In previous chapters we discussed geometrical properties of fractals characterized by fractal dimensions D f . However, the fractal dimension Df is not sufficient to describe all features of complex structures or distributions. For example, a distribution of mineral resources on earth is one such case. Gold is found in high densities only at a few rich places, whereas an extremely small amount of gold exists almost everywhere. Assume that areas with gold density larger than ρ1 are colored in red on a world atlas, and that areas with gold density larger than ρ2 (< ρ1) are colored in blue. Even if the distribution of red portions on the atlas is characterized by a fractal dimension D f , the fractal dimension of the blue region might differ from D f .It can be understood intuitively that the fractal dimension of the red region with very large ρ1 is close to zero, while the dimensionality of the blue region with very small ρ2 is almost two [4.1].
KeywordsFractal Dimension Voltage Drop Multifractal Spectrum Parabolic Approximation Growth Probability
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- 4.1C.J. Allègre, E. Lewin: Earth Planet. Sci. Lett. 132, 1 (1995)Google Scholar
- 4.10B.B. Mandelbrot: Fractals and Multifractals: Noise, Turbulence and Galaxies (Springer, New York 1988)Google Scholar
- 4.13J.F. Gouyet: Physics and Fractal Structures (Springer, New York 1996)Google Scholar
- 4.17J.P. Bouchaud, M. Potters: Theory of Financial Risk (Cambridge University Press, Cambridge 1999)Google Scholar
- 4.18B.B. Mandelbrot: Multifractals and 1/f Noise (Springer, New York 1998)Google Scholar
- 4.22G.A. Edgar: Integral, Probability, and Fractal Measures (Springer, New York 1997)Google Scholar