The Fractalist

I had expected to be able to leaf through The Fractalist: Memoir of a Scientific Maverick, by Benoit Mandelbrot. I had expected that I would spot a couple of interesting nuggets along the way, but that the story couldn’t possibly sustain my interest enough to support a close read.

Boy, was I wrong!

This memoir turned out to be utterly absorbing, capturing my close attention deeply enough to reward reading every single word and pausing to think as I did so. Not that it is without flaws — far from that. On the whole it reads like a good second draft, not like a polished, well-edited narrative. But there’s a good reason for that: Mandelbrot died before he could finish it, so he never had the chance to go through the manuscript one last time, and I don’t suppose any editor was willing to touch it substantially (not that Random House or any other publisher seems to do much actual editing these days).

Before I continue, I want to be sure that you know who Mandelbrot was. You’ve probably heard of the Mandelbrot Set, but if you haven’t recently studied precalculus in recent years you might not really know what it is or why it’s significant.

You won’t find out from this memoir.

You’ll see some beautiful pictures of it and will read about the mathematical world’s reactions (pro and con) to it discovery, but this isn’t a math book and isn’t the place to learn about the Mandelbrot Set. There are hundreds of other sources for that.

In fact, you have to read through 248 pages before you get any inkling of the Mandelbrot Set. That’s OK, because the first three quarters of the memoir are absolutely fascinating. Actually, it is more an autobiography than a memoir, since it thoroughly covers all of Benoit Mandelbrot’s life from his birth in 1924 until a few months before his death in 2010. In the first chapters of the book we learn a lot about Mandelbrot’s experiences growing up Jewish in Poland and in France, particularly in Nazi-occupied France. And we learn a lot about his long scientific career, which spanned much more than fractals. And we learn a lot about the intellectual sources of that career.

It’s particularly interesting to read about the early days of somebody who is quintessentially modern, somebody whose research was ongoing for many recent decades (up until three years ago!), because those early days now seem so long ago. The 1930’s and most of the ’40’s were before I was born, after all. And yet here we have a contemporary looking back and recalling the following:

The Jews’ situation in Poland was clearly desperate, but what could be done? … My brilliant cousin Mirka, a year older than I, faced her own set of difficulties…She placed first on the fiercely difficult entrance exam to the only suitable girls’ high school in Warsaw, but was bounced from the Jewish quota by others with better-connected parents. [My uncle] Szolem spoke to colleagues in Paris. Letters to influential contacts in Warsaw went up, up, and up — and Mirka was admitted.

What mighty person “fixed” Mirka’s admission? He was Poland’s most political and powerful mathematician, Wacław Sierpiński, whose role in my life, always indirect and never planned, cannot be overestimated.

Sierpiński, who seems to my students to be a long-dead mathematician, someone who might have been a contemporary of Newton’s for all they know, first pops up here and then reappears at many points in Mandelbrot’s book. Another famous 20th-Century mathematician, Gaston Julia, also pops up here and there. Neither is really a surprise, given the best-known content of Mandelbrot’s work. Perhaps more of a surprise is the presence of Jean Piaget for several pages. Piaget is known for having great ideas but poor writing, so it’s much better to read explanations of his work by people other than Piaget himself. Here’s what Mandelbrot has to say about him:

Early in the school year, he asked me to look at his current book and handed me a chapter. I found it interesting but asked him to explain a few lines in greater detail. Piaget apologized and obliged: in no time obscure lines became obscure whole pages.

As I suggested above, I also learned a lot about Mandelbrot’s research outside of the areas of iteration and fractals. He did early work in economics, linguistics, and geometry, all of which he tied together with theories of “roughness.” This interdisciplinary content resonated deeply with me, since my career has combined math teaching, linguistics, technical writing, and software development, all tied together with ideas of “representation.” I was especially struck by Mandelbrot’s anecdote concerning a talk about personal incomes that he was giving at Harvard in 1962 or so [dates are often hard to determine in this autobiography]. He walks into the office of the professor running the seminar, and saw the following:

A peculiar diagram on his blackboard seemed to me nearly identical to one I was about to draw in my lecture! How was it, I soon asked, that something I had just discovered about personal incomes was already on display? “I have no idea what you are talking about. This diagram concerns cotton prices.”

Two different topics with the same visual representation — “the same pattern of concavity and convexity.” Representations are powerful, sometimes independently of what they are representing.

