Saturday, June 18, 2011

Fractals : geometric objects

The term "fractal" was coined by Benoit Mandelbrot in 1975. Derived from the Latin fractus, participle of the verb to break open, which means "break, break open." In fact, fractal images are considered by the mathematical objects of fractional dimension.

The mathematicians had begun to describe fractals for over a century, but their ideas were largely ignored until Mandelbrot has framed the argument in a consistent discipline and rich in fruits:
"Fractal geometry plays two roles. And 'the geometry of deterministic chaos and can also describe the geometry of the mountains, clouds and galaxies."

 Fractals are geometric objects, exactly like the circle or triangle, they have some different properties. A substantial difference between a Euclidean and a fractal geometric object is the way it builds. The Euclidean plane with a curve, the Fractal, however, is not based on an equation, but on an algorithm. This means that there is a method, not necessarily numeric, which must be used to draw the curve.

In addition, the algorithm is never used once: the procedure is iterated a number of times infinity: each iteration, the curve gets closer and closer to the end result and after a certain number of iterations the human eye is no longer able to distinguish the changes, so when you actually draw a fractal, you can stop after a reasonable number of iterations.

A curve is called fractal if it has the property of 'autosimilitudine: enlarging any part of the curve displays a particular set of equally rich and complex than the previous, this process of "zoom" can be continued indefinitely. In general we consider a fractal set that enjoys all or many of the following properties:

    Self-similarity, ie the union of copies of itself at different scales. As mentioned above, there is the same figure (and the same mathematical formula) proceeding towards different levels of size, smaller or larger;
    Fine structure, for which details are revealed at every magnification;
    Irregularities, which points to the local conditions do not meet simple geometric or analytic;

Fractals are often found in the study of dynamical systems and chaos theory and are often described recursively by simple equations, written with the help of complex numbers plays a significant role in the case and still the only tool capable of providing a solution to the problem is statistical. The case can generate irregularities and is capable of generating an irregularity so intense as that of the coasts, even in many situations it is difficult to prevent the case to go beyond our expectations. There are several families of fractals, divided by the degree of the equation generating content in the algorithm

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