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Fractals

October 17, 2010 by staff 

Fractals, Benoit Mandelbrot died a few Days on October 14, 2010. Since 1987, Mandelbrot was a member of the Yale Department of Mathematics. This chaptered of my book “Gina said: Adventures in the Blogosphere String war” on fractals is presented here on this sad occasion.

A small demonstration of the impact of Mandelbrot: when you search Google for an image of “Mandelbrot” does not get images of Mandelbrot himself but rather the creation of images of Mandelbrot. You get pages full of beautiful images of Mandelbrot sets

Physics modeling continuous and smooth mathematical objects have led to outstanding achievements of science in recent centuries. The interaction between geometry and smooth stochastic processes is also a very powerful and fruitful idea. Realization of the importance of Mandelbrot fractals and work on their study can be added to this short list of the main paradigms of mathematical modeling of real world phenomena.
Fractals are beautiful mathematical objects whose study dates back to the late 19th century. The Sierpinski triangle and the Koch snowflake are the first examples of fractals, which are built by simple rules and recursive.
Other examples are based on iterations study of simple functions, particularly functions defined on complex numbers.
Still other examples are from different stochastic (random) processes. For example, the outer limit of a Brownian motion in the plane, and the limit process of percolation (random Thurs Hex).
We have already mentioned the importance of the concept of “dimension” in mathematics. A point has dimension 0, a line has dimension 1, and the plane is 2 dimensional and 3 dimensional spaces is. Fractals often have “fractional dimension”. The Koch snowflake has a dimension of 1.2619, and the Sierpinski triangle has dimension 1.5850. The frontier of Brownian motion in the plane is a fractal; Mandelbrot conjectured that its size is 4 / 3, which was recently proven by Lawler, Schramm and Werner.

Beloit Mandelbrot coined the term fractal in his book, which has also informally proposed the following definition of a fractal, “a rough or fragmented geometric shape that can be divided into two parts, each of which is (at least approximately) a reduced size copy of the whole. “One important property of fractals is called” self-similarity, whereby a small part of the big picture is very similar to the whole picture. Mandelbrot also understood and promoted the importance of fractals in various fields of physics. Indeed, fractals are now playing an important role in many areas of modern physics (and there is also some controversy about their role). Mandelbrot has also written an important document concerning the applications of fractals in finance. The notion of self-similarity is also important in other areas. In computing, the self-similarity of a problem is referred to as “self-reducibility, and this property facilitates the design of efficient algorithms to solve the problem.

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