You noticed that I specifically mentioned geometry above. Mandelbrot attributes much of his success to his emphasis on geometry rather than algebra. Here is a typical passage describing his college days:

I would raise my hand and describe my findings: “Monsieur, I see an obvious geometric solution.” I quickly grasped the most abstract problem that the teacher could contrive. And then — with no effort, conscious search, or delay — I continued along a path that somehow avoided every difficulty…. I managed to be examined on the basis of speed and good taste in, first, translating algebra back into geometry, and then thinking in terms of geometric shapes. My analytic skills remained so-so, but that did not matter — the hard work was done by geometry, and it sufficed to fill in short calculations that even I could manage.

Don’t be fooled by the false modesty. The arrogant portions reveal Mandelbrot’s true self, here and elsewhere. For instance:

I was racing across the Latin Quarter when my mathematics teacher, M. Pons, hailed me in the street, and we had our first and last private conversation. “Let’s talk about the big math problem at Polytechnique. I could not solve it in the time allowed, but examiners say that — in the whole of France — one student did solve it, and he is from my class. Could it be you?”

“Well, I did solve the entire problem — including every optional question at the end.”

“How did you manage? No human could resolve that triple integral in the time allowed!”

“I saw that it is the volume of the sphere. But you must first change the given coordinates to the strange but intrinsic coordinates I thought the underlying geometry suggested.”

“Oh!” And he walked away, repeating, “But of course, of course, of course!”

When the exam ordeal ended, my grade was 19.75/20. Nobody ever received 20/20 — ever! For this and other top mathematics marks, rumor appointed me the best math student in the country that year. Everyone seemed to know of my skimpy formal preparation, so I was credited with a feat that would be remembered for years to come.

I don’t think that passage needs any further commentary. But I should follow it with a passage that comes near the end of the book:

I did not plan any general theory of roughness, because I prefer to work from the bottom up, not from the top down. So even though I didn’t try to create a field, now, long after the fact, I am enjoying this enormous unity and emphasize it in every new publication.

I reach beyond arrogance when I proclaim that fractals had been pictured forever but their true role remained unrecognized and waited to be uncovered by me.

As Mandelbrot wrote at a different point:

I don’t feel I “invented” the Mandelbrot Set: like all of mathematics, it has always been there, but a peculiar life orbit made me the right person at the right place at the right time to be the first to inspect this object, to begin to ask many questions about it, and to conjecture many answers. Though it had not been seen before, I had a very strong feeling that it existed but remained hidden because nobody had the insight to identify it.

The development of technology plays an important part in the later chapters of this story. Many fractals and related objects, especially the Mandelbrot Set, are impossible to draw 0r investigate without the use of computers. So it’s not surprising that Mandelbrot did not fare well in Harvard’s Math Department. When I was there in the late ’60s, no one in the Math Department would touch those “greasy machines,” and apparently things were no better in the early ’80s:

Computers and their use were not welcome at Harvard. Hence, there was a near-total absence of both equipment and skills among the students and faculty. And because personal computers had not yet become ubiquitous, those who absolutely needed them went elsewhere or had private and well-hidden facilities.

I think things are very different today in the Harvard Math Department.

My only real complain about The Fractalist is an unavoidable one: as I suggested above, it reads like a good second draft, not like an edited final copy. That was unavoidable because Mandelbrot died before he could finish it. On a related point, here is his description of how he writes a book:

I never begin with a table of contents and then write chapters, sections, and sentences in the order in which they appear… Every so often I wake up in the morning with the overwhelming feeling that a chunk of the book is in the wrong place and had better be brought forward or back. Quite literally, a book does not approach completion until I know it by heart. As a young man, I had no access to a typist or time for careful successive handwritten texts. So I often sent the printer something that in truth was an immature early draft… Word processing has made this syndrome incomparably easier to live with but has not cured it. One day, my programmer, watching my assistant suffer with an especially messy draft, asked how I had managed before electronics and without an assistant. My answer: “Extremely painfully.”

In closing, let me discuss Mandelbrot’s middle name. Or the lack thereof. After he was born, you see, he was named “Benoit Mandelbrot.” Later in life he became “Benoit B. Mandelbrot.” And what does the “B” stand for, since he actually had no middle name? It stands for “Benoit B. Mandelbrot,” of course.



Categories: Books, Math