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Fractals are the geometry of the natural world. They're about the broken, wrinkled, wiggly world- the uneven shapes of nature, unlike the idealised forms of Euclidean geometry. We see fractals everywhere; indeed, we are fractals ourselves. Fractal geometry is an extension of classical geometry which can make precise models of physical structures, from ferns to galaxies. It can describe the shape of a cloud as precisely as an architect can describe a house. Introducing Fractals traces the historical development of this mathematical discipline, explores its descriptive powers in the natural world, and then looks at the applications and the implications of the discoveries it has made. As John Archibald Wheeler, protégé of Niels Bohr, friend of Albert Einstein and mentor of Richard Feynman has said, 'No one will be considered scientifically literate tomorrow, who is not familiar with fractals.'
Author : K. J. Falconer
ISBN : 9780199675982
Genre : Mathematics
File Size : 29.38 MB
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An essential discussion of the popular science and mathematics behind fractals reveals how fractal shapes can be found everywhere in nature from clouds to coastlines, explaining how basic concepts in fractal geometry produced a revolution in mathematical understandings of patterns in the 20th century. Original.
Author : Andrzej Katunin
ISBN : 9781351801218
Genre : Computers
File Size : 85.38 MB
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This book presents concisely the full story on complex and hypercomplex fractals, starting from the very first steps in complex dynamics and resulting complex fractal sets, through the generalizations of Julia and Mandelbrot sets on a complex plane and the Holy Grail of the fractal geometry – a 3D Mandelbrot set, and ending with hypercomplex, multicomplex and multihypercomplex fractal sets which are still under consideration of scientists. I tried to write this book in a possibly simple way in order to make it understandable to most people whose math knowledge covers the fundamentals of complex numbers only. Moreover, the book is full of illustrations of generated fractals and stories concerned with great mathematicians, number spaces and related fractals. In the most cases only information required for proper understanding of a nature of a given vector space or a construction of a given fractal set is provided, nevertheless a more advanced reader may treat this book as a fundamental compendium on hypercomplex fractals with references to purely scientific issues like dynamics and stability of hypercomplex systems.
Author : David P. Feldman
ISBN : 9780199566433
Genre : Mathematics
File Size : 22.10 MB
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For students with a background in elementary algebra, this book provides a vivid introduction to the key phenomena and ideas of chaos and fractals, including the butterfly effect, strange attractors, fractal dimensions, Julia Sets and the Mandelbrot Set, power laws, and cellular automata. The book includes over 200 end-of-chapter exercises.
Fractals for the Classroom breaks new ground as it brings an exciting branch of mathematics into the classroom. The book is a collection of independent chapters on the major concepts related to the science and mathematics of fractals. Written at the mathematical level of an advanced secondary student, Fractals for the Classroom includes many fascinating insights for the classroom teacher and integrates illustrations from a wide variety of applications with an enjoyable text to help bring the concepts alive and make them understandable to the average reader. This book will have a tremendous impact upon teachers, students, and the mathematics education of the general public. With the forthcoming companion materials, including four books on strategic classroom activities and lessons with interactive computer software, this package will be unparalleled.
Author : John C. Russ
ISBN : 0306447029
Genre : Science
File Size : 25.81 MB
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The author integrates discussions of fractal geometry, surface modeling techniques, and applications to real world problems to provide a comprehensive, accessible overview of the field. His work will equip researchers with the basic tools for measurement and interpretation of data, stimulating more work on these problems and, perhaps, leading to an understanding of the reasons that Nature has adopted this geometry to shape much of our world.
