Alex Delogu · Rating 4 out of 5]

This book really gets into the theoretical stuff that was missing in Gleick's book on chaos. It still doesn't go heavily into the math but I struggled with some of the more technical material. I will certainly come back to. Stewart is a gifted expositor.

[Kobdik · Rating 4 out of 5]

To jedna z tych książek, które jeśli ocenisz zbyt nisko jest ryzyko, to nie książka jest słaba, tylko ty za głupi żeby ją zrozumieć. Ode mnie 4 ;-)

[Charles · Rating 5 out of 5]

The best mathematical models for many physical events rely on chaotic formulas and the number continues to grow rapidly. It now appears that some exposure to chaos and fractals will be a necessary component of the education of all future applied mathematicians. Given the simplicity of many of the equations, it can be strongly argued that chaos should be an early component of all mathematics education. Also, programming a computer to generate the images is very simple and a lot of fun.
To study chaos, you need a place to start, and this book will point you in the right direction and give you a brisk tail wind. The author, best known for his mathematics columns in Scientific American, writes with exceptional clarity. There are very few equations, as Stewart relies extensively on the verbal explanation. While computer generation is mentioned, only one very short BASIC program is given.
The material is pretty standard for introductory chaos and could serve as a textbook for a non-mathematical course in the subject. It would also be valuable reading for a course in the philosophy of science. Fairly extensive historical backgrounds are given for many of the initial discoveries.
If you have heard about chaos and want to know what all the excitement is about or are looking for reading material for a class you are teaching, this book is an excellent place to explore.

Published in Journal of Recreational Mathematics, reprinted with permission and this review appears on Amazon.

[Koen Crolla · Rating 4 out of 5]

Nothing particularly new, but I guess I've read enough of these now that that was pretty likely. It's a very good overview of the whats and whys of chaos theory, comparable to Gribbin's Deep Simplicity, though maybe slightly less accessible. The final chapter is marred by an ill-conceived rant at a straw man of reductionism, but nobody is perfect.

[Diego Fernández · Rating 5 out of 5]

There subjects were interesting I did not know. I learnt a lot from it.

There mere interesting is I grasped on "God plays dice" - it was a referential to Einstein's quotes when he was working quantum mechanics. But yeah, it doesn't mean we refer to God popular culture. I think the law of natures has to see with chaos. Well, I have no words to say about a review of this book. Took me few days or weeks to finish due to I bought in Spanish and was quite larger.

This kind of theory was new to me. I hope for this year I learn more of mathematics than I'd learnt at school or university. Well, thanks for reading my reviews, even if it was too short. :)

[Pete Wung · Rating 4 out of 5]

Please read my book review on my blog. Thank you.

https://polymathtobe.blogspot.com/202...

[José González · Rating 4 out of 5]

A good introduction to chaos. It’s rewarding both as a mathematics book and a philosophical inquiry of the nature of our relation to knowledge of the natural world. This relation is established through the human language of math.

[rebecca · Rating 5 out of 5]

i haven't read every chapter of this book, but i read most. and the book is excellent. despite the tagline, the book contains minimal maths (which is quite refreshing). would recommend to anyone interested in supplementing their learning or just curious bc it's quite accessible. somehow manages to include both depth and breadth to the subject, without being much over 400 pages long. incredible.

[Babis · Rating 3 out of 5]

Υπερβολικά επιστημονικό για τα γούστα μου.
Θέλει οπωσδήποτε να έχεις κάποιες βάσεις ώστε να το ευχαριστηθείς.

[Julius · Rating 4 out of 5]

En un pasado remoto la naturaleza se nos aparecía como algo incomprensible, gobernado por el antojo de los dioses. Durante siglos los científicos fueron rastreando regularidades en ella, hasta que creyeron haber descubierto unas leyes que prescribían el movimiento de cada partícula del universo con exactitud y para siempre: el mundo era como un mecanismo de relojería.

En el siglo XX esta visión comenzó a cuartearse, y la incertidumbre y el azar se introdujeron en ella. Los sistemas no siempre actuaban como estaba previsto y las nociones de predicción y experimento adquirieron aspectos intranquilizadores. El reloj se estaba desarmando, y la ley y el orden eran reemplazados, como diría Einstein, por «un Dios que juega a los dados».

