IBM100: Memories of Benoit Mandelbrot, by Michael Frame

An early cosmology placed the earth on the backs of four elephants, 
all standing on a turtle. The question, "On what did the turtle
 stand?" always was answered, "It's turtles all the way down." Calculus 
and all of 19th century analysis worked because the functions studied
 are more closely approximated by their tangent lines the more closely we 
look. When we zoom in, the curves appear simpler. By the start of the 
20th century, mathematicians knew some examples where this was false,
 but these were regarded as monsters. Benoit Mandelbrot recognized the
 mathematics of these monsters described much of nature, and expanded 
this idea into fractal geometry. Very often, nature does not get simpler
 under magnification; Benoit gave us a way to quantify the fact that
 it's complicated all the way down.



Benoit's geometric intuition was remarkable. Although he never learned 
to program computers, he understood their workings well enough to keep
 busy several programmers at IBM and at Yale. Studying the graphical
 output led him to remarkable discoveries: the Mandelbrot set, of course,
 but also multifractals, dimension as a measure of roughness, distributions
 of signal errors, realistic models of financial data, and on and on. Much
 of this work was done at IBM, which provided Benoit with then uncommon 
computing resources, and the freedom to follow his imagination. Almost 
all areas of contemporary science and art bear the imprint of Benoit's vision.

 I worked with Benoit for 20 years. Although I'll never understand why 
he brought me into his world, being there did give me a close view of
 how he thought. Because he believed that history is more about stories 
than large themes, and also because others have written about these 
large themes, I'll tell a few stories about working with Benoit. Any
 conclusions are yours to draw.



The first collaboration Benoit suggested involved binary fractal trees: 
each branch splits into two, at a fixed angle from the previous branch, 
and scaled by a constant factor. Benoit was interested in trees for
 which the left and right branches just touched. He showed me some
 fairly complicated formulas relating the angle and scaling factor, and
 asked if I could find a derivation. Several days of messy algebra
 gave the result. Benoit understood my approach almost immediately, but
 when I asked how he got the formulas, he said he could "see the geometry. "
This was fascinating, and more than a little unsettling.



Our next project involved the distributions of gaps in fractals. This
 collaboration convinced each of us we wanted to continue working with the
 other. Benoit described the problem, and how he thought the answer would
 look. I went to work on the calculations, harder this time, but got 
the answer and found it was the opposite of what Benoit had expected. 
In all our interactions, Benoit had been very kind, gentle almost. Yet
 I'd heard stories of a terrible arrogance, so I was nervous showing him 
the results. Again Benoit got the point very quickly, grinned broadly,
 and said, "Marvelous! This problem is more interesting than I'd expected. "
I saw he was a scientist, interested in what's correct, not in validating
 any preconceptions. I could work with such a person. Years later Benoit 
told me (couched in a story about Diogenes and his lamp - Benoit's 
conversations were filled with footnotes to footnotes to footnotes, hmmm, 
a habit I seem to have picked up from him) that he decided he could work
 with me because I was willing to disagree with him and explain my
 reasoning. At this point, we were comfortable working together. This
 comfort persisted right up to the end.



We worked on many other projects together, though mostly related to 
teaching fractals. I am not a talented programmer, so most scientific
 investigations were carried out with others. I remember Henry Kaufman
 working with Benoit on statistics of very large DLA clusters, pushing
 the limits of the IBM computers he was using. Often by seeing something 
unexpected in the graphs, Benoit suggested parts of the algorithm he and
 Henry should revisit. Again, pretty scary, at least to me.

 Benoit supported all my teaching efforts at Yale. He and I got an NSF
 grant to run summer teacher training workshops. In this we were 
helped by Nial Neger, a retired high school math teacher recruited by
 Benoit for this purpose.

Benoit and I coedited a book on fractals and 
education, wrote a paper on negative dimensions, started another paper
 on that topic, started a book, Fractal Gems, for the Mathematical
 Association of America, started a book on multifractals. Then Benoit
 got distracted by working on his memoirs, and then he died. This was
 completely numbing. I'm a cancer patient, too, and he was supposed to
 outlive me. Mandelbrot in a world without Frame makes much more sense 
than Frame in a world without Mandelbrot.



The projects we began won't be finished. I have no idea how to continue 
them without his guidance. I am helping Aliette Mandelbrot, Benoit's
 widow, and his assistant Merry Morse finish Benoit's memoirs, but beyond
 that, I don't know.



The most fun I had with Benoit happened when one of us, usually him,
 got an idea for a new direction to explore. Then I saw his mind working
 at full speed, at least as full speed as could be managed with the
 requirement of speech. We were playing. The same image keeps coming 
back to my mind. We were little kids, running through an open field 
under a blue sky, curious about every single thing we saw, anxious to 
share each surprise with the other. That never again will I answer the
 phone and hear, "Hello, Michael. This is Benoit. I was wondering ... "
is soul-crushingly sad. A treasured friend is gone. The architect of 
a scientific revolution has departed.



Still, sorrow is tempered by my memories of twenty years of these
 explorations together. Working at IBM nourished the imagination of 
this remarkable man. For that, I am profoundly grateful.

Michael Frame is a Yale University professor, who worked alongside Benoit Mandelbrot for over 20 years.