Introduction to Chaos, Fractals and Dynamical Systems

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This book offers a fun and enriching introduction to chaos theory, fractals and dynamical systems, and on the applications of fractals to computer generated graphics and image compression. Introduction to Chaos, Fractals and Dynamical Systems particularly focuses on natural and human phenomenon that can be modeled as fractals, using simple examples to explain the theory of chaos and how it affects all of us. Then, using straightforward mathematic and intuitive descriptions, computer generated graphics and photographs of natural scenes are used to illustrate the beauty of fractals and their importance in our world. Finally, the concept of Dynamical Systems, that is, time-dependent systems, the foundation of Chaos and Fractal, is introduced. Everyday examples are again used to illustrate concepts, and the importance of understanding how these vital systems affect our lives. Throughout the fascinating history of the evolution of chaos theory, fractals and dynamical systems is presented, along with brief introductions to the scientists, mathematicians and engineers who created this knowledge. Introduction to Chaos, Fractals and Dynamical Systems contains ample mathematical definitions, representations, discussions and exercises, so that this book can be used as primary or secondary source in home schooling environments. The book is suitable for homeschooling as a focused course on the subject matter or as a classroom supplement for a variety of courses at the late junior high or early high-school level. For example, in addition to a standalone course on Chaos, Fractals and Dynamical Systems (or similar title), this book could be used with the following courses: Precalculus Geometry Computer programming (e.g. Rust, C, C++, Python, Java, Pascal) Computer graphics The text can also be used in conjunction with mathematics courses for undergraduates for non-science majors. The book can also be used for informal and lively family study and discussion. For each chapter, exercises and things to do are included. These activities range from simple computational tasks to more elaborate computer projects, related activities, biographical research and writing assignments. Sample Chapter(s) Introduction Chapter 1: What is Chaos? What are Fractals? Contents: What is Chaos? What are Fractals? Foundations of Chaos and Fractal Theory Chaos and Fractals in Nature Chaos and Fractals in Human-Made Phenomena Dynamical Systems and Systems Theory Readership: The market consists of mathematically inclined and/or homeschooled students from grades 6 or 7 through 12, and even early undergraduate, as well as computer science students the same grade levels as above. Hobbyists of all ages. Rust community — while this is not a book that intends to teach the Rust language, it is an impactful showcase of the language. We think we could generate some buzz for it on social media, for example by posting links to the program source code and full-color images (which are freely available online).

KITCHEN SCIENCE FRACTALS

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This book provides a collection of 44 simple computer and physical laboratory experiments, including some for an artist's studio and some for a kitchen, that illustrate the concepts of fractal geometry. In addition to standard topics — iterated function systems (IFS), fractal dimension computation, the Mandelbrot set — we explore data analysis by driven IFS, construction of four-dimensional fractals, basic multifractals, synchronization of chaotic processes, fractal finger paints, cooking fractals, videofeedback, and fractal networks of resistors and oscillators. In order to benefit fully from this book, readers should have access to either Mathematica or Python and be able to run programs in one of those languages. No programming skill is needed: the programs can be copied from the supplementary tab of https://www.worldscientific.com/worldscibooks/10.1142/11774#t=suppl upon publication release of the book. Request Inspection Copy Contents: IFS Labs: Finding IFS for Fractal Images Spiral Fractals from IFS Finding IFS Rules from Images of Points A Fractal Leaf by IFS Fractal Wallpaper Cumulative Gasket Pictures IFS and Addresses Decimals as Addresses IFS with Memory IFS with More Memory Data Analysis by Driven IFS Dimension and Measurement Labs: Dimension by Box-Counting Paper Ball and Bean Bag Dimensions Calculating Similarity Dimension Sierpinski Tetrahedron Koch Tetrahedron Sierpinski Hypertetrahedron Basic multifractals: ƒ(α) Curves Iteration Labs: Visualizing Iteration Patterns Synchronized Chaos Domains of Compositions Fractals and Pascal's Triangles Fractals and Pascal's Triangle Relatives Mandelbrot Sets and Julia Sets Circle Inversion Fractals Fractal Tiles Labs in the Studio and in the Kitchen: Fractal Painting: Decalcomania 1 Fractal Painting: Decalcomania 2 Fractal Painting: Bleeds Fractal Painting: Mixing Fractal Painting: Dripping Fractal Paper Folds A Closer Look at Leaves Structures of Vegetables Cooking Fractals Labs in the Lab: Magnetic Pendulum Optical Gasket Video Feedback Fractals Electrodeposition Viscous Fingering Crumpled Paper Patterns Fractal Networks of Resistors Fractal Networks of Magnets Synchronization in Fractal Networks of Oscillators What Else?: Building Block Fractals Non-Euclidean Tilings Fractal Perimeters Multifractal Finance Fractal Music Other Ideas Why Labs Matter Readership: Advanced high school and beginning college students, including those in the humanities, with an interest in how fractal mathematics can reveal patterns in the natural world. Teachers and researchers interested in broadening their understanding of fractal modeling will also find this book of interest.