Random Fractal Forgeries
Mandelbrot’s fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature. Such shapes often possess a remarkable invariance under changes of magnification. This statistical self-similarity may be characterized by a fractal dimension, a number that agrees with our intuitive notion of dimension but need not be an integer. In section I, a series of computer generated and rendered random fractal shapes provide a visual introduction to the concepts of fractal geometry. These complex images, with details on all scales, are the result of the simplest rules of fractal geometry. Their success as forgeries of the natural world has played an important role in the rapid establishment of fractal geometry as a new scientific discipline and exciting graphic technique. Section II presents a brief mathematical characterization of these forgeries, as variations on Mandelbrot’s fractional Brownian motion. The important concepts of fractal dimension and exact and statistical self-similarity and self-affinity will be reviewed. Finally, section III will discuss independent cuts, Fourier filtering, midpoint displacement, successive random additions, and the Weierstrass-Mandelbrot random function as specific generating algorithms.
KeywordsBrownian Motion Fractal Dimension Spectral Density Fractal Geometry Fractional Brownian Motion
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