Fractals in nature: From characterization to simulation

Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex forms found in nature. Shapes such as coastlines, mountains and clouds are not easily described by traditional Euclidean geometry. Nevertheless, they often possess a remarkable simplifying invariance under changes of magnification. This statistical self-similarity is the essential quality of fractals in nature. It may be quantified by a fractal dimension, a number that agrees with our intuitive notion of dimension but need not be an integer. In Section 1.1 computer generated images are used to build visual intuition for fractal (as opposed to Euclidean) shapes by emphasizing the importance of self-similarity and introducing the concept of fractal dimension. These fractal forgeries also suggest the strong connection of fractals to natural shapes. Section 1.2 provides a brief summary of the usage of fractals in the natural sciences. Section 1.3 presents a more formal mathematical characterization with fractional Brownian motion as a prototype. The distinction between self-similarity and self-affinity will be reviewed. Finally, Section 1.4 will discuss independent cuts, Fourier filtering, midpoint displacement, successive random additions, and the Weierstrass-Mandelbrot random function as specific generating algorithms for random fractals. Many of the mathematical details and a discussion of the various methods and difficulties of estimating fractal dimensions are left to the concluding Section 1.6.