Mandelbrot the Magnificent Read online

  Begin Reading

  Table of Contents

  About the Author

  Copyright Page

  The author and publisher have provided this e-book to you without Digital Rights Management software (DRM) applied so that you can enjoy reading it on your personal devices. This e-book is for your personal use only. You may not print or post this e-book, or make this e-book publicly available in any way. You may not copy, reproduce or upload this e-book, other than to read it on one of your personal devices.

  Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author’s copyright, please notify the publisher at: http://us.macmillanusa.com/piracy.

  To Benoît,

  Thank you for showing me the story so blatantly hiding between your lines.

  Acknowledgments

  I am grateful to my editor, Ann VanderMeer, for embracing this unsung mathematical superhero, and to my agent, Renee Zuckerbrot, for guiding him to such a perfect home. Special thanks to the people who helped me shape this book, including Dinah Lenney, Robert Kerbeck, Shubha Venugopal, Lisa Alexander, and Jeff Stockwell. And to Ethan, for his mathematical expertise and textbook loans.

  ALIETTE IS COOKING CAULIFLOWER once again. My wife favors an old recipe from Brittany. First she blanches it in rapidly boiling water with salt, butter, and clove. Then she fries it in mouton fat with parsley, chervil, and thyme. The curves and angles of her face as she works under the unforgiving light of the kitchen are still beautiful, even after five decades of marriage. I have always been a great lover of geometry. A dash of vinegar and white pepper completes the dish, so much more palatable than the way Mother used to prepare that hateful vegetable. My job was to chop the head into bite-sized pieces (pallid brain leached clean of thought), but it was cauliflower that saved my family that summer in 1944, so over the years I have taught myself to be fond of it.

  Aliette sets the plate down next to my laptop, a gift from our children for my eightieth birthday, so I can finally finish my memoirs (or begin them).

  “Eat,” she says, “while it’s still hot.”

  I take a forkful and admire the curlicues of steam coming off the tiny florets, each part so like the whole, only smaller (an infinite downward repetition). Cauliflower: my madeleine, memento mori, Mandelbulb, the model for the fractal theory of nature that bears my name. I take a bite and the past returns to me with pitiless clarity.

  Warsaw

  I WAS BORN ON November 20, 1924, at Ulica Muranowska 14, a street that would soon become part of the Warsaw Ghetto. My brother, Léon, was born fifteen months later. We lived in a nice fourth-floor apartment filled with dark wood paneling, richly upholstered furniture, and our most precious possessions, books. The front entrance and sitting room were dedicated to Mother’s dental practice. All day long patients would come to our home, everyone from the poorest peddler to the wealthiest diamond merchant. “Teeth, a more effective leveler of society than Bolshevism,” Mother liked to say.

  I have many happy memories of my brilliant Uncle Szolem coming over for dinner with his wife. Father would be working late at his wholesale ladies’ hosiery business, Mother and Aunt Gladys would be busy in the kitchen, and Uncle Szolem would entertain us with stories about the many mathematicians he idolized: Euclid and his geometry, Fibonacci and his integers, Poincaré and his unsolvable theorems, Gaston Julia and his rational functions; but it was the story of Kepler’s ellipses that truly captured my imagination.

  “Johannes Kepler discovered a brand-new law of nature,” Uncle Szolem held forth from our best armchair, his manicured fingers pulling shapes out of the air like some metaphysical magician. “Kepler borrowed the conical slice from Apollonius of Perga, and produced a curved shape with not one, but two foci.

  “Then Kepler applied that shape to Aristotle’s classical theory of planetary motion, whereby all heavenly bodies, including the sun, orbit the earth in perfectly circular paths …

  “ … and instantly all those ‘anomalies’ that had previously bedeviled astronomers—Mercury retrograde, Saturn return—disappeared, just by replacing a circle with an ellipse.

  “So simple!” Uncle Szolem snapped his fingers.

  Suddenly I found myself astride one of those painted carousel horses at the Warsaw Zoo, the ones that Mother had never allowed me to ride for fear that I might fall off and break my head. Round and round we rode to the plinking sounds of the calliope, until my horse broke free from its circular orbit and began galloping along a tangent line, gaining speed as we shot off into the distance, the wind tossing my hair, tossing the horse’s no-longer-wooden mane into my face, and just as we reached the outer perimeter of the park, we were snapped back by the invisible force of that second focal point. Relentlessly, our path curved inward, centrifugal forces tugging at my belly button, as we were pulled back in the direction of the carousel.

  I came to rest once again in our living room. The chandelier above my head tinkled in harmony to the molecules that had been displaced upon my reemergence into this world. Uncle Szolem hadn’t noticed anything; in fact he was still talking, though I was no longer so interested in what he was saying.

  “I want to make a discovery just like Kepler’s,” I announced, my life’s purpose suddenly clear to me, “a discovery so simple, so obvious, that no one else has thought of it.”

  Uncle Szolem squinted down at me. “Have you been sitting here the entire time?”

  I hesitated. “Yes?”

  Uncle Szolem shook his head. “What you wish for is nothing but a childish dream,” he said dismissively.

  I looked at Léon, who was busy running a toy truck through the interweaving vines of the Persian carpet. He was a child, I was not.

  “Why can’t I be like Kepler?” I insisted. No doubt Kepler had also once ridden the carousel horse to the land of curves.

  “Mathematics needs men who are willing to dedicate their lives to her without thought of reward,” said Uncle Szolem, selfless mathematician. He stood up and smoothed the creases from his trousers. “Yearning for fame is childish,” he said as he left the room.

  At six years of age, I had disappointed my uncle, and he had lost interest in me. But I had learned something about myself that day: opposition made me only more determined. (Also, shapes can have very curious properties.)

  *

  The Depression hit Poland especially hard and awakened ethnic strife. I was only eight years old, but I already knew that the Jewish situation in Warsaw was desperate. Uncle Szolem left Poland for Paris, where he had been offered a teaching position at one of the universities. I was sorry to see him go, even though I was still tender about his dismissal of my Keplerian dreams.

  Father joined his brother to see if he could build a better life for us in France. Unfortunately, there were no prestigious academic positions waiting for him. Father had been sixteen years old when Szolem was born. Their mother died soon thereafter, so Father had been forced to leave school to take care of his little brother, doing anything he could to make money. Eventually he settled into the rag trade. He never complained about the work he had to do to support his family, though the Mandelbrots, originally from Vilnius, were descended from a long line of Talmudic scholars. Who knows what Father would have become had he been able to continue his studies?

  Not long after Father left Warsaw, Hitler became chancellor of Germany, President Hindenburg died, and the political landscape began to deteriorate. Soon there was talk of another war. Mother had grown up in St. Petersburg and survived the Russian Revolution. She knew what was coming, knew the price of hesitating. In 1936, three full years before Hitler invaded Poland, Mother, Léon, and I left Warsaw, taking nothing with us bu
t some essential clothing, family photographs, and the dental equipment that could be easily packed and carried. (Friends who had been reluctant to leave their park-view apartments, their Meissen china, their illusory dreams of status, did not survive.)

  Paris

  FATHER HAD RENTED TWO narrow rooms in the 19th Arrondissement, set end to end, like a railroad compartment. There was no hot running water, no bath. The first time Mother entered the apartment, she sobbed inconsolably. By the second day she had recovered and taken control of the household. From then on we were all forbidden to speak Polish. Mother brushed up on her schoolgirl French. Soon she was able to write flawlessly and speak with almost no accent. Father lugged home an obsolete multivolume Larousse Encyclopedia and I read it cover to cover. (My accent, however, remained atrocious, like French filtered