Monday, October 18, 2010

Fractals, Chaos Theory and the Butterfly Effect: Benoit Mandelbrot is dead. By Geniusofdespair

In 1982 Mandelbrot published "The Fractal Geometry of Nature" a classic on chaos theory. I read it many years ago. The term fractal was coined by Benoît Mandelbrot in 1975. Read about Chaos theory and Mandelbrot's set, here is an excerpt from this brief history:

Many scientists were exploring equations that created fractal equations. The most famous fractal image is also one of the most simple. It is known as the Mandelbrot set (pictures of the mandelbrot set). The equation is simple: z=z2+c. To see if a point is part of the Mandelbrot set, just take a complex number z. Square it, then add the original number. Square the result, then add the original number. Repeat that ad infinitum, and if the number keeps on going up to infinity, it is not part of the Mandelbrot set. If it stays down below a certain level, it is part of the Mandelbrot set. The Mandelbrot set is the innermost section of the picture, and each different shade of gray represents how far out that particular point is. One interesting feature of the Mandelbrot set is that the circular humps match up to the bifurcation graph. The Mandelbrot fractal has the same self-similarity seen in the other equations. In fact, zooming in deep enough on a Mandelbrot fractal will eventually reveal an exact replica of the Mandelbrot set, perfect in every detail (Figure left).

Benoit B. Mandelbrot has died at 85. This was a guy who fascinated me in the 80's with his brilliance. Here is more from the New York Time Obit link:

Dr. Mandelbrot traced his work on fractals to a question he first encountered as a young researcher: how long is the coast of Britain? The answer, he was surprised to discover, depends on how closely one looks. On a map an island may appear smooth, but zooming in will reveal jagged edges that add up to a longer coast. Zooming in further will reveal even more coastline.

“Here is a question, a staple of grade-school geometry that, if you think about it, is impossible,” Dr. Mandelbrot told The New York Times earlier this year in an interview. “The length of the coastline, in a sense, is infinite.”

In the 1950s, Dr. Mandelbrot proposed a simple but radical way to quantify the crookedness of such an object by assigning it a “fractal dimension,” an insight that has proved useful well beyond the field of cartography.

Over nearly seven decades, working with dozens of scientists, Dr. Mandelbrot contributed to the fields of geology, medicine, cosmology and engineering. He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains fold as they grow, among other phenomena.

His influence has also been felt within the field of geometry, where he was one of the first to use computer graphics to study mathematical objects like the Mandelbrot set, which was named in his honor.

1 comment:

Anonymous said...

As these great minds leave us, one hopes in addition to leaving documents, that they leave some young researchers who are influenced by them on the scene to carry his work to the next level. The universities would be the best place, but a lot of these great minds are not even attached to a university. Many universities are no longer into pure knowledge creation any more, but are instead focused on football and other fiancial generators. We need to capture some of the thinking of these creative people along with their greatness, and their quest for knowledge for a new generation of inventors. This is what has made America great.