文档介绍：5 Fractal and Brownian Motion
Fractal
Mandelbrot (1982) first introduced an idea, called the ªfractalº, to statistically de-
scribe geometrically random shapes, including coast lines, rivers, mountains,
clouds, lightning tracks, etc., which are widely observed in nature. According to
Mandelbrot, there are many examples including everywhere nondifferentiable fig-
ures in nature and he called them fractals. The classification of those figures may
be made in terms of invariance and dimensions. More specifically, a conditional
requirement under which the invariance holds in fractal is self-similarity. Fractal
dimensions can be fractional numbers, which are characteristic of figures under
self-similarity (Stauffer and Stanley, 1990; Schroeder, 1991). The idea has been
widely extended (see, for instance, Pietronero and Tosatti, 1986).
First, we consider a non-random fractal as an example. In a right triangle, let
us draw the altitude above the hypotenuse (see Fig. ). Then, we have three
similar right triangles, DABC, DDBA, and DDAC. In Euclidean geometry, be-
cause they are similar, the areas of these triangles are proportional to the squares
of the corresponding sides with the same proportionality constant, k.
AreaDABCka2 ; AreaDDACkb2 ; AreaDBAkc2 :
5:1
These relations are invariant with the same value of k. We can repeat the similar
procedure