Author : Giorgio Franceschetti
ISBN : 9780080469010
Genre : Technology & Engineering
File Size : 34.62 MB
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Scattering, Natural Surfaces, and Fractals provides a comprehensive overview of electromagnetic scattering from natural surfaces, ranging from the classical to the more recent (fractal) approach. As remote sensing applications become increasingly important, this text provides readers with a solid background in interpretation, classification and thematization of microwave images. The “scattering problem is discussed in detail with emphasis on its application to electromagnetic wave propagation, remote sensing, radar detection, and electromagnetic diagnostics. Natural surface and fractals complete this treatise focusing on how the fractal model represents our natural environment and other planets in our solar system, most recently as used to research the planet Venus and Titan, one of the moons of Saturn. An example of how scattering, fractals, and natural surfaces are of great importance is the following: Natural oil slicks in the ocean have been found to be fractal while man-made ones (generated by illegal washing of oil carrying ships) are not. Processing of an ocean image from space may detect the latter by means of a fractal analysis. An elegant and clear treatment of a rigorous topic with informative prose and realistic illustrations of scattering Provides readers with a solid background in interpretation, classification, and thematization of microwave images The only book available on fractal models and their application to scattering
Author : Gabriele A. Losa
ISBN : 3764371722
Genre : Mathematics
File Size : 39.25 MB
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This volume is number four in a series of proceedings volumes from the International Symposia on Fractals in Biology and Medicine in Ascona, Switzerland which have been inspired by the work of Benoît Mandelbrot seeking to extend the concepts towards the life sciences. It highlights the potential that fractal geometry offers for elucidating and explaining the complex make-up of cells, tissues and biological organisms either in normal or in pathological conditions.
Author : Paul S. Addison
ISBN : 0849384435
Genre : Science
File Size : 48.7 MB
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Fractals and Chaos: An Illustrated Course provides you with a practical, elementary introduction to fractal geometry and chaotic dynamics-subjects that have attracted immense interest throughout the scientific and engineering disciplines. The book may be used in part or as a whole to form an introductory course in either or both subject areas. A prominent feature of the book is the use of many illustrations to convey the concepts required for comprehension of the subject. In addition, plenty of problems are provided to test understanding. Advanced mathematics is avoided in order to provide a concise treatment and speed the reader through the subject areas. The book can be used as a text for undergraduate courses or for self-study.
The essential question that fractal dimensions attempt to answer is about the scales in Nature. For a system as non-idealistic and complex as a protein, studying scale-invariance becomes particularly important. Fractal Symmetry of Protein Exterior investigates the diverse facets of the various scales at which we describe protein biophysical and biochemical phenomena. Although these ideas are entirely mathematical, mathematical expositions have been avoided, unless the use of some expressions becomes absolutely obligatory. A first chapter introduce into fractal dimensions, protein exteriors and to methods to study the roughness of surfaces. The main topics covered in the following chapters include: protein-protein interaction interfaces; protein surface-roughness and local shape as well as adhesion on protein and other rough biomolecular surfaces.
Author : Lui Lam
ISBN : 038740614X
Genre : Science
File Size : 67.78 MB
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This textbook provides an introduction to the new science of nonlinear physics for advanced undergraduates, beginning graduate students, and researchers entering the field. The chapters, by pioneers and experts in the field, share a unified perspective. Nonlinear science developed out of the increasing ability to investigate and analyze systems for which effects are not simply linear functions of their causes; it is associated with such well-known code words as chaos, fractals, pattern formation, solitons, cellular automata, and complex systems. Nonlinear phenomena are important in many fields, including dynamical systems, fluid dynamics, materials science, statistical physics, and paritcel physics. The general principles developed in this text are applicable in a wide variety of fields in the natural and social sciences. The book will thus be of interest not only to physicists, but also to engineers, chemists, geologists, biologists, economists, and others interested in nonlinear phenomena. Examples and exercises complement the text, and extensive references provide a guide to research in the field.
L-System Fractals covers all the fundamental aspects of generating fractals through L-system. Also it provides insight to various researches in this area for generating fractals through L-system approach & estimating dimensions. Also it discusses various applications of L-system fractals. Fractals generated from L-System including hybrid fractals Dimension calculation for L-system fractals Images and codes for L-system fractals Research directions in the area of L-system fractals Usage of various freely downloadable tools in this area
Author : Michael F. Barnsley
ISBN : 9780486320342
Genre : Mathematics
File Size : 57.13 MB
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Up-to-date text focuses on how fractal geometry can be used to model real objects in the physical world, with an emphasis on fractal applications. Includes solutions, hints, and a bonus CD.
