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Foreword | |
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Introduction: Causality Principle, Deterministic Laws and Chaos | |
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The Backbone of Fractals: Feedback and the Iterator | |
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The Principle of Feedback | |
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The Multiple Reduction Copy Machine | |
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Basic Types of Feedback Processes | |
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The Parable of the Parabola--Or: Don't Trust Your Computer | |
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Chaos Wipes Out Every Computer | |
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Classical Fractals and Self-Similarity | |
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The Cantor Set | |
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The Sierpinski Gasket and Carpet | |
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The Pascal Triangle | |
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The Koch Curve | |
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Space-Filling Curves | |
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Fractals and the Problem of Dimension | |
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The Universality of the Sierpinski Carpet | |
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Julia Sets | |
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Pythagorean Trees | |
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Limits and Self-Similarity | |
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Similarity and Scaling | |
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Geometric Series and the Koch Curve | |
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Corner the New from Several Sides: Pi and the Square Root of Two | |
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Fractals as Solutions of Equations | |
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Length, Area and Dimension: Measuring Complexity and Scaling Properties | |
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Finite and Infinite Length of Spirals | |
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Measuring Fractal Curves and Power Laws | |
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Fractal Dimension | |
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The Box-Counting Dimension | |
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Borderline Fractals: Devil's Staircase and Peano Curve | |
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Encoding Images by Simple Transformations | |
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The Multiple Reduction Copy Machine Metaphor | |
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Composing Simple Transformations | |
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Relatives of the Sierpinski Gasket | |
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Classical Fractals by IFSs | |
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Image Encoding by IFSs | |
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Foundation of IFS: The Contraction Mapping Principle | |
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Choosing the Right Metric | |
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Composing Self-Similar Images | |
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Breaking Self-Similarity and Self-Affinity: Networking with MRCMs | |
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The Chaos Game: How Randomness Creates Deterministic Shapes | |
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The Fortune Wheel Reduction Copy Machine | |
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Addresses: Analysis of the Chaos Game | |
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Tuning the Fortune Wheel | |
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Random Number Generator Pitfall | |
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Adaptive Cut Methods | |
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Recursive Structures: Growing Fractals and Plants | |
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L-Systems: A Language for Modeling Growth | |
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Growing Classical Fractals with MRCMs | |
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Turtle Graphics: Graphical Interpretation of L-Systems | |
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Growing Classical Fractals with L-Systems | |
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Growing Fractals with Networked MRCMs | |
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L-System Trees and Bushes | |
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Pascal's Triangle: Cellular Automata and Attractors | |
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Cellular Automata | |
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Binomial Coefficients and Divisibility | |
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IFS: From Local Divisibility to Global Geometry | |
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HIFS and Divisibility by Prime Powers | |
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Catalytic Converters, or How Many Cells Are Black? | |
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Irregular Shapes: Randomness in Fractal Constructions | |
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Randomizing Deterministic Fractals | |
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Percolation: Fractals and Fires in Random Forests | |
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Random Fractals in a Laboratory Experiment | |
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Simulation of Brownian Motion | |
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Scaling Laws and Fractional Brownian Motion | |
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Fractal Landscapes | |
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Deterministic Chaos: Sensitivity, Mixing, and Periodic Points | |
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The Signs of Chaos: Sensitivity | |
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The Signs of Chaos: Mixing and Periodic Points | |
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Ergodic Orbits and Histograms | |
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Metaphor of Chaos: The Kneading of Dough | |
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Analysis of Chaos: Sensitivity, Mixing, and Periodic Points | |
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Chaos for the Quadratic Iterator | |
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Mixing and Dense Periodic Points Imply Sensitivity | |
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Numerics of Chaos: Worth the Trouble or Not? | |
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Order and Chaos: Period-Doubling and Its Chaotic Mirror | |
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The First Step from Order to Chaos: Stable Fixed Points | |
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The Next Step from Order to Chaos: The Period-Doubling Scenario | |
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The Feigenbaum Point: Entrance to Chaos | |
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From Chaos to Order: A Mirror Image | |
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Intermittency and Crises: The Backdoors to Chaos | |
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Strange Attractors: The Locus of Chaos | |
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A Discrete Dynamical System in Two Dimensions: Henon's Attractor | |
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Continuous Dynamical Systems: Differential Equations | |
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The Rossler Attractor | |
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The Lorenz Attractor | |
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Quantitative Characterization of Strange Chaotic Attractors: Ljapunov Exponents | |
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Quantitative Characterization of Strange Chaotic Attractors: Dimensions | |
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The Reconstruction of Strange Attractors | |
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Fractal Basin Boundaries | |
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Julia Sets: Fractal Basin Boundaries | |
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Julia Sets as Basin Boundaries | |
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Complex Numbers--A Short Introduction | |
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Complex Square Roots and Quadratic Equations | |
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Prisoners versus Escapees | |
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Equipotentials and Field Lines for Julia Sets | |
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Binary Decomposition, Field Lines and Dynamics | |
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Chaos Game and Self-Similarity for Julia Sets | |
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The Critical Point and Julia Sets as Cantor Sets | |
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Quaternion Julia Sets | |
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The Mandelbrot Set: Ordering the Julia Sets | |
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From the Structural Dichotomy to the Binary Decomposition | |
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The Mandelbrot Set--A Road Map for Julia Sets | |
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The Mandelbrot Set as a Table of Content | |
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Bibliography | |
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Index | |