Fractals Everywhere - Michael Barnsley - Academic Press

Description: Fractals Everywhere by Michael Barnsley Academic Press, 1988, 0120790629, Corrected sixth printing, Hardcover with pictorial laminated cover, Fine condition, no underlining, no highlighting. 394 pages. INTRODUCTION Fractal geometry will make you see everything differently. There is danger in reading further. You risk the loss of your childhood vision of clouds, forests, galaxies, leaves, feathers, flowers, rocks, mountains, torrents of water, carpets, bricks, and much else besides. Never again will your interpretation of these things be quite the same. The observation by Mandelbrot [Mand 1982] of the existence of a "Geometry of Nature" has led us to think in a new scientific way about the edges of clouds, the profiles of the tops of forests on the horizon, and the intricate moving arrangement of the feathers on the wings of a bird as it flies. Geometry is concerned with making our spatial intuitions objective. Classical geometry provides a first approximation to the structure of physical objects; it is the language which we use to communicate the designs of technological products, and, very approximately, the forms of natural creations. Fractal geometry is an extension of classical geometry. It can be used to make precise models of physical structures from ferns to galaxies. Fractal geometry is a new language. Once you can speak it, you can describe the shape of a cloud as precisely as an architect can describe a house. This book is based on a course called "Fractal Geometry" which has been taught in the School of Mathematics at Georgia Institute of Technology for two years. The course is open to all students who have completed two years of calculus. It attracts both undergraduate and graduate students from many disciplines, including mathematics, biology, chemistry, physics, psychology, mechanical engineering, electrical engineering, aerospace engineering, computer science, and geophysical science. The delight of the students with the course is reflected in the fact there is now a second course entitled "Fractal Measure Theory." The courses provide a compelling vehicle for teaching beautiful mathematics to a wide range of students. CONTENTS Acknowledgments 1 Introduction 2 Metric Spaces; Equivalent Spaces; Classification of Subsets; and the Space of Fractals 2.1. Spaces 2.2. Metric Spaces 2.3. Cauchy Sequences, Limit Points, Closed Sets, Perfect Sets, and Complete Metric Spaces 2.4. Compact Sets, Bounded Sets, Open Sets, Interiors, and Boundaries 2.5. Connected Sets, Disconnected Sets, and Pathwise Connected Sets 2.6. The Metric Space (H(X), h): The Place Where Fractals Live 2.7. The Completeness of the Space of Fractals 2.8. Additional Theorems about Metric Spaces 3 Transformations on Metric Spaces; Contraction Mappings; and the Construction of Fractals 3.1. Transformations on the Real Line 3.2. Affine Transformations in the Euclidean Plane 3.3. Mobius Transformations on the Riemann Sphere 3.4. Analytic Transformations 3.5. How to Change Coordinates 3.6. The Contraction Mapping Theorem 3.7. Contraction Mappings on the Space of Fractals 3.8. Two Algorithms for Computing Fractals from Iterated Function Systems 3.9. Condensation Sets 3.10. How to Make Fractal Models with the Help of the Collage Theorem 3.11. Blowing in the Wind: Continuous Dependence of Fractals on Parameters 4 Chaotic Dynamics on Fractals 4.1. The Addresses of Points on Fractals 4.2. Continuous Transformations from Code Space to Fractals 4.3. Introduction to Dynamical Systems 4.4. Dynamics on Fractals: Or How to Compute Orbits by Looking at Pictures 4.5. Equivalent Dynamical Systems 4.6. The Shadow of Deterministic Dynamics 4.7. The Meaningfulness of Inaccurately Computed Orbits is Established by Means of a Shadowing Theorem 4.8. Chaotic Dynamics on Fractals 5 Fractal Dimension 5.1. Fractal Dimension 5.2. The Theoretical Determination of the Fractal Dimension 5.3. The Experimental Determination of the Fractal Dimension 5.4. Hausdorff-Besicovitch Dimension 6 Fractal lnterpolation 6.1. Introduction: Applications for Fractal Functions 6.2. Fractal Interpolation Functions 6.3. The Fractal Dimension of Fractal Interpolation Functions 6.4. Hidden Variable Fractal Interpolation 6.5. Space-Filling Curves 7 Julia Sets 7.1. The Escape Time Algorithm for Computing Pictures of IFS Attractors and Julia Sets 7.2. Iterated Function Systems Whose Attractors Are Julia Sets 7.3. The Application of Julia Set Theory to Newton's Method 7.4. A Rich Source of Fractals: Invariant Sets of Continuous Open Mappings 8 Parameter Spaces and Mandelbrot Sets 8.1. The Idea of a Parameter Space: A Map of Fractals 8.2. Mandelbrot Sets for Pairs of Transformations 83. The Mandelbrot Set for Julia Sets 8.4. How to Make Maps of Families of Fractals Using Escape Times 9 Measures on Fractals 9.1. Introduction to Invariant Measures on Fractals 9.2. Fields and Sigma-Fields 9.3. Measures 9.4. Integration 9.5. The Compact Metric Space (P(X), d) 9.6. A Contraction Mapping on (P(X) 9.7. Elton's Theorem 9.8. Application to Computer Graphics References Additional Background References Index nthdegree books

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