El profesor Stewart parte de ahí para llevarnos hacia una nueva concepción de la regularidad -la de la matemática del caos- que da sentido a la complejidad de la vida real: desde las inexplicables volteretas de un satélite de Saturno a los latidos de nuestro corazón, desde la previsión meteorológica al crecimiento de las poblaciones de insectos. Conceptos como el de «fractal» no sólo sirven para entender cómo se ha creado la cosmografía imaginaria de La guerra de las galaxias, sino que resultan hoy indispensables para «captar la textura de la realidad». Lo admirable es que Stewart acierta a explicarlo con sencillez, recurriendo al ejemplo o a la anécdota, sin utilizar formulaciones matemáticas ni recurrir a la jerga del especialista.

Este es un libro que resulta francamente entretenido, pero que nos enriquecerá además, con una visión enteramente renovada del mundo y de la ciencia. Un libro dividido en conceptos por capítulos, y un libro en definitiva, para releer.

[Paulo · Rating 5 out of 5]

Um excelente livro sobre o Caos.

Esse livro fez-me recordar sobre a formação de Engenheiro de Controle de Processos, versando sobre inflexões, pontos de sela, autovalores (eigenvalues) e outros aspectos fascinantes à existência de um ponto de equilíbrio ou tendente à instabilidade.

Deve ser lido como um livro de idéias, sem as fórmulas matemáticas complexas para determinar se um ponto leva à instabilidade ou não, além das questões da passagem de um mundo analógio (mundo real) para o digital (computadores).

Ao final, grande quantidade de referências bibliográficas. Para os aficionados.

[Maria · Rating 4 out of 5]

Me encanto bastante, me dio un insight a esta parte de las matematicas de la que no tenia ni idea y pues haber si escribo un resumen propio como debe de ser cuando este siendo menos vaga. Pero de momento, lo acabo de terminar, y de los mas simple statements me acuerdo de como simple systems can give rise to very complex behaviour, de cmoo chaos no quiere decir que algo no se pueda predecir, pero si que teniendo finitely accurate initial measurements, means errors grow exponentially when making predictions for the weather for example. Del descubrimiento del madelbrot set hasta la aplicacion de estas matematicas para construir una maquina para measure el quality del material para ser puesto into springs. Como todos los libros esto, rompe la concepcion de que el hacer matematicas es completamente irrelevante a la vida real. Cuando sin las matematicas, casi todo lo que tenemos simplemente no podria haberse developed.

Y aquí vienen los notes que he tardado un poqutín demasioda para sentarme y escribirlo o ejem más bien copiar los highlights del kindle y editarlo para acabar con un resumen que es un poquitin demasiado largo y estonces aca solo pongo lo q cabe y el resto esta guardado en el forder de books en el escritorio

Chaos occurs when a deterministic (that is, non-random) system behaves in an apparently random manner.
An inherent feature of mathematical equations in dynamics. The ability of even simple equations to generate motion so complex, so sensitive to measurement, that it appears random.
Slight change in the starting value: lose track completely of where it's going.
Out of chaos emerges pattern. Determinism for simple systems with few degrees of freedom, statistics for complicated systems with many degrees of freedom.

Poincaré founded the modern qualitative theory of dynamical systems. His greatest creation was topology – the general study of continuity. He called it analysis situs – the analysis of position.
Topology has been characterized as ‘rubber sheet geometry’. More properly, it is the mathematics of continuity. Continuity is the study of smooth, gradual changes, the science of the unbroken. Discontinuities are sudden, dramatic: places where a tiny change in cause produces an enormous change in effect.
In other words, instead of looking at all initial states, you can look at just Figure 24 If a point in phase space traces out a closed loop, then it will repeat the same motion periodically for ever. a few. Imagine a whole surface of initial states, and follow the evolution of each until (if it ever does) it comes back and hits the surface again (Figure 25). Can you find one state that returns exactly to where it started? If so, you've bagged a periodic solution.

Neptune, Pluto, and a grain of interstellar dust, the three -body problem
The piece of dust moves within the rotating mutual gravitational field of the two planets. It thinks of itself not as a member of a three-body system, but as a tiny ball rolling around on a rotating but fixed landscape. That's Hill's reduced model.

