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Fractals to create nature with the computer

Posted on: Jul, 23rd 2010
Fractal of the Mandelbrot set
The fractal geometry, makes reference to geometrical objects fractured, is a mathematical tool very powerful to represent in a computer and, with fidelity, structures that we can find, either in the nature or some things created by the man. Furthermore allows realize simulations more accurate of all kind of natural phenomenons. Also it's applied, in certain situations, to resolve all kind of problems. Would you like know how are made?

The fractals are generated from recursive algorithms. But, What is the recursion?. It's a difficult concept to understand because to comprehend the recursion first you have to understand the recursion. Jokes apart, the recursion is the reference of something to itself. A very simple example to appreciate the recursion is put a mirror in front of another. In an algorithm consists in make that, in some point of the algorithm, will call to itself. This allows create behaviour and complex structures with very simple algorithms. A good example of this is the Koch fractal, that consist in create an geometric structure with the same structure repeated inside itself.

Koch's fractal

This is a way to proceed that was to expect in the nature. This explains how a few genes can generate the richness of the circulatory system of the body and a lot of more complex structures of the living beings, this is because this genes are encoding a simple recursive algorithm.

The name of fractal was coined by Benoît Mandelbrot. He developed and popularize this discipline thanks to the first works in this field of Gaston Julia. Mandelbrot could generate his first fractals why, in that moment, was one of the few persons in the world that could use a computer, due to he was working at IBM.

The introduction to his book "The Fractal Geometry of Nature" of 1982 says:

"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

We can find fractal everywhere, the coasts, the affluents of the rivers, the sea horses, the leaves, branches and roots of the trees, the snow flakes, the broccoli, the cauliflower, the romanesco, etc.

Below you can enjoy a recent video of Benoît Mandelbrot:

Watch on TED

In the Video, Mandelbrot talks about the Hausdorff's dimension or fractal dimension. At the ordinary geometry we have objects of one, two or three dimension while at the fractal geometry, the dimensions have decimals. For instance, say that a fractal object has a dimension of 1.33, indicates that have one dimension (it's a line) and the decimals, as Mandelbrot explains in the video, represents the roughness of this. As much higher is the decimal number more rough will be the line.

The way to draw the Mandelbrot set it's relatively simple. It use this recursive formula:

Actual result = (Prior result)^2 + constant (where the initial prior result it's cero)

In this page, I have made a little program for you can see the Mandelbrot set. It works of the following way: Take as value of the constant a complex number (it's composed by an imaginary part and a real part). If this chosen value for the constant made that the formula tends to infinity, doesn't belong to the set, then we paint this pixel with black. In this way, we will go drawing the result, going through the coordinates "x" and "y" of the screen assigning values to the complex constant for every position.

At the practice, as you can't calculate infinite values to check if this kind of serie have a limit, it's consider that the formula tends to infinity if arrive to surpass a certain value during a determinate number of iterations.

As a reward for reach the end of this post, here you have a 3D travel by the Mandelbrot set:

Watch on Youtube

The fractals also appears when we represent graphically the variables of chaotic dynamical systems, although at this situations it's more correct call it strange attractor. Represent this attractors in a computer allows study how will behave the system in a near future, but about this I will talk in other post.

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