Wednesday, March 5, 2014

Fractals in a fairy tale

We were all watching the Oscars last Sunday and my son and I noticed that the song "Let it go" from the movie Frozen mentions fractals.  Fractal is a mathematical term and we were quite surprised but delighted to see it used in a children's song.  One of the most famous fractals is the Mandelbrot set, named after Benoit Mandelbrot, the father of fractals.  When I got my Commodore VIC-20 in the 1980's, my brother and I (and I am sure countless other kids playing with their personal computers) wrote a program to compute the Mandelbrot set, and zoom in and display various sections of it, (not knowing that 15 years later, I will work at the same place where Mandelbrot worked and have a chance to have lunch with him on several occasions).  At the time, it takes hours to compute a picture of the Mandelbrot set.

The definition of the Mandelbrot set is quite simple.  Points on the plane correspond to complex numbers and each point is used as a parameter $c$ in the equation $x_{n+1}= x_n^2 + c$.  The iterates $x_i$ starting from $x_0 = 0$ remain bounded if and only if $c$ belongs to the Mandelbrot set.  Since it may take many iterations before the iterates start to diverge, practically the computer program assumes the iterates are bounded if it hasn't diverged after a large number of iterations.

1 comment:

  1. great post!

    I'm doing blog about the Mandelbrot set accompanying an ebook designed for school-level readers - it'll take you from basic arithmetic, through the idea of iteration, give a very gentle introduction to complex numbers, and hold your hand through coding your own Mandelbrot and Julia sets.

    Comments welcome on both the blog and the ebook (published soon) - the aim is to maximise understanding and I believe anyone with school maths can do it.