The Fractalist

by Benoit Mandelbrot

  After Benoit told me of his illness, I tried to get him to reflect on his accomplishments, his amazing legacy—enough to satisfy a dozen brilliant scientists. Instead, he talked about the work that remained to be done, work that would go unfinished, at least by him. What he regretted most, other than not completing this memoir, was leaving his ideas about negative dimensions at such an early stage. As with so much of Benoit’s work, this began with a simple but elegant question. Starting with the familiar formula for the dimension of the intersection of two sets, Benoit asked, “Can we make sense of a negative result from the intersection formula?” From simple questions … to calculations of clever examples … to validation by experimental work.

  Another unfinished area, lacunarity, began when Benoit noted that many fractals appearing quite different have the same dimension, like the fractals you see below. Benoit wondered if the distribution of gaps in a fractal could be measured by some number. This proved challenging, but early steps have been taken. More needs to be done. Benoit talked about how the project could continue. In the end, he cared about the work, not about his reputation. I believe this was true throughout the long arc of his remarkable life.

  (Illustration Credit aft.3)

  One summer long ago, my grandfather and I were lying on his driveway watching the stars come out as the sky darkened. “Darkened” isn’t the right word—it isn’t that the evening sky is darker; it’s deeper. I was looking into impossible depths. I had a glimpse of an amazing surprise just slightly out of reach, as if I were waking up. What was it? Half a century later, I learned it from Benoit. People often have brief hints of radically different ways to organize what they see, hear, and feel. Very few have more than a glimpse. Benoit shifted the whole world under our feet, giving thousands of people the tools to see the world in a new way. Learning how to recognize this is the clearest example I know of waking up.

  Benoit’s lesson is this: Find the thing you love and follow it with all your heart. What you are following may not always be clear, but if you persist, you will find it, and when you do, you will wake up. What we find is ours, and what each of us finds enriches all of us. This, I think, is Benoit’s last, best lesson. Follow your curiosity, your passion, wherever it leads. Whether you find a new world or a new snowflake, it doesn’t so much matter. Like fractals, life is better understood as a process than as a result.

  Often Benoit said a fractal is defined as well by what has been removed as it is by what remains. Benoit’s dying has left a hole. His wonderful curiosity, sense of kindness, fierce loyalty to friends, and unbounded love for his family have dissolved into the night sky. Memories remain, of course, and a monumental list of accomplishments.

  The last word is Benoit’s, from what was to be his last major talk, at the February 2010 Technology, Entertainment, and Design conference in Long Beach, California:

  Bottomless wonders spring from simple rules … repeated without end.

  Real fern (Illustration Credit bm1.1)

  Fractal fern using an L-system (Illustration Credit bm1.2)

  Real clouds (Illustration Credit bm1.3)

  Fractal clouds (Illustration Credit bm1.4)

  Real coastline (Illustration Credit bm1.5)

  Fractal coastline (Illustration Credit bm1.6)

  Real or fractal?

  Imitation: the first step to understanding

  Surface

  dimension 2.15 (Illustration Credit bm1.7)

  Surface

  dimension 2.5 (Illustration Credit bm1.8)

  Surface

  dimension 2.8 (Illustration Credit bm1.9)

  Fractal forgeries showing the relationship between fractal dimensions and roughness

  “Cool Afternoon” (Illustration Credit bm1.10)

  “Lethe” (Illustration Credit bm1.11)

  Artistic renderings of fractal landscapes

  Fractal painting of flowers, Augusto Giacometti (Illustration Credit bm1.12)

  Cast of a human lung (Illustration Credit bm1.13)

  The Great Wave, Hokusai (Illustration Credit bm1.14)

  Rough deposit of gold (Illustration Credit bm1.15)

  Turbulence on Jupiter (Illustration Credit bm1.16)

  Science, Art, and Nature

  Zooming into the Mandelbrot set (Illustration Credit bm1.17)

  “Pharoah’s Breastplate,” limit set of circle inversions (Illustration Credit bm1.18)

  Variation of the Mandelbrot set (Illustration Credit bm1.19)

  Deep into the Mandelbrot set (Illustration Credit bm1.20)

  “Cave painting,” modified Mandelbrot set fragment (Illustration Credit bm1.21)

  Quarternion Julia sets (Illustration Credit bm1.22)

  Illustration Credits

  Reprinted from The Fractal Geometry of Nature: 29.1

  Courtesy Michael Frame: aft.1, aft.3

  Augusto Giacometti, 1912: bm1.12

  Sigmund Handelman: itr.1, 21.6, 23.2, bm1.6

  Eriko Hironaka: 25.7

  Katsushika Hokusai, ca. 1829–33: bm1.14

  Mark Laff: itr.1

  Mark R. Laff and Sigmund Handelman: 29.4

  Shaun Lovejoy: bm1.4

  Benoit B. Mandelbrot Archives: Title page, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2.1, 2.2, 2.3, 3.1, 4.1, 4.2, 5.1, 9.1, 9.2, 10.1, 11.2, 11.1, 18.1, 18.2, 21.4, 21.5, 22.1, 22.2, 23.3, 25.6, 26.1, 26.2, 26.3, 27.1, 28.2, 28.3, 29.2, 29.5, 29.6, 29.8 29.7, bm1.15

  Peter Moldave: 25.1, 25.2, 25.3

  Produced by Ken G. Monks: bm1.18

  Merry Morse: bm1.1, bm1.3

  © 1988 F. Kenton Musgrave: bm1.10 bm1.11

  NASA: bm1.16

  NASA Website public image: bm1.5

  Alan Norton: 258, 28.1, bm1.22

  Jean Louis Oneto: 21.1

  © H. O. Peitgen: 25.4

  © H. O. Peitgen and H. Richter: aft.2

  Przemyslaw Prusinkiewicz, 1986: bm1.2

  Courtesy Bernard Sapoval: 9.3

  Courtesy Nassim N. Taleb: 22.3

  Photograph by John Tate: 29.3

  UC San Diego Center for Astrophysics: 231

  Richard F. Voss: itr.1, 21.2, 21.3, itr.2, bm1.7 bm1.8 bm1.9, bm1.17, bm1.19, bm1.20 bm1.21

  Courtesy E. R. Weibel, Institute of Anatomy, University of Bern: bm1.13

  ABOUT THE AUTHOR

  A graduate of the École Polytechnique, Benoit Mandelbrot received his doctorate in mathematics from the University of Paris and spent thirty-five years at IBM as a research scientist and seventeen years as a professor at Yale University. Best known as the father of fractal geometry, he transformed our understanding of phenomena characterized by extreme variability and roughness in a wide array of fields including economics, data visualization, fluid turbulence, biology, geology, and material science. He died on October 14, 2010.