Benoit Mandelbrot, whose magical creations gripped the imagination, has died at the age of 85. The famed Mandelbrot set demonstrated how a simple, recursive formula can create endless complexity.
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.
The most famous images of fractals are computer generated. But it was a sculptor friend, Rick Rothrock, who first sparked my interest in how fractals can describe the natural world.
Mandelbrot had a restless imagination, and would move from subject to subject:
In his 2004 book, The (mis)Behavior of Markets, Mandelbrot applied his techniques to finance. He criticized the use of differential equations that assume continuous change in prices. (Think back on your calculus class and the invention of infinitesimals.) The famous Black-Scholes equation is a powerful tool, but its assumption that prices move continuously proved fatal when Long-Term Capital Management hired the brain trust behind the equation. LTCM famously built a fortune and lost it in a financial storm that the equations said should almost never happen.
Instead of rigorously proving his insights in each field, he said he preferred to “stimulate the field by making bold and crazy conjectures” — and then move on before his claims had been verified. This habit earned him some skepticism in mathematical circles.
“He doesn’t spend months or years proving what he has observed,” said Heinz-Otto Peitgen, a professor of mathematics and biomedical sciences at the University of Bremen. And for that, he said, Dr. Mandelbrot “has received quite a bit of criticism.”
“But if we talk about impact inside mathematics, and applications in the sciences,” Professor Peitgen said, “he is one of the most important figures of the last 50 years.”
His insights have helped lead scientists to tackle phenomena that don't neatly fit into classical mathematical models. We don't necessarily have models for how things behave beyond the boundaries, but we at least have a better sense of where those boundaries are.