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.
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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
.
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