## Benoît’s Fractals

Mandelbrot set

Benoît Mandelbrot passed away a few dayes ago on October 14, 2010. Since 1987, Mandelbrot was a member of the Yale’s mathematics department. This chapterette from my book “Gina says: Adventures in the Blogosphere String War”   about fractals is brought here on this sad occasion.

A little demonstration of Mandelbrot’s impact: when you search in Google for an image for “Mandelbrot” do not get pictures of Mandelbrot himself but rather pictures of Mandelbrot’s creation. You get full pages of beautiful pictures of Mandelbrot sets

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Benoit Mandelbrot (1924-2010)

Modeling physics by continuous smooth mathematical objects have led to the most remarkable achievements of science in the last centuries. The interplay between smooth geometry and stochastic processes is also a very powerfull and fruitful idea. Mandelbrot’s realization of the prominence of fractals and his works on their study can be added to this short list of major paradigms in mathematical modeling of real world phenomena.

## Fractals

Fractals are beautiful mathematical objects whose study goes back to the late 19th century. The Sierpiński triangle and the Koch snowflake are early examples of fractals which are constructed by simple recursive rules.

Koch snowflake  (Note: The Koch snowflake is just the “boundary” of the blue shape in the picture.)

Sierpiński triangle

Other examples are based on the study iterations of simple functions, especially functions defined over the complex numbers.

Mandelbrot set (top of the post) and Julia set (above).

Still other examples come from various stochastic (random) processes. For instance, the outer boundary of a Brownian motion in the plane, and the boundary of the percolation process (random Hex game) .

Brownian motion in the plane. The boundary is the “border” between the white areas and the colored areas. (If you get the impression that “boundary” is an important notion in many areas you are correct!)

We already mentioned the importance of the notion of “dimension” in mathematics. A point has dimension 0, a line has dimension 1, the plane has dimension 2 and the space is 3 dimensional.  Fractals often have “fractional dimension”. The Koch snowflake has dimension 1.2619 , and the Sierpiński triangle has dimension  1.5850. The “boundary” of the Brownian motion in the plane is a fractal; Mandelbrot conjectured that its dimension is 4/3 and this was recently proved by Lawler, Schramm, and Werner.

The term fractal was coined by Benoît Mandelbrot, who in his book also proposed the following informal definition of a fractal: “a rough or fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of the whole.” An important property of fractals is referred to as “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 areas of physics. Indeed, today fractals play an important role in many areas of modern physics (and there is also some controversy regarding their role).  Mandelbrot also wrote an important  paper concerning applications of fractals in finance. The notion of self similarity is also important in other areas. In computer science, the self similarity of a problem is referred to as “self-reducibility”, and this property facilitates the design of efficient algorithms for solving the problem.

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### 6 Responses to Benoît’s Fractals

1. Simon says:

I’ve always wanted to get an overview of what people study about fractals and Mandelbrot/Julia sets. It can’t be the study of pretty pictures.

I have this impression (based on nothing concrete, and I would be happy to be proved wrong) that fractals and self-similarity are like perfect numbers: the problems are pretty, but there’s not much general theory and not much of a real connection to other areas of mathematics. Perhaps it provides some new questions to ask.

Maybe someone could give an application of these concepts to some problem that does not contain the word “fractal” in it ?

• Gil says:

Hi Simon, It is hard to think of an impression which is so further away from the truth. (But it is nevertheless sort of interesting how you have reached this impression.) The area that studies Fractals close to Mandelbrot and Julia sets is called complex dynamics. It is a deep and important area of mathematics with a developed general theory and many exciting problems, with a lot of connections to other areas.

• gowers says:

Bodil Branner’s article “Dynamics” in the Princeton Companion to Mathematics explains very well why the Mandelbrot set is much more than just a pretty picture.

2. Pingback: Fractals

3. What I really love about fractals is the wide range of applications that fractals have. Perhaps the most interesting use of fractals I’ve heard about is that of antennas to catch multiple frequency ranges – an engineer attended a talk about fractals, and had an idea – what if I make an antenna like a fractal. Turns out that it was the most efficient design – and was proved theoretically later.