Sunday, October 17, 2010

RIP Benoit Mandelbrot

A portion of the Mandelbrot Set
The world lost a major scientific figure this week. Benoit Mandelbrot, who recognized that the geometry of nature is fractal, died on Thursday at the age of 85. The New York Times published a nice obituary, describing his life and work and recounting his story of how he got started on the research that led to the fractal concept.

This has always been one of my favorite stories about Mandelbrot, and I often told it to the students in my classes on nonlinear dynamics and complexity science, since it also provides a great illustration of what a fractal is.

The story begins when Mandelbrot tried to measure the length of the coastline of Britain and soon realized that the answer varied depending on the resolution of the map or photograph he was measuring. At a high altitude, say from the vantage of a satellite, the coastline appeared to be a certain length, but as the camera zoomed in, more bays and estuaries were revealed and the coastline length increased.

Mandelbrot knew that this could not continue indefinitely, because, if it did, the length of the coastline of Britain would be infinite and this was, obviously, impossible. He eventually hit upon a way out of this puzzling paradox, and the solution was a very clever idea: the coastline is best thought of, he realized, not as a one-dimensional curve with a length but as an object with a dimension somewhere between one and two.


Romanesco broccoli
A two-dimensional object would be a filled-in area, such as the land mass of Britain itself, so what Mandelbrot was suggesting is that the coastline is neither one-dimensional, like a curve or straight line, nor two-dimensional, like the entire island, but a "fuzzy" object with a fractional dimension. 

He called this type of object a fractal and soon found that nature is filled with fractals. Coastlines are definitely fractal, but so are trees, mountain ranges, the network of blood vessels in our bodies and even vegetables. One of my favorite examples is the Romanesco broccoli, which I wrote about in one of my earliest posts on this blog.

The BBC has published a beautiful photo essay of Mandelbrot and the fractals he discovered, including the beautiful one which bears his name: the Mandelbrot set, a tiny portion of which is shown at the beginning of this post.

Rest in Peace, Benoit. You will be deeply missed.

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