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A Math Genius’s Sad Calculus

Benoit Mandelbrot, the father of fractal geometry, pens a disturbing new memoir on mathematics—and survival

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“Nearly all common patterns in nature are rough,” writes the mathematician Benoit Mandelbrot at the beginning of The Fractalist: Memoir of a Scientific Maverick published posthumously this month by Pantheon. “They have aspects that are exquisitely irregular and fragmented—not merely more elaborate than the marvelous ancient geometry of Euclid but of massively greater complexity.”

As Mandelbrot illustrates in his memoir, there are many rewards out there for an elite mathematician—university chairs, corporate research jobs, and conference junkets—but fame is not usually one of them. His uncle, Szolem Mandelbrojt, was a star of French mathematics in the 20th century, but he looked with suspicion on the kind of acclaim Benoit received when he published his 1975 book Fractals and its popular successor The Fractal Geometry of Nature in 1982. “There are 15 people in the world who read everything I write,” Szolem told his nephew. “That is enough. I find that very comforting.”

What made Benoit Mandelbrot something of a cult figure was the fact that his key discovery, fractal geometry, generates a weird kind of visual beauty. The set of mathematical objects known as the Mandelbrot set is produced using a formula that he describes as “very plain”: “Pick a constant c and let the original z be at the origin of the plane; replace z by z times z; add the constant c; repeat.” A graph of the set produces an image of fantastic complexity and strange magnificence. It exhibits the key property of a fractal, which is that it is self-similar—it repeats its own pattern on every scale.

This kind of pattern, Mandelbrot found, is common in nature and even in art. The illustrations in The Fractalist include an image of the human lung, a photograph of atmospheric turbulence on Jupiter, and Hokusai’s famous painting “The Great Wave”: All display the kind of regular irregularity that Mandelbrot taught the world to call fractal. (He came up with the name, he writes, after consulting his son’s Latin dictionary and finding the word fractus, “broken.”) More, his mathematical work had applications to a surprising range of fields: He made contributions to the study of linguistics, financial markets, geography, aeronautics, and cosmology, which were united by what Mandelbrot calls “roughness.” Or, as he put it in The Fractal Geometry of Nature, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

Mandelbrot’s life work was to develop mathematical tools able to measure that kind of fiendishly difficult, real-world complexity. The challenge facing The Fractalist is that it is almost impossible for a non-mathematician to advance beyond these generalities and understand what precisely it is that Mandelbrot accomplished. Knowing this, he allows no mathematical formulas or notation in the book—the formula for the Mandelbrot set is the sole exception. It is clear enough, however, that the mathematics Mandelbrot worked with has nothing to do with the kind most of us learned in school; it is infinitely more creative and exciting. His own gift, he writes, was an intuitive ability to “see” complex shapes. As a student, he could solve difficult problems much faster than the rest of the class by turning equations into mental geometry: “In no time, searching for and studying symmetry became central to my work … hopelessly complicated problems of integral calculus could be ‘reduced’ to familiar shapes that made them easy to resolve.”

For this reviewer, reading The Fractalist is rather like reading about a poet who wrote in a foreign language for which no adequate translation is available. You know Mandelbrot is up to exciting things, but you have to take them mostly on faith. What he can share, and does copiously, are the steps of his worldly career: the professorial appointments, the job as a researcher at IBM, the papers published and colleagues courted and impressed. There is so much of this kind of thing in the second half of The Fractalist that it comes to read like an annotated CV, and it has the effect of making Mandelbrot seem very vain. But then, this is a man who decided early in life that he wanted to be a second Kepler, founding a new field of study and revolutionizing humanity’s picture of the world. (In his own view, he accomplished this: “In my Keplerian quest I faced many challenges. The good news is that I succeeded.”) All of this sits oddly with his later declaration that “a memoir is a lesson in humility.”

What makes The Fractalist compelling, even for a non-mathematician, is its first half, which describes Mandelbrot’s childhood and adolescence as a Jew in wartime Europe. The fact that he survived to adulthood is itself something of a miracle. He was born in Warsaw in 1924 and raised in a modernizing Jewish family where Polish was the language of choice. (He could not talk directly to his Yiddish-speaking grandfather, he recalls.) The Mandelbrot family’s roots were in Lithuania, where it was said to have produced “men of great learning, some even famous within Jewry.” That it went on to produce Szolem and Benoit lends support to the idea that the intellectual skills prized by traditional Jewish learning are the basis of modern Jewish achievement in secular fields.

