The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation

by Gary William Flake

Overview

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 topics: 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.

Product Details

ISBN-13:	9780262561273
Publisher:	MIT Press
Publication date:	01/27/2000
Series:	A Bradford Book
Edition description:	Reprint
Pages:	520
Product dimensions:	8.00(w) x 9.00(h) x 0.98(d)
Age Range:	18 Years

Table of Contents

	Preface	xiii
	How to Read This Book	xiv
	Dealing with Difficult Subjects	xv
	Personal Motivation	xvi
	Acknowledgments	xvi
1	Introduction	1
1.1	Simplicity and Complexity	2
1.2	The Convergence of the Sciences	5
1.3	The Silicon Laboratory	6
I	Computation	9
2	Number Systems and Infinity	11
2.1	Introduction to Number Properties	12
2.2	Counting Numbers	14
2.3	Rational Numbers	15
2.4	Irrational Numbers	16
2.5	Further Reading	22
3	Computability and Incomputability	23
3.1	Godelization	25
3.2	Models of Computation	26
3.3	Lisp and Stutter	30
3.4	Equivalence and Time Complexity	36
3.5	Universal Computation and Decision Problems	40
3.6	Incomputability	42
3.7	Number Sets Revisited	45
3.8	Further Reading	48
4	Postscript: Computation	51
4.1	Godel's Incompleteness Result	52
4.2	Incompleteness versus Incomputability	53
4.3	Discrete versus Continuous	55
4.4	Incomputability versus Computability	56
4.5	Further Reading	57
II	Fractals	59
5	Self-Similarity and Fractal Geometry	61
5.1	The Cantor Set	62
5.2	The Koch Curve	65
5.3	The Peano Curve	66
5.4	Fractional Dimensions	67
5.5	Random Fractals in Nature and Brownian Motion	71
5.6	Further Exploration	75
5.7	Further Reading	76
6	L-Systems and Fractal Growth	77
6.1	Production Systems	78
6.2	Turtle Graphics	80
6.3	Further Exploration	81
6.4	Further Reading	92
7	Affine Transformation Fractals	93
7.1	A Review of Linear Algebra	94
7.2	Composing Affine Linear Operations	96
7.3	The Multiple Reduction Copy Machine Algorithm	98
7.4	Iterated Functional Systems	103
7.5	Further Exploration	105
7.6	Further Reading	106
8	The Mandelbrot Set and Julia Sets	111
8.1	Iterative Dynamical Systems	112
8.2	Complex Numbers	112
8.3	The Mandelbrot Set	114
8.4	The M-Set and Computability	118
8.5	The M-Set as the Master Julia Set	120
8.6	Other Mysteries of the M-Set	125
8.7	Further Exploration	125
8.8	Further Reading	127
9	Postscript: Fractals	129
9.1	Algorithmic Regularity as Simplicity	130
9.2	Stochastic Irregularity as Simplicity	132
9.3	Effective Complexity	134
9.4	Further Reading	136
III	Chaos	137
10	Nonlinear Dynamics in Simple Maps	139
10.1	The Logistic Map	141
10.2	Stability and Instability	144
10.3	Bifurcations and Universality	148
10.4	Prediction, Layered Pastry, and Information Loss	150
10.5	The Shadowing Lemma	153
10.6	Characteristics of Chaos	154
10.7	Further Exploration	156
10.8	Further Reading	158
11	Strange Attractors	159
11.1	The Henon Attractor	160
11.2	A Brief Introduction to Calculus	165
11.3	The Lorenz Attractor	168
11.4	The Mackey-Glass System	173
11.5	Further Exploration	176
11.6	Further Reading	180
12	Producer-Consumer Dynamics	181
12.1	Producer-Consumer Interactions	182
12.2	Predator-Prey Systems	183
12.3	Generalized Lotka-Volterra Systems	186
12.4	Individual-Based Ecology	187
12.5	Unifying Themes	197
12.6	Further Exploration	198
12.7	Further Reading	201
13	Controlling Chaos	203
13.1	Taylor Expansions	204
13.2	Vector Calculus	205
13.3	Inner and Outer Vector Product	207
13.4	Eigenvectors, Eigenvalues, and Basis	209
13.5	OGY Control	211
13.6	Controlling the Henon Map	215
13.7	Further Exploration	218
13.8	Further Reading	219
