University of Torino - Department of Computer Science
 
   
What is a fractal?

      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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