What is a fractal?
B. B. Mandelbrot calls fractal
(from latin frangere i.e. to break) a very complex structure that
looks like itself at any level of detail.
Often such structures show an underlying
geometrical regularity, called invariance w.r.t. scale change or
self-similarity, that seems to describe natural shapes and configurations
in a better and more synthesised way than traditional Euclidean geometry.
One way for building fractal images
is to apply the concept of iteration, by computing a sequence of
points in the complex plane, defined as follows by a function f:
z
n+1 = f
(z n)
Many natural phenomena can be described
by mathematical models of this kind. The evolution of a population, the
behavior of the atmosphere and the growth of a capital are just a few examples.
How can one draw a fractal?
In order to draw a fractal one must
decide how to assign colors to the points in the plane. There are different
ways of doing it, one is the following. Given a point Z in the plane,
a function f, and a value for the constants in f , one starts
an iterative process in which, first, f is applied to Z and,
subsequently, to the result of the previous iteration. The color of the
point Z will be decided depending on the number of iterations
produced before f 's value grows higher than a predefined threshold.
Observe that different images of
a same fractal can be obtained simply by changing the function f and
the color palette.
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