14	Postscript: Chaos	221
14.1	Chaos and Randomness	222
14.2	Randomness and Incomputability	224
14.3	Incomputability and Chaos	226
14.4	Further Reading	227
IV	Complex Systems	229
15	Cellular Automata	231
15.1	One-Dimensional CA	232
15.2	Wolfram's CA Classification	236
15.3	Langton's Lambda Parameter	242
15.4	Conway's Game of Life	245
15.5	Natural CA-like Phenomena	251
15.6	Further Exploration	255
15.7	Further Reading	258
16	Autonomous Agents and Self-Organization	261
16.1	Termites	262
16.2	Virtual Ants	264
16.3	Flocks, Herds, and Schools	270
16.4	Unifying Themes	275
16.5	Further Exploration	276
16.6	Further Reading	278
17	Competition and Cooperation	281
17.1	Game Theory and Zero-Sum Games	282
17.2	Nonzero-Sum Games and Dilemmas	288
17.3	Iterated Prisoner's Dilemma	293
17.4	Stable Strategies and Other Considerations	295
17.5	Ecological and Spatial Worlds	297
17.6	Final Thoughts	303
17.7	Further Exploration	303
17.8	Further Reading	304
18	Natural and Analog Computation	307
18.1	Artificial Neural Networks	309
18.2	Associative Memory and Hebbian Learning	312
18.3	Recalling Letters	316
18.4	Hopfield Networks and Cost Optimization	318
18.5	Unifying Themes	324
18.6	Further Exploration	325
18.7	Further Reading	326
19	Postscript: Complex Systems	327
19.1	Phase Transitions in Networks	328
19.2	Phase Transitions in Computation	332
19.3	Phase Transitions and Criticality	334
19.4	Further Reading	336
V	Adaptation	337
20	Genetics and Evolution	339
20.1	Biological Adaptation	340
20.2	Heredity as Motivation for Simulated Evolution	342
20.3	Details of a Genetic Algorithm	343
20.4	A Sampling of GA Encodings	348
20.5	Schemata and Implicit Parallelism	353
20.6	Other Evolutionary Inspirations	355
20.7	Unifying Themes	356
20.8	Further Exploration	358
20.9	Further Reading	360
21	Classifier Systems	361
21.1	Feedback and Control	363
21.2	Production, Expert, and Classifier Systems	364
21.3	The Zeroth Level Classifier System	370
21.4	Experiments with ZCS	373
21.5	Further Exploration	379
21.6	Further Reading	380
22	Neural Networks and Learning	383
22.1	Pattern Classification and the Perceptron	385
22.2	Linear Inseparability	390
22.3	Multilayer Perceptrons	392
22.4	Backpropagation	393
22.5	Function Approximation	398
22.6	Internal Representations	404
22.7	Other Applications	409
22.8	Unifying Themes	410
22.9	Further Exploration	411
22.10	Further Reading	413
23	Postscript: Adaptation	415
23.1	Models and Search Methods	416
23.2	Search Methods and Environments	419
23.3	Environments and Models	422
23.4	Adaptation and Computation	423
23.5	Further Reading	424
	Epilogue	425
24	Duality and Dichotomy	427
24.1	Web of Connections	428
24.2	Interfaces to Hierarchies	429
24.3	Limitations on Knowledge	431
	Source Code Notes	435
	Glossary	443
	Bibliography	469
	Index	483

What People are Saying About This

Greg Chaitin

This delightful book illustrates beautifully the paradigm shift in physics from writing equations and solving them to computer modeling and experimentation.

Barak Pearlmutter

This book is a delight.

Endorsement

This delightful book illustrates beautifully the paradigm shift in physics from writing equations and solving them to computer modeling and experimentation.

—Greg Chaitin, author of The Limits of Mathematics

From the Publisher

This book is a delight.

—Barak Pearlmutter, University of New Mexico

This delightful book illustrates beautifully the paradigm shift in physics from writing equations and solving them to computer modeling and experimentation.

—Greg Chaitin, author of The Limits of Mathematics