Wednesday, January 28, 2009

The Mandelbrot Set

Before reading this post, be sure you understand the previous post i: The Most Mysterious of All Numbers

One of the most famous examples of chaos in mathematics is the Mandelbrot Set. The Mandelbrot Set, discovered by Benoit Mandelbrot, has infinite complexity. But complexity is not necessarily rooted in complexity. The Mandelbrot Set is produced by by a method called iteration. Iteration is applying the same thing (or equation) over and over. The Mandelbrot Set is infinite iterations of a simple equation:

z → z^2+c

Where c is a complex number (as described in the post i: the most mysterious of all numbers). If c^2+c is continually applied than the first few terms in the sequence are:

0, c, c^2+c, (c^2+c)^2+c^2+c, (c^2+c)^2+c^2+c)^2+(c^2+c)^2+c^2+c

Every point on the complex plane is put into this equation, and if the result at each step escapes to infinity (grows larger and larger), than the point is left blank. If the steps of the sequence stay within a defined region and do not escape to infinity, the point is colored black. For example, the sequence for 1+i:

Step 1 0
Step 2 1+i
Step 3 (1+i)^2+(1+i)=1+2i-1+1+i=1+3i

As you can see, the sequence extends to infinity and is therefore colored white. Although the black regions and white regions are solid and uninteresting, the edges are filled with complex patterns.

The "home" of the Mandelbrot Set.

For more images of the Mandelbrot Set, go to this site.


Anonymous said...

Very interesting post, really "opened" my eyes about the complexity of the Mandelbrot set.
Cheers from Sweden.

Louis said...

Thanks for your comment. I do find that the complexities of fractals among the most beautiful forms of mathematics