The global depression combined with intense anti-Semitism made Poland during the 1920s and 1930s a very difficult place for Jews. Mandelbrot’s father, who had forsaken advanced schooling in order to become a garment dealer and support his family, emigrated to Paris in 1931, hoping to make a better living. Five years later, his family joined him—spurred, Mandelbrot writes, by the experience of a cousin who was denied admission to high school out of anti-Semitism. The decision was made just at the right time: If they had waited a few years, the Mandelbrots would have been trapped by the war and almost certainly would have died in the Holocaust, as did most of their Warsaw friends.

Life in France was not easy, either—the family was extremely poor and lived in a tenement in the Paris slum of Belleville. But Mandelbrot’s parents believed strongly in the promise of France, and especially in the meritocratic school system. By insisting that Benoit acculturate quickly—learning to speak French without an accent and get along in French society—they gave him skills that would be indispensable after the fall of France in 1940. The Mandelbrots fled Paris for the southern, unoccupied zone, where they survived in the small town of Tulle thanks to the help of friends of uncle Szolem’s. “Our constant fear,” Mandelbrot writes,”was that a sufficiently determined foe might report us to an authority and we would be sent to our deaths. … We escaped this fate. Who knows why?”

One reason why, he suggests, is that his academic brilliance won him special consideration. “Xenophobia lost, meritocracy won,” he writes, and this would become the motto of his French experience:

The chronicles of Eastern Europe included a growing number of stories in which a would-be “butcher” is oversupplied with potential victims and a person perceived to be special is somehow spared. Father must have felt it was very bad to be overly conspicuous, but very good to be seen as rare and special. This attitude, which he probably brought from Warsaw, created in me an elevated level of commitment and ambition.

Thus motivated, Mandelbrot advanced to excellent high schools and then, after the Liberation, to the École Polytechnique, one of the ultra-elite “grands écoles.” More than 60 years later, he still writes with pride about the score he received on his entry examination—the highest in France, not just in that year but possibly ever. From then on, he was more or less guaranteed a cushy career in French academia—though he ended up spending most of his adult life in the United States, drawn by the intellectual freedom offered by IBM’s pure research division.

This back story gives a certain pathos to Mandelbrot’s late-life boasting. Hannah Arendt wrote in The Origins of Totalitarianism about the plight of the “exception Jew”—the Jew who is recognized as an equal in European society, but only because of his exceptional achievements or charm, like Benjamin Disraeli. For Mandelbrot, his mathematical genius might literally have saved his life during the war years; it certainly raised him from the immigrant poverty in which he was born and introduced him to an international fraternity of elite scientists. But there is also something tragic about Mandelbrot’s conviction he felt that he had to be a genius simply in order to survive.


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If boasting has been perpetrated, it is justified. There were undoubtedly many Holocaust victims whose incipient genius was never realized because they were dead. As a brilliant mathematician, Mandelbrot surely recognizes that genius alone does not guarantee maturity, and that the laws of probability were more applicable to his survival. That has got to be some maddening knowledge.

“The Mandelbrot family’s roots were in Lithuania, where it was said to have produced “men of great learning, some even famous within Jewry.” That it went on to produce Szolem and Benoit lends support to the idea that the intellectual skills prized by traditional Jewish learning are the basis of modern Jewish achievement in secular fields.” It’s obvious that there is something else which is the basis of Jewish achievement. 4 out of 9 Nobel’s this year (ignoring the European Union). If you are afraid of this because you think we will all be rounded up and killed or because you think it makes other groups look bad, don’t just make something up.

Boychic says:

A wonderful and inspiring article. Imagine how many “exception Jews” might have perished in the camps. Considering how we Jews have been mistreated we have indeed earned our bragging rights.

gwhepner says:


“You had to be a genius to survive,” thought

showed with his new mathematical technique, fractal

irregularity is not beyond

by a brilliant mind that is contractile.

the problems that irregularity

for Jews when they’re regarded as irregular

gentiles, Jewish genius can reduce disparity

Jews and the gentiles with great work that’s secular.

Miha Ahronovitz says:

Adam , what an extraordinary essay disguised as book review . This sentence reverberates in me; ” there is also something tragic about Mandelbrot’s conviction he felt that he had to be a genius simply in order to survive.”