Author : Andre HECK
ISBN : 9783540543534
Genre : Science
File Size : 29.22 MB
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'Fractal geometry addressesitselfto questions that many people have been asking themselves. It con cerns an aspect of Nature that almost everybody had been conscious of, but could not address in a formal fashion. ' 'Fractal geometry seems to be the proper language to describe the complezity of many very compli cated shapes around us. ' (Mandelbrot, 1990a) 'I believe that fractals respond to a profound un easiness in man. ' (Mandelbrot, 1990b) The catchword fractal, ever since it was coined by Mandelbrot (1975) to refer to a class of abstract mathematical objects that were already known at the turn ofthe 19th century, has found an unprecedented resonance both inside and outside the scientific community. Fractal concepts, far more than the concepts of catastrophe theory introduced a few years earlier, are currently being applied not only in the physical sciences, but also in biology and medicine (Goldberger and West 1987). In the mid-eighties, Kadanoff (1986) asked the question: 'Why all the fuss about /ractals'! '. He offered a twofold answer: in the first place, it is 'because of the practical, technological importance of fractal objects'. Indeed he emphasised the relevance of these structures for materials scientists and oil drilling engineers, in search of structures with novel properties, or models for the flow of oil through the soil. His second answer was: 'Because of the intellectual interest of fractals '.
This well-illustrated volume reviews the many applications of fractal geometry used to study earth science patterns and processes. Designed as an introduction to this new area of study, the papers represent a broad spectrum of earth science disciplines where patterns have proven to be fractal over many orders of magnitude in time or space. This volume features an exhaustive list of published fractal dimensions measured for earth science patterns.
Author : Ronald Goldman
ISBN : 9781439803356
Genre : Computers
File Size : 28.14 MB
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Taking a novel, more appealing approach than current texts, An Integrated Introduction to Computer Graphics and Geometric Modeling focuses on graphics, modeling, and mathematical methods, including ray tracing, polygon shading, radiosity, fractals, freeform curves and surfaces, vector methods, and transformation techniques. The author begins with fractals, rather than the typical line-drawing algorithms found in many standard texts. He also brings the turtle back from obscurity to introduce several major concepts in computer graphics. Supplying the mathematical foundations, the book covers linear algebra topics, such as vector geometry and algebra, affine and projective spaces, affine maps, projective transformations, matrices, and quaternions. The main graphics areas explored include reflection and refraction, recursive ray tracing, radiosity, illumination models, polygon shading, and hidden surface procedures. The book also discusses geometric modeling, including planes, polygons, spheres, quadrics, algebraic and parametric curves and surfaces, constructive solid geometry, boundary files, octrees, interpolation, approximation, Bezier and B-spline methods, fractal algorithms, and subdivision techniques. Making the material accessible and relevant for years to come, the text avoids descriptions of current graphics hardware and special programming languages. Instead, it presents graphics algorithms based on well-established physical models of light and cogent mathematical methods.
Author : Michael F. Barnsley
ISBN : 0387955216
Genre : Computers
File Size : 43.20 MB
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This IMA Volume in Mathematics and its Applications FRACTALS IN MULTIMEDIA is a result of a very successful three-day minisymposium on the same title. The event was an integral part of the IMA annual program on Mathemat ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley (Department of Mathematics and Statistics, University of Melbourne), Di etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward R. Vrscay (Department of Applied Mathematics, University of Waterloo) for their excellent work as organizers of the meeting and for editing the proceedings. We take this opportunity to thank the National Science Foundation for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume grew out of a meeting on Fractals in Multimedia held at the IMA in January 2001. The meeting was an exciting and intense one, focused on fractal image compression, analysis, and synthesis, iterated function systems and fractals in education. The central concerns of the meeting were to establish within these areas where we are now and to develop a vision for the future.
