Appreciating Benoit Mandelbrot: Fractalist & Nomad-by-choice

I suppose it was fitting that I heard of Benoit Mandelbrot’s death while taking part in a panel discussion about fractals in New York City. The panel was in conjunction with a new opera, “Cracked Orlando,” with music composed by Jonathon Dawes, and libretto (by Terry Marks-Tarlow), a work which was partly structured by fractal patterns. I was invited to the panel because I’d written a book, “Heaven’s Fractal Net,” about fractal patterns in cultures. It was Benoit Mandelbrot who coined the term and introduced the concepts of fractals, a new geometry of self-similar shapes on various scales. It shocked me that day to hear of his death–his mind was ever-youthful and exploratory.

I became interested in fractals gradually. First, in the ‘60s, in the sunshine up in Vermont’s Green Mountains in the autumn, I became fascinated with experiencing the inherent structure of the eyes. I was visiting Vermont after living in the New York slums for a few years. As brilliant leaves fell curlicuing down against a vivid blue sky, I watched the light dancing in the field of my eyesight, making spirals in patterns like square-dancers doing allemande left, or interlocking quilt patches tessellated together. Psychedelic entheogens reveal vision’s wonders structured in our human physiology. And for several years I tried to create art works—collages, drawings and paintings to capture the orderly interplaying swirls of tiny lights pin-wheeling like dancing comets in colors and subtle geometry. I still have some of the collages. M. C. Escher fascinated me with his fields of transforming shapes, like variations on themes of tessellated tiles.

While studying at Harvard on a Danforth grad school fellowship ten years later, I saw in the Harvard newspaper a piece about Mandelbrot’s fractal research and was fascinated by the order and exuberance, the patterns of self-similarities, and the spirals and shapes suggesting infinity. Too busy to follow up on it I stashed the paper in a folder. I earned my PhD degree and a job in Indiana, and worked extremely hard learning to teach and being a new father. I had no time to spare in my busy life teaching courses, researching and publishing, going to conferences and doing committee work to keep my job.

Then on my sabbatical year I rediscovered Mandelbrot’s fractal research, and read more about it in Mandelbrot’s own writings and in books like James Gleick’s “Chaos.” I recalled the mystery of the structure of vision that intrigued me in Vermont, and caught a sense of the excitement in new kinds of research on chaos, fractals, strange attractors, nonlinear systems. I began to get ideas about explored these areas of thinking, examining these dynamics not only in nature, but also in cultures. There were fractals in temple architecture, and archetypes were strange attractors; there were self-similarities in stories.

Mandelbrot developed the first “theory of roughness,” to mathematically describe structures like tree bark, mountains, plants, blood vessels and lungs, clouds and galaxy clusters. Rivers and tree branches. He sought and developed a formula to measure the roughness in nature, geometrically visualizing principles of order, wholeness, infinity.

Fractals are whole shapes made up of different sized versions of the overall shape itself. For example, a large triangle composed of various scaled smaller ones, or a tree branch made of increasingly smaller similar branches. Mandelbrot was my kind of maverick. Not a typical specialist, he contributed original thinking to diverse fields. This made him an interloper, a rebel, someone who broke the usual rules. He explored geography—the coastlines, and economics—Wall Street patterns. He didn’t look at boundaries the way most people do. He wrote, “Science would be ruined if (like sports) it were to put competition above everything else, and if it were to clarify the rules of competition by withdrawing entirely into narrowly defined specialities. The rare scholars who are nomads-by-choice are essential to the intellectual welfare of the settled disciplines.” A nomad or pioneer has a wide horizon to explore.

Mandelbrot was a refugee in Europe eluding the Nazis during WWII. Stories about him intrigued me. He found a clue to one of his discoveries in a wastebasket. He never learned the alphabet in the usual ABC sequence. In France he had trouble with math as it was being taught there at the time, but was bright and could solve problems no one else could, in his own way. He worked at IBM and explored a variety of interests by experimenting, and taught at Yale. “He used the geometry of fractals to explain how galaxies cluster, how wheat prices change over time and how mammalian brains folds as they grow, among other phenomena,” as Jascha Hoffman wrote in Mandelbrot’s New York Times obituary. He didn’t fit the usual mold of an academic, but influenced many fields. I thought (and still think) he should win the Nobel Prize for his original fractal theory.

Like Benjamin Franklin, Mandelbrot followed his inquisitiveness, and was happy to study as an amateur in areas where he was not an authority, pursuing ideas from sheer interest, curiosity, and the wonderment of exploration. Mandelbrot didn’t confine himself to himself, as the Tao Te Ching describes the character of the sage, but he took an active interest in the work of others, sharing his reflections, and contributing to various fields, consciously playing parts in the vast community of human learning. He was not exactly “re-inventing” himself, but was learning new things, exploring new aspects of the world around him, retooling, seeing the same world the experts see, but with fresh eyes and a fresh mind.

Thanks to Mandelbrot I explored patterns in religious artifacts, from Hindu temple architecture to daily Rangoli patterns—rice-powder designs which housewives in India make on their thresholds each dawn. I explored fractals in poetry and philosophy, mythology and music. I thoroughly enjoyed thinking about fractal patterns and making art works with them, seeing self-similarities in great art works and in nature—clouds, trees, nervous systems, and so on. And I experienced first-hand how difficult it is to get a new idea across, the sacrifices it takes to try something new, when I wrote a book, Heaven’s Fractal Net. (“Heaven” in the title referrs to Lao Tzu’s line in the Tao Te Ching: “The net of heaven (Tao) has large meshes, but nothing escapes from it.” (Chapter 73) Probably a better title for my book would have been Fractals in Cultures.

Some thinkers are painful to encounter. They seem to want to punish you for thinking differently from them. Others are stimulating, encouraging, sharing and liberating. They nourish you and impel you on your path. Mandelbrot, for me, was the second kind. As a Spanish poet wrote, “There are no paths; paths are made by walking.” Mandelbrot made his own path by walking, and his example has been a great inspiration. The term fractals appears all over the world now. The song from the Disney movie “Frozen” won an Oscar in 2014—The words “frozen fractals” are sung in the lyrics to call to mind the shapes of snowflakes. So useful and beautiful, ubiquitous fractals provide many images of natural order and wholeness. My thanks go to Benoit Mandelbrot for providing the means for developing what I call “a fractal sensibility.” That skill is something basic which human beings can develop—identifying and appreciating the beauty of fractals all over the place. It’s an honor to have known Benoit, a privilege to have spoken with him on the phone a couple of times. Did you know that the antennas in cell phones are configured in a fractal structure? That mountains in Hollywood films are created with fractal formulas? That fractals still have many more potential uses and beauties to be discovered?


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