The history of describing natural objects using geometry is as old as the advent of science itself, in which traditional shapes are the basis of our intuitive understanding of geometry. However, nature is not restricted to such Euclidean objects which are only characterized typically by integer dimensions. Hence, the conventional geometric approach cannot meet the requirements of solving or analysing nonlinear problems which are related with natural phenomena, therefore, the fractal theory has been born, which aims to understand complexity and provide an innovative way to recognize irregularity and complex systems. Although the concepts of fractal geometry have found wide applications in many forefront areas of science, engineering and societal issues, they also have interesting implications of a more practical nature for the older classical areas of science. Since its discovery, there has been a surge of research activities in using this powerful concept in almost every branch of scientific disciplines to gain deep insights into many unresolved problems. This book includes eight chapters which focus on gathering cutting-edge research and proposing application of fractals features in both traditional scientific disciplines and in applied fields.

Historically, science has developed by reducing complex situations to simple ones, analyzing the components and synthesizing the original situation. While this 'reductionist' approach has been extremely successful, there are phenomena of such complexity that one cannot simplify them without eliminating the problem itself. Recently, attention has turned to such problems in a wide variety of fields. This is in part due to the development of fractal geometry. Fractal geometry provides the mathematical tools for handling complexity. The present volume is a collection of papers that deal with the application of fractals in both traditional scientific disciplines and in applied fields. This volume shows the advance of our understanding of complex phenomena across a spectrum of disciplines. While these diverse fields work on very different problems, fractals provide a unifying formalism for approaching these problems.

"The best accomplishment in human research is feasibly they understand the natural phenomena can be exhibited by mathematical models. The history of describing natural objects using geometry is as old as the advent of science itself. Traditionally lines, squares, rectangles, circles, spheres, etc. have been the basis of our intuitive understanding of the geometry. However, nature is not restricted to such Euclidean objects which are only characterized typically by integer dimensions. Hence, the conventional geometric approach cannot meet the requirements of solving or analysing nonlinear problems which are related with natural phenomena, therefore, the fractal theory has been born, which aims to understand complexity and provide an innovative way to recognize irregularity and complex systems"--

For almost ten years chaos and fractals have been enveloping many areas of mathematics and the natural sciences in their power, creativity and expanse. Reaching far beyond the traditional bounds of mathematics and science to the realms of popular culture, they have captured the attention and enthusiasm of a worldwide audience. The fourteen chapters of the book cover the central ideas and concepts, as well as many related topics including, the Mandelbrot Set, Julia Sets, Cellular Automata, L-Systems, Percolation and Strange Attractors, and each closes with the computer code for a central experiment. In the two appendices, Yuval Fisher discusses the details and ideas of fractal image compression, while Carl J.G. Evertsz and Benoit Mandelbrot introduce the foundations and implications of multifractals.

From the reviews: "In the world of mathematics, the 1980's might well be described as the "decade of the fractal". Starting with Benoit Mandelbrot's remarkable text The Fractal Geometry of Nature, there has been a deluge of books, articles and television programmes about the beautiful mathematical objects, drawn by computers using recursive or iterative algorithms, which Mandelbrot christened fractals. Gerald Edgar's book is a significant addition to this deluge. Based on a course given to talented high- school students at Ohio University in 1988, it is, in fact, an advanced undergraduate textbook about the mathematics of fractal geometry, treating such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. However, the book also contains many good illustrations of fractals (including 16 color plates), together with Logo programs which were used to generate them. ... Here then, at last, is an answer to the question on the lips of so many: 'What exactly is a fractal?' I do not expect many of this book's readers to achieve a mature understanding of this answer to the question, but anyone interested in finding out about the mathematics of fractal geometry could not choose a better place to start looking." #Mathematics Teaching#1

Many statistical and methodological developments regarding fractal analyses have appeared in the scientific literature since the publication of the seminal texts introducing Fractal Physiology. However, the lion’s share of more recent work is distributed across many outlets and disciplines, including aquatic sciences, biology, computer science, ecology, economics, geology, mathematics, medicine, neuroscience, physics, physiology, psychology, and others. The purpose of this special topic is to solicit submissions regarding fractal and nonlinear statistical techniques from experts that span a wide range of disciplines. The articles will aggregate extensive cross-discipline expertise into comprehensive and broadly applicable resources that will support the application of fractal methods to physiology and related disciplines. The articles will be organized with respect to a continuum defined by the characteristics of the empirical measurements a given analysis is intended to confront. At one end of the continuum are stochastic techniques directed at assessing scale invariant but stochastic data. The next step in the continuum concerns self-affine random fractals and methods directed at systems that entail scale-invariant or 1/f patterns or related patterns of temporal and spatial fluctuation. Analyses directed at (noisy) deterministic signals correspond to the final stage of the continuum that relates the statistical treatments of nonlinear stochastic and deterministic signals. Each section will contain introductory articles, advanced articles, and application articles so readers with any level of expertise with fractal methods will find the special topic accessible and useful. Example stochastic methods include probability density estimation for the inverse power-law, the lognormal, and related distributions. Articles describing statistical issues and tools for discriminating different classes of distributions will be included. An example issue is distinguishing power-law distributions from exponential distributions. Modeling issues and problems regarding statistical mimicking will be addressed as well. The random fractal section will present introductions to several one-dimensional monofractal time-series analysis. Introductory articles will be accompanied by advanced articles that will supply comprehensive treatments of all the key fractal time series methods such as dispersion analysis, detrended fluctuation analysis, power spectral density analysis, and wavelet techniques. Box counting and related techniques will be introduced and described for spatial analyses of two and three dimensional domains as well. Tutorial articles on the execution and interpretation of multifractal analyses will be solicited. There are several standard wavelet based and detrended fluctuation based methods for estimating a multifractal spectrum. We hope to include articles that contrast the different methods and compare their statistical performance as well. The deterministic methods section will include articles that present methods of phase space reconstruction, recurrence analysis, and cross-recurrence analysis. Recurrence methods are widely applicable, but motivated by signals that contain deterministic patterns. Nonetheless recent developments such as the analysis of recurrence interval scaling relations suggest applicability to fractal systems. Several related statistical procedures will be included in this section. Examples include average mutual information statistics and false nearest neighbor analyses.