Fractals



Fractals are self-similar objects: the closer you look the more you see. One result of being self-similar is that fractals can be described as having a fractional dimension.

A fraction of a dimension? Here is one way to understand this idea. If you try to measure a non-exotic object, like a watering hose randomly tossed onto the garden lawn, with a ruler, then the number you get for the length will depend on the length of the ruler.

A six foot ruler will give only a crude approximation to the length of the hose. A six inch ruler "flip-walked" along the hose does a better job.

As you use smaller and smaller rulers you converge on (get closer and closer to) the actual length of the hose.

With a fractal object, like a coastline, the smaller you make your ruler, the longer the coastline appears. This is because smaller and smaller rulers measure smaller and smaller jigs and jags in the coastline. Fractal objects have jigs and jags on all scales. They do not start to look smooth as you magnify them.



In other words, fractals are infinitely complicated: the closer you look the more detail you see. Most fractals are generated by relatively simple equations where the results are fed back into the equations again and again. The recursiveness of this procedure is why one sees structure at one scale in a fractal repeated, perhaps shrunk, rotated, and slightly distorted, on another smaller scale.

The mathematician Benoit Mandelbrot, long a student of unusual statistical processes, coined the name "fractal" in the mid-1970s for this class of self-similar complicated objects that emerge out of simple recursive rules.

The procedures that produce fractals can either be probabilistic, like Michael Barnsley's Iterated Function Systems, or determinstic processes that produce chaotic attractors.


For some great fractal demos try the Fractal Explorer Kit.

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