The Computational Beauty of Nature

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1 Gary William Flake The Computational Beauty of Nature Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation A Bradford Book The MIT Press Cambridge, Massachusetts London, England

2 Preface xiii How to Read This Book xiv Dealing with Difficult Subjects Personal Motivation xvi Acknowledgments xvi xv 1 Introduction Simplicity and Complexity The Convergence of the Sciences i 1.3 The Silicon Laboratory 6 I Computation 9 2 Number Systems and Infinity Introduction to Number Properties Counting Numbers Rational Numbers Irrational Numbers Further Reading 22 3 Computability and Incomputability Godelization Models of Computation Lisp and Stutter Equivalence and Time Complexity Universal Computation and Decision Problems Incomputability Number Sets Revisited Further Reading 48

3 4 Postscript: Computation Godel's Incompleteness Result Incompleteness versus Incomputability Discrete versus Continuous Incomputability versus Computability Further Reading 57 II Fractals 59 5 Self-Similarity and Fractal Geometry The Cantor Set The Koch Curve The Peano Curve Fractional Dimensions Random Fractals in Nature and Brownian Motion Further Exploration Further Reading 76 6 L-Systems and Fractal Growth Production Systems Turtle Graphics Further Exploration Further Reading 92 7 Affine Transformation Fractals A Review of Linear Algebra Composing Affine Linear Operations The Multiple Reduction Copy Machine Algorithm Iterated Functional Systems Further Exploration Further Reading The Mandelbrot Set and Julia Sets Iterative Dynamical Systems Complex Numbers The Mandelbrot Set The M-Set and Computability The M-Set as the Master Julia Set Other Mysteries of the M-Set Further Exploration 125

4 8.8 Further Reading 127 Postscript: Fractals Algorithmic Regularity as Simplicity Stochastic Irregularity as Simplicity Effective Complexity Further Reading 136 III Chaos Nonlinear Dynamics in Simple Maps The Logistic Map Stability and Instability Bifurcations and Universality Prediction, Layered Pastry, and Information Loss The Shadowing Lemma Characteristics of Chaos Further Exploration Further Reading Strange Attractors The Henon Attractor A Brief Introduction to Calculus The Lorenz Attractor The Mackey-Glass System Further Exploration Further Reading Producer-Consumer Dynamics Producer-Consumer Interactions Predator-Prey Systems Generalized Lotka-Volterra Systems Individual-Based Ecology Unifying Themes Further Exploration Further Reading Controlling Chaos Taylor Expansions Vector Calculus Inner and Outer Vector Product 207

5 13.4 Eigenvectors, Eigenvalues, and Basis OGY Control Controlling the Henon Map Further Exploration Further Reading Postscript: Chaos Chaos and Randomness Randomness and Incomputability Incomputability and Chaos Further Reading 227 IV Complex Systems Cellular Automata One-Dimensional CA Wolfram's CA Classification Langton's Lambda Parameter Conway's Game of Life Natural CA-like Phenomena Further Exploration Further Reading Autonomous Agents and Self-Organization Termites Virtual Ants Flocks, Herds, and Schools Unifying Themes Further Exploration Further Reading Competition and Cooperation Game Theory and Zero-Sum Games Nonzero-Sum Games and Dilemmas Iterated Prisoner's Dilemma Stable Strategies and Other Considerations Ecological and Spatial Worlds Final Thoughts Further Exploration Further Reading 304

6 18 Natural and Analog Computation Artificial Neural Networks Associative Memory and Hebbian Learning Recalling Letters Hopfield Networks and Cost Optimization Unifying Themes Further Exploration Further Reading Postscript: Complex Systems Phase Transitions in Networks Phase Transitions in Computation Phase Transitions and Criticality Further Reading 336 V Adaptation Genetics and Evolution Biological Adaptation Heredity as Motivation for Simulated Evolution Details of a Genetic Algorithm A Sampling of GA Encodings Schemata and Implicit Parallelism Other Evolutionary Inspirations Unifying Themes Further Exploration Further Reading Classifier Systems Feedback and Control Production, Expert, and Classifier Systems The Zeroth Level Classifier System Experiments with ZCS Further Exploration Further Reading Neural Networks and Learning Pattern Classification and the Perceptron Linear Inseparability Multilayer Perceptrons 392

7 xii Contents 22.4 Backpropagation Function Approximation Internal Representations Other Applications Unifying Themes Further Exploration Further Reading Postscript: Adaptation Models and Search Methods Search Methods and Environments Environments and Models Adaptation and Computation Further Reading 424 Epilogue Duality and Dichotomy Web of Connections Interfaces to Hierarchies Limitations on Knowledge 431 Source Code Notes 435 Glossary 443 Bibliography 469 Index 483

Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science

Hei nz-ottopeitgen. Hartmut Jürgens Dietmar Sau pe. Chaos and Fractals. New Frontiers of Science Hei nz-ottopeitgen Hartmut Jürgens Dietmar Sau pe Chaos and Fractals New Frontiers of Science Preface Authors VU X I Foreword 1 Mitchell J. Feigenbaum Introduction: Causality Principle, Deterministic