Goodbye to the Father of Fractals

Oliver Wendell Holmes famously once wrote, "A mind that is stretched by a new experience can never go back to its old dimensions." My mind, and assuredly those of countless others, never did after pondering some of the key concepts of the mathematician Benoit Mandelbrot, who died last week at the age of 85.

Mandelbrot invented the term fractal to describe the "roughness" he saw all around him in nature -- the jagged shape of a cloud, the rugged indentations of a coastline. Classical Euclidean mathematics, the kind we learn in school, serves well for the human-made world of straight lines, circles, and squares. But nature's non-linear shapes were generally considered unmeasurable -- until Mandelbrot developed fractal geometry.

"In the whole of science, the whole of mathematics, smoothness was everything," Mandelbrot says in "Hunting the Hidden Dimension." "What I did was open up roughness for investigation."

Suddenly, something as ragged as a coastline could come under mathematical scrutiny. While he couldn't actually measure a coastline, Mandelbrot found, he could measure its roughness. It required rethinking one of the basic concepts in math -- dimension.
In Mandelbrot's view, another dimension exists between two and three dimensions. It's a fractal dimension, and the rougher something is, the higher its fractal dimension. This roughness, this fractal dimension, he discovered, could be measured quite well using fractal geometry.

Fractals -- and a coastline is a fractal, Mandelbrot said -- have another distinctive quality: self-similarity. Look at the branches of a tree. Big branches give rise to smaller branches that give rise to yet smaller branches, yet all look essentially the same; they're just at different scales. They're self-similar.

Natural selection, as Mandelbrot pointed out in his classic 1982 book The Fractal Geometry of Nature, has favored fractal self-similarity over and over throughout evolutionary history. From the self-similar stalks on a head of broccoli, to the ever-smaller branching of blood vessels in the human body, fractals are everywhere in nature.

And thanks to Mandelbrot, they're everywhere in the human world now, too. The antenna in your cell phone? Fractal. Those noise barriers along highways? Fractal.

All told, since Mandelbrot introduced fractal geometry to the world in the 1970s, his new math has informed fields as diverse as biology and physics, ecology and engineering, medicine and cosmology. And like the endlessly self-similar Mandelbrot set (at left) -- the iconic fractal, named in his honor -- the end of its applicability is nowhere in sight.

Feel your mind stretching yet?

NOVA's "Hunting the Hidden Dimension" will re-air on Tuesday, December 14 at 8pm on PBS.

Image credits: (Mandelbrot) Courtesy Benoit Mandelbrot; (fractal) © WGBH Educational Foundation

User Comments:

Having read James Gleick's book 'Chaos', with more than one mention of Benoit.. my previously stifled world was opened up to the randomness that becomes fractal, and even more fractal (if that's possible)chaos is order, and vice versa... if I might be allowed to be so bold (a la Mr Roddenberry!!)

A giant thinker.

Thank You Benoit Mandelbrot Your work has inspired me in an artistic and intellectual way I will never forget.

I am a research scholar in mathematics from India. Here I bow my head before the Great Personality who transcended mathematics to a "mystical" dimension with rational roots.

May his soul rest in a 'dimensionless world'.

ahh thats a real shame i will miss him

We should also note Mandelbrot's impact on the study of financial markets.

Tom Murray, Ph.D. (UF '06) introduced me to the term "fractal" in '04 in reference to Natural Family Systems Theory's founding genius, Dr. Murray Bowen, M.D.'s principals of "self-differentiation" and transgenerational emotional reactivity. Single behaviors are often, if not most frequently, indicators/evidence/fractals of patterns/sequences/habits of behaviors we homo sapiens display and possess.

Thank You Benoit Mandelbrot,I want to study financial markets.

You can use the online fractal generator to zoom the mandelbrot fractal.

In Benoit Mandelbrot's passing, who's work on Fractal Geometry helped create Pixar and inspire so many an appropriate tribute would be that he helped us imagine "To infinity and beyond."

I first heard some vague information about fractals from a friend working on his Masters in Computer Science a few years before they completely transformed the world of computer graphics.

Fascinated ever since by the topic and the man. Even now I like to play with a good fractal explorer application.

Nova should do an update on where the study of fractal geometry is leading and has lead. I'd like to hear more about the fractal encoding of images.

blog comments powered by Disqus