Fractals

Fractals are not just a matter of geometry but have a number of applications for the well-being of life. Fractal properties are useful in medical science. Its applicability in medical science paves the way to identify fatal diseases, for instance, the fractal properties of the blood vessels in the retina may be useful in diagnosing the diseases of the eye or in determining the severity of the disease. Herein we begin with a detailed study of fractals including its difference with the Euclidean geometry. Next, some constructions of mathematical fractals will be given, and finally, the various methods of calculating dimensions of fractal structures are discussed in the subsequent sections. Finally, one of the goals of this book is to explore the connection of chaotic orbit and fractal cascading and the determination of well-organized structures or strange attractors in chaotic regimes. There is a loss of information or loss of predictability in chaotic motion, and so several quantifying measures are discussed. For example, the quantitative measure of the strangeness of a strange attractor is its fractal dimension. On the other hand, the boundary between chaotic/turbulence and regular motions may have fractal properties, and the determination of fractal basin is a new research area of nonlinear dynamics. There are processes or objects which are not quantified by a single fractal, objects are nonhomogeneous in nature. Multifractals are discussed for measuring nonhomogenous objects or chaotic attractors or energy-eddies cascade in fully developed turbulence.