How long is the coastline of Britain?
It sounds like a simple question. We have maps. We have rulers. So just take out a map and measure! But which map do you use? The problem is, the more detailed a map you use, the greater your answer will be, because a coastline is not smooth. It is jagged. There are river mouths and bays and promontories to account for. The answer changes as the resolution of the map increases. Do you chart a path around each rock? Each stone? What about every grain of sand?
This is not the simple math problem of computing the circumference of a circle given the diameter. As a math problem, this is turned out to be an entirely new kind of beast that its discoverer Benoit Mandelbrot called “fractal geometry.” Mandelbrot assigned a “fractal dimension” to measure jaggedness in general.
Since most things scientists want to study in nature, such as clouds, proteins, galaxies, earthquake faults, are neither tiny particles nor smooth Platonic objects, of course it was important to have some kind of mathematical language for describing them.
Thanks to Mandelbrot, now they have one.
Fractals are also great for generating cool pictures like this one:
Mandelbrot passed away Thursday in Cambridge, Massachusetts. He was an instigator rather than a rigorous investigator, a rogue mathematician who left the proofs for those who followed in his maverick footsteps. He will be missed, but hopefully he’s inspired other rogues and mavericks out there to come and do their bit to shake things up.
One last thing — in the name of alumni pride, I’d like to point out that in addition to all the other famous places in America and France where Mandelbrot worked and studied, he found time to earn a master’s degree in aeronautics at my own alma mater Caltech.