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© Copyright 2000, Jim Loy
On the left, we see a Julia set. I have drawn red
brackets to show where part of the set is endlessly repeated, smaller and
smaller and smaller. This self-similarity is usually the major identifying
feature of a fractal. Mandelbrot originally defined fractals using the "fractal
dimension," which I will write about some day. A much simpler definition
involves a kind of loose self-similarity.
An example of this self-similarity is a cloud. Without other objects to compare with, it is very hard to tell the size of a cloud by looking at a photo. Is it a small picture of a big cloud or a big picture of a small cloud, or what? Different parts of the cloud, of vastly different sizes, are more or less similar to each other.
The self-similarity of this fractal fern is obvious.
Some of the parts are mirror images of other parts, as well as being bigger or
smaller.
The self-similarities of the above fractals are exact. Each
smaller portion is an exact copy of the larger portions. In many fractals, the
self-similarity is not exact, just as it is not exact in clouds. The Mandelbrot
set (left), for example, contains many copies of itself. None of these copies
is an exact duplicate, however. There are actually infinitely many copies of
the entire set in the image below (the northernmost part of the Mandelbrot
Set). If only I could give you infinite resolution. Anyway, most of the copies
appear to be tiny black dots, here:
Also see the animation at the end of The Koch Curve.