The Fractal Geometry of Nature by Benoit B. Mandelbrot

Author:Benoit B. Mandelbrot [Mandelbrot, Benoit B.]
Language: eng
Format: epub
Publisher: Apress - A
Published: 2010-07-15T21:00:00+00:00

Plate 255 FRACTIONAL BROWN TRAILS (DIMENSIONS D~1.1111, D~1.4285)

The Figure on the left constitutes an example of a statistically self-similar fractal curve with D = 1 / 0.9000 ~ 1.1111. Its coordinate functions are independent fractional Brown functions of exponent H=0.9000, which accounts for the Joseph Effect for the Nile. The fact that H is close to 1 does not suffice to prevent self-intersections, but greatly discourages them by forcing the curve’s “trend” to persist in any direction upon which it has embarked. Thinking of complicated curves as the super-impositions of large, medium, and small convolutions, it may be said that in the case of high persistence and dimension close to 1, small convolutions are barely visible.

The Figure to the right uses the same computer program with D ~ 1/0.7000~1.4285. The pseudo-random seed is not changed, hence the overall shape is recognizable. But the increase in the value of D increases the relative importance of the small convolutions, and to a lesser extent, of the medium ones. Previously invisible details become very apparent.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.