Tuesday, October 19, 2010

Benoît Mandelbrot - A Mathematician You Should Know

Yesterday, I heard of the passing of Benoît Mandelbrot on October 14th.  Mandelbrot is most famous for coining the word fractal, and one of the most famous fractals is named after him.  A picture is below.
The Mandelbrot Set
 To generate this image, you need to look at the one-parameter family of functions
with complex variable z and parameter c.  The process that one performs with this function is iteration of some starting value.  That means that the function is evaluated for the starting value, and then the function is repeatedly evaluated at the output from the last step of the iteration.  This iteration process can continue forever.  The Mandelbrot set is the set of all parameters c for which iteration of a starting value of 0 does not become infinite.  This is the black region in the picture above.

One of the properties for which fractals are famous is that you can zoom in on one part of the fractal and that part will look like the whole fractal.  This is the property of self-similarity.  You can see a close-up of part of the mandelbrot set below.
A close-up of part of the Mandelbrot Set

Benoît Mandelbrot was born in Warsaw, Poland in 1924.  His family moved to Paris in 1936.  He studied in France, and briefly at Cal Tech.  He moved to the United States in 1958.  He spent 35 years at IBM's Watson Research Center, and then taught at Yale Univerity, retiring in 2005.

Mandlebrot is also famous for his book The Fractal Geometry of Nature, which describes how nature produces objects with self-similarity.  Tree branches, blood vessels, and romanesco broccoli are a few examples.

I first encountered Mandelbrot while writing a paper on fractals in high school.  I read about him in James Gleik's book Chaos: Making of a New Science.  I learned more about his work on fractals in graduate school, where I was studying dynamical systems.  I certainly count him as one of my influences in mathematics.

Here is Mandelbrot giving a talk on the TED website.