Information Technology Reference
In-Depth Information
Figure 2.14.
Geometric fractals constitute a family of shapes containing infinite levels of detail. In the fractals
reproduced here, the tip of each branch continues branching over many generations, on smaller
and smaller scales, and each magnified, smaller-scale structure is similar to the larger form
(self-similarity). As the fractal dimension increases between one and two, the tree sprouts new
branches more and more vigorously.
shown in Figure
2.14
, which bursts out of a one-dimensional space but does not fill a
two-dimensional Euclidean plane. Again the precise character of the bursting depends
on the value of
D
. If the fractal dimension is barely above one the tree is wispy
and broom-like; as the dimension increases from one to two, the canopy of branches
becomes more and more lush.
Fractal objects are ubiquitous in the physical world, being seen, for example, in the
self-similar accretions of cumulus clouds, in turbulent fluid flows, in the folds of the
surface of the human brain and in the patterning of magnetic spins when iron reaches a
critical temperature. The organic, tree-like fractal shown in Figure
2.14
bears a striking
resemblance to many physiologic structures.
Compared with a smooth, classical geometric form, a fractal curve (surface) appears
wrinkled. Furthermore, if the wrinkles of a fractal are examined through a microscope
more wrinkles become apparent. If these wrinkles are now examined at higher magnifi-
cation, still smaller wrinkles (wrinkles on wrinkles on wrinkles) appear, with seemingly
endless levels of structure emerging. The fractal dimension provides a measure of the
degree of irregularity. A fractal structure as a mathematical entity has no characteris-
tic scale size and so the emergence of irregularity proceeds downward to ever smaller
scales and upward to every larger scale. A natural fractal, on the other hand, always ends
at some smallest scale as well as at some largest scale and whether or not being fractal
is a useful concept depends on the size of the interval over which the process appears