Wall Street treated Mandelbrot in a similar way. They did not like his maths because he told them that future is unpredictable. I would say all financial crashes, are a proof for Mandelbrot fractals. See Fractals , JP Morgan; how a CIO can become the scapegoat.

Interesting piece on what sounds like an interesting memoir, but within mathematics, Mandelbrot’s stature is controversial, and not only because so many of his colleagues look askance on the self-propagating quality of those fractal images, which have found their way onto the covers and within the pages of countless pop-sci books. I believe I’m correct in saying that not one of the ideas commonly associated with Mandelbrot actually originated with him. Nor was he the one who developed them in a mathematically rigorous way. He was the first to use computer technology to get good images of these algorithms, and he was the evangelist who persuaded others of their significance, especially in applied science. In fact, the Mandelbrot Set described in the article was named in his honor for his study and popularization of it, not because he created it.

    Mandelbrot made some a pretty major contributions to economics, too. I think in 100 years he is more likely to be remembered as the father of experimental economics than the guy that coined the term “fractal”.

      His mentor was Paul Levy, right? Levy’s contributions to probability form the basis of a lot mathematical finance, and I think Mandelbrot built on his results. He also won the Wolf Prize for Physics, awarded by Israel, so yeah, I think the nature of his contributions really has been in applying mathematical ideas.

So Mandelbrot was also Jewish? Man, those guys are taking over!

“The global depression combined with intense anti-Semitism made Poland during the 1920s and 1930s a very difficult place for Jews.

For a more balanced description of Polish-Jewish relations in the two decades preceeding WW II see:

A tendency on the part of Ashkenazi Americans to exaggerate and decontextualize Polish anti-Semitism — and thereby to stereotype the Polish peoples as anti-Semitic — seems unfair to me. You might suppose it doesn’t matter three-quarters of a century later, except that it poisons contemporary Jewish — or, rather, American Ashkenazi — attitudes towards the Poles and Poland.

On decontextualization — is that a word? –see this compilation of (mostly) Jewish sources:

Better understanding of history, origins, and dimensions of anti-Semitism in eastern Europe would be a good thing for all concerned in my opinion.

I post this with last link with a certain amount of trepidation, so please Google my name before jumping to conclusions. Concentrate on the sources. thanks,

    It’s all our fault that we were hated and killed, right Luke, you f–king nazi

    My maternal grandparents immigrated from Poland during the first decade of the 20th Century. They didn’t like to discuss their experiences in Poland, but I did hear from my grandfather that the local school set their exams on Saturday so the Jewish kids couldn’t take them and pass to the next class. He had to drop out and take an apprenticeship to a tailor, which was pretty much how he spent the rest of his life. He had a basic Jewish education and was fluent in Yiddish and Hebrew. He conducted our seders every year.

    The Poles were anti-Semites through all their years as Catholics, just like the rest of Catholic Europe. I remember the American Irish Catholics of my childhood on Long Island — the ones who attended Catholic schools were exactly as anti-Semitic. The nuns at Cure of Ars told them the Jews killed Jesus.

    My best friend from public school, however, was Catholic (not Irish), and her parents were great.

pkbrandon says:

Is this any more than an extreme case of what we Jews have always known: we have to do better to do as well.

Benoit did NOT discover “fractal”
geometry ( a small part of Geometric Measure Theory).

The work of L.C. Young, Koch, Besocovitch and a host of other long predates him.

The so-called Mandlebrot set, was constructed and published long before by Besocovitch,

Benoit proved no theorems, though this area is full of wonderful deep theorems.

In short, he was a fraud, and a plagiarist.

Безико́вич, with several english transliterations, was a jew too. I forgot Georg Kantor
(Cantor), who published “fractal sets” in the 19th century.

Or Karl Wierstrass, who constructed such sets, also in the 19th century. Unlike Kantor,
he was not jewish.

ToNYC_EWR says:

It’s always interesting to read examples of science exceptionalism and then blend in the racial theme of WW II as if it meant something other than the human race and human brain being exceptional. Interesting in a tragically pathetic manner to non-racists I should add.

‘You had to be a genius to survive’
How many geniuses and potential geniuses were murdered by the Nazis?


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A Math Genius’s Sad Calculus

Benoit Mandelbrot, the father of fractal geometry, pens a disturbing new memoir on mathematics—and survival