|A portion of the Mandelbrot Set|
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.
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.