Complex pattern formation

TL;DR: Fractal geometry as mentioned in this paper is a geometrical structure that at first seems to be complicated, irregular, and random, but when viewed carefully, one begins to realize the presence of tractable properties that are inherent in it and help us to systematically study them.

Abstract: This chapter discusses the pattern formation governed by the process of mass flow and diffusion related to precipitation as crystals, electro-deposits, bacterial colonies, and diffusion. In this case it is possible to rationalize the complex structure with use of new mathematical concepts of fractal geometry. A fractal is a geometrical structure that at first seems to be complicated, irregular, and random. A fractal pattern is one that repeats itself at smaller and smaller scales. When viewed carefully, one begins to realize the presence of tractable properties that are inherent in it and help us to systematically study them. The principal objectives of fractal growth studies are characterization and quantification of hidden order in complex pattern and analysis of correlation in the development of order in a seemingly disordered state. A fractal is a subject associated with the discipline of non-linear dynamics. Fractals enjoy widespread attention not only in science but also in popular culture. The reason why the fractal geometries are of physical interest is that in nature a number of random processes build such geometries naturally. A fractal is composed of similar structure of finer details. Its length is not well defined. A fractal consists of geometric fragments of varying size and orientation but similar shape. This property of self-similarity or scale invariance means that if we take part of a fractal object and magnify it by the same magnification factor in all directions, the magnified picture is indistinguishable from the original. The dissemination of the concept of fractal geometry and related ideas have provided us the ways of describing a very broad range of irregular structures generated by living and non-living systems.