Chaotic motion obeys exactly the same laws as simple motion, it is just that even small internal variables and force must be accounted for precisely and individually

The nature of randomness and superpositions has meant it is physically impossible to know both velocity and position of particles. There must always be uncertainty.
in quantum mechanics uncertainty is physically manifested and probability waves are thought by some to have physical existence in our reality.

the arrows on nearby curves are fairly closely aligned. This means that the notional fluid, whose flow is represented by the lines, doesn't get torn apart: the motion is continuous.
First, on the left-hand side, there's a point towards which all nearby Figure 36 Phase portrait of a flow in the plane, showing (left to right) a sink, a saddle, a limit cycle, and a source. flow-lines spiral. This is known as a sink. It's rather like a plughole down which the fluid is gurgling, hence perhaps the name. Over on the right-hand side is a plughole in reverse, a point from which fluid spirals away. This is called a source. Think of fluid bubbling up from a spring. In between is a place where flow lines appear to cross. This is known as a saddle. Actually the lines don't cross; something more interesting happens, which I'll describe below. If two jets of a real fluid run into each other, you see saddles. Finally, surrounding the source on the right is a single closed loop. This is a limit cycle. It resembles an eddy, where fluid goes round and round. A whirlpool. In a few pages' time we'll see that, roughly speaking, flows in the plane possess (some or all of) these features, and typically nothing else. There can be several of each feature, but you won't find anything more complicated. I'll also explain why I use the word ‘typically’ here. But first, let's acquaint ourselves more closely with these four fundamental features of flows in the plane – differential equations with two degrees of freedom. Figure
Saddles (Figure 39) are more interesting. They're also the sort of thing that only a mathematician would think of – except that Mother Nature has an even more vivid imagination. In a sense, they're steady states that are stable in some directions and unstable in others.
The one shown is a stable limit cycle: nearby points move towards it. There is also an unstable limit cycle: nearby points move away.

limit cycles are really interesting. If you start on one (Figure 40), you go round and round and round forever, repeating the same motion over and over again. The motion is periodic. There are two basic kinds of limit cycle. The one shown is a stable limit cycle: nearby points move towards it. There is also an unstable limit cycle: nearby points move away.
quasiperiodicity. Here several different periodic motions, with independent frequencies, are combined together.
the criterion for the combination to be periodic is that the ratio of the periods should be a rational number – an exact fraction p/q where p and q are whole numbers.
structurally stable to mean a flow whose topology doesn't change if the equations describing it are altered by a small enough amount. This is a quite different idea from a stable state of a given equation.
So what does a general dynamical system do in the long run? It settles down to an attractor. An attractor is defined to be… whatever it settles down to!

‘Almost all’ numbers in the interval 0 to 1 have decimal expansions that are random. This was proved by an American mathematician called Gregory Chaitin, who studied the limitations of computability. It's believable if you say it right: a number chosen ‘at random’ will have random digits. So the deterministic dynamical system that we've constructed behaves in this random fashion, not just for a few weird initial points, but for almost all of them!
every continuous mapping of a line segment to itself must have at least one fixed point: a point that maps to itself.
Let me recap. If there's a line segment, such that every point starting on it eventually comes back to it, then there is at least one periodic solution passing through that segment.

to build a Cantor set you start with an interval of length 1, and remove its middle third (but leaving the end points of this middle third). This leaves two smaller intervals, one-third as long: remove their middle thirds too.
are now ready for an electrifying discovery. Not only does the wrap-ten-times mapping have the four curious properties noted – sensitivity to initial conditions, existence of random itineraries, common occurrence of random itineraries, and cake-mix periodicity/aperiodicity. So do the solenoid and its corresponding differential equation.

In 1950 the American ENIAC computer made the first successful calculations in weather prediction. By 1953 the Princeton MANIAC machine had made it clear that routine weather-prediction was entirely feasible.
Numerical weather-prediction is like a huge game of three-dimensional chess. Imagine a fine grid of points drawn on the surface of the Earth, at several heights to track the up–down motion of the atmosphere as well as north–south and east–west.
what are the rules of the game? The rules are the equations of motion of the atmosphere.
Thousands upon thousands of repetitive calculations based on explicit and deterministic rules.
the number of variables involved in the atomic model is far too large for a computer to handle. It can't track each individual atom of the atmosphere.
Institute of Technology in 1944, is cunning and surprising. The model is deliberately coarsened, to filter out the sound waves. You don't use the most accurate possible equations: you deliberately make them less accurate – to

the convection cells, going round and round in a periodic fashion. In a manner typical of classical applied mathematics, Saltzman guessed an approximate form of the solution, substituted it into his equations, ignored some awkward but small terms, and took a look at the result. Even his highly truncated equations were too hard to solve by a formula, so he put them on a computer. He noticed that the solution appeared to undergo irregular fluctuations: unsteady convection. But it didn't look at all periodic.
system of equations that has now become a classic: Here x, y, z are his three key variables, t is time, and d/dt is the rate of change. The constants 10 and 8/3 correspond to values chosen by Saltzman; the 28 represents the state of the system just after the onset of unsteady convection,
is paper shows the first 3,000 iterations of the value of the variable y (Figure 54). It wobbles periodically for the first 1,500 or so, but you can see the size of the wobble growing steadily. Lorenz knew from his linear stability analysis that this would happen: but what happened next? Madness. Violent oscillations, swinging first up, then down; but with hardly any pattern to them.

Using the curve, you can predict the value of the next peak in z provided you know the value of the current peak. In this sense, at least some of the dynamics is predictable. But it's only a short-term prediction.
Figure 57 The butterfly effect: a numerical simulation of one variable in the Lorenz system. The curves represent initial conditions differing by only 0.0001. At first they appear to coincide, but soon chaotic dynamics leads to independent, widely divergent trajectories.

any physical system that behaved nonperiodically would be unpredictable.

from its attractor, then it rapidly homes back on to it. So chaos is a strange and beautiful combination of stability
If you want to predict whereabouts on its attractor a chaotic system will lie in the distant future, and all you know is where it is now, then you've got problems. On the other hand, you can safely predict that even after a random disturbance the system will quickly return to its attractor – or, if it has several attractors, it will return to one of them.

What the butterfly does is disturb the motion of the point in phase space that represents the Earth's weather. Assuming that this point lies on an attractor, albeit a highly complex multidimensional one, then the tiny flapping of the butterfly can divert the point off the attractor only very briefly, after which it rapidly returns to the same attractor. However, instead of returning to the point A that it would have reached if undisturbed, it returns to some nearby point B. The trajectories of A and – then diverge exponentially, but because they lie on the same attractor, they generate time-series with the same texture. In particular, a hurricane – which is a characteristic weather motif – cannot occur in the perturbed time-series unless it was (eventually) going to occur in the original one. So what the butterfly does is to alter the timing of a hurricane that – in a sense – was going to happen anyway.
Given all this, it is an exaggeration to claim the butterfly as the cause of the big changes that its flapping wing sets in train. The true cause is the butterfly in conjunction with everything else. There are billions of butterflies in the world, and the lazy agitation of their wings is just one source of tiny vortices in our atmosphere. The weather is determined by the combined effect of all such influences. The proverbial butterfly is just as likely to cancel out a hurricane as to create one – and it may just raise the average temperature of India by a hundredth of a degree, or generate a small grey cloud over Basingstoke.
In there somewhere is a butterfly, but it is just one of a trillion factors (I underestimate) that have contributed, since the dawn of time and the birth of space, to the presence of that mass of humid air now, here, using these molecules of water and gas. It is no more sensible to blame the flapping of a butterfly's wing than it is the flipping of a quark's quantum state.

Where does this sensitivity come from? It's a mixture of two conflicting tendencies in the dynamics. The first is stretching. The mapping x → 10x expands distances locally by a factor of ten. Nearby points are torn apart. The second is folding. The circle is a bounded space, there isn't room to stretch everything. It gets folded round itself many times, that's the only way to fit it in after you've expanded distances by ten. So, although points close together move apart, some points far apart move close together.

[Mangoo · Rating 3 out of 5]

Chaos represents the third great scientific revolution of last century, after Einstein's relativity and (among the earliest) Plank's and Nernst's quantum field theory. As the others two, chaos is endowed with a veil of mistery and fantasy and remoteness, though appealing in this case, even though its rules are by now quite known and its growing applications are very disparate. This notwithstanding, chaos remains more a curiosity or an abused metaphor among college students, not talking about youngsters, while it should have all rights to belong to high school curricula even. This because it is very interesting and essentially easy to understand if well presented. And it is highly entartaining.
Ian Stewart has produced more than two decades ago this good popularization, an introduction to non-linear system dynamics for laymen, naming it after a reputedly Einsteinian sentence which, truth said, was originally referred to the epistemic a-causality that quantum mechanics seemed to purport and that he did not digest (not till its very late years, that is).
Stewart's is a real introduction to chaos facts and its manifold ramifications (maps, weather forecasts, maths), as compared to the more famous yet more gossipy "Chaos" by Gleick, which deals more with the epic history of the development of ideas behind chaos theory.
Suggested to neophites, and to high school students, too.

[Rosalind · Rating 4 out of 5]

It's a thankless task trying to write for a general audience about a subject as rich, varied and profound as mathematics. Especially in a culture where maths is so badly taught many adults take great pride in not being any good at it (hint: there's a lot more to maths than endlessly adding and dividing fractions!) Some fail by being too superficial, but Ian Stewart can't be accused of that. Here he takes on the relatively new field of chaos, the mathematics of systems where very small changes in parameters lead to huge differences in outcome that, to the uninformed, appear random. These chaotic systems are the true building blocks of the real world rather than the neat, straightforward formulae that create smooth, regular shapes, yet generations of mathematicians and physicists have shied away from them until very recently. Stewart shows how even the most shapeless systems, when looked at from the right angle, exhibit the most exquisite patterns and symmetries. His style is informal, chatty, sometimes iconoclastic, but be warned: it's not a book for mathematical novices. Some of the concepts are mind-twisting!

[Brian Powell · Rating 3 out of 5]

I enjoyed this book initially but felt like it kinda ran out of steam about half-way through. The author's writing style is friendly and engaging, and the material is most definitely interesting. Overall, it was light in details -- which I guess is the point of a popular-level account -- but I found it generally lacking. The title is a bit sensationalized too -- god does play dice but not through chaos. The subtitle is also misleading -- this is not a book about the mathematics of chaos per se... it's more about the conceptual essence of the phenomenon.

[Franck Chauvel · Rating 4 out of 5]

Ian explains in, I think, a very accessible way the mathematics of "Chaos". I found the selected examples simple enough and yet compelling and I liked the stories about the scientists who pioneered the field. Ian also outlook various applications of chaos theory to practical issues, including spring manufacturing.

A very nice introduction, which I recommend to those, like me, who run away when Greek symbols show up. I eventually did not understood whether God plays dice or not, but the I learned a lot about chaotic systems on the way.

[Eric Hertenstein · Rating 3 out of 5]

More Mathy than Chaos but covering much of the same ground. An excellent introduction to the wonders of phase space, and there's a fantastic little bit about Pointcaré near the beginning. Stewart's kind of a cad, though.

[Josh Holland · Rating 4 out of 5]

Lovely introduction to chaos and its discovery and applications in nonlinear dynamics. The style of writing is accessible but not patronising, and there is a nice amount of Ian Stewart's wit scattered amongst the pages.

It focuses more on chaos than the quantum mechanics the titular quote refers to, but there is a chapter on QM at the end. It is a very good read for people interested in chaos and how the world really works.

[Mark Yashar · Rating 3 out of 5]

A good introduction to chaos theory, and some of the mathematics behind it, and possible applications. Topics include strange attractors, self-similarity, and fractals. The book includes some helpful illustrations. The book seems to go into a bit more mathematical detail (some actual equations) than a typical book about chaos theory for a general audience. This particular edition of the book seems to be a bit out-dated in some places (given that it was published in 1990).

[T Campbell · Rating 4 out of 5]

Entertaining and clearly the basis for a lot of modern understanding of the concept. I highly recommend it for anyone who wants to brush up their own mathematical knowledge. It's an older book with added chapters, and could stand to be updated to be a bit more contemporary in terms of the modern implications of its subjects, but that's a tough compromise to manage: get too meaty and you lose accessibility. Still, I'm picking up another Ian Stewart book now.

[Joe · Rating 4 out of 5]

excellent introduction to chaos theory and nonlinear dynamics, including their relationship to topology, history of discovery, and projections into the future. in particular enjoyed learning about poincare sections and how they relate to phase portraits and attractors. also chaotic control (von neuman's dream).

[Michal · Rating 5 out of 5]

A great and lean introduction to the topics of non-linear dynamics and chaos. Ian Stewart is an excellent mathematics popularizer.

The book offers multitude of examples of nonlinearity's existence and utilization which quickly segway to how the mathematical theories explaining them were created.

I recommend this position to anyone casually interested in mathematics.

[Trystan Hopkins · Rating 1 out of 5]

Read this and one other book on Chaos theory. The author misrepresents Einstein's theories at many points in the book, i presume in order to promote his own relativistic views as if they were supported by Einstein himself. Ultimately, it reads like a book whose author didn't fully understand the topic at hand.

[Bradley Gram-hansen · Rating 4 out of 5]

Ian Stewart writes many books, including this one, which cater to both mathematicians and non-mathematicians. Although, no mathematical knowledge is required, they are very insightful books and are a great bit of fun!