| Here are four examples of fractals found in nature. |
| Once you know the
visual signature, finding other examples is very easy. |
| Click each picture to magnify in a new window. |
|
| Fractals found in nature differ from our first mathematical examples in
two important ways: |
the self-similarity of natural fractals is approximate or statistical and |
this self-similarity extends over only a limited range of scales. |
| To understand the first point, note that many forces scuplt and grow natural fractals, while
mathematical fractals are built by a single process. |
| For the second point, the forces responsible for a natural fractal structure are effective
over only a limited range of distances. |
The waves carving a fractal coastline are altogether
different from the forces holding together the atoms of the coastline. |
Or, as a student commented in
the very first offering of this course, "I thought I would learn the atoms making up a sheep
looked like little sheep. I was surprised to learn Nature is much more complicated than this." |
| An excellent way to understand this limited range of scales for natural fractals can be
seen in the photographs http://www.public.asu.edu/~starlite
of Gayla Chandler. |
| Here, against a background of natural fractals, we see models of the
Sierpinski tetrahedron,
rendered over only four or five levels, yet obviously things we'd call fractal. |
| This comparison makes very clear that natural objects possessing
similar structures on only a few levels still can be called fractal, though we must be careful
to find enough levels. Too few
and the object can't be called fractal. |
