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.
KeywordsQuartz Beach Autocorrelation Sine Bark
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- 1.Mandelbrot, B.B. The Fractal Geometry of Nature, (Freeman, New York) 1982 and references therein. See also, Fractals: Form, Chance, and Dimension, W. H. Freeman and Co., San Francisco (1977).Google Scholar
- 2.Voss, R. F. “1/f (flicker) noise: a brief review”, Proc. 32rd Annual Symposium on Frequency Control,Atlantic City, (1979), 40–46 and references therein.Google Scholar
- 7.Gardner, M. “White and brown music, fractal curves, and one-over-f noise”, Mathematical Games column in Scientific American, April 1978 p16.Google Scholar
- 9.For example see: Freeman, J. J. Principles of Noise, John Wiley & Sons, Inc., New York, (1958), Chapter 1, “Fourier Series and Integrals.” or Robinson, F.N.H. Noise and Fluctuations, Clarendon Press, Oxford, (1974).Google Scholar
- 10.A good discussion is found in Reif, F. Statistical and Thermal Physics,McGraw-Hill Book Co., New York, (1965), Chapter 15, “Irreversible Processes and Fluctuations.”Google Scholar
- 13.Carpenter, L. “Computer rendering of fractal curves and surfaces”, SIGGRAPH ‘80 Conference Proceedings, (1980) 109.Google Scholar
- 15.Mandelbrot, B.B. “Comment on Computer rendering of Fractal Stochastic Models”, Comm. of the ACM, 25,(1982) 581–584, and the response by Fournier, Fussell, and Carpenter.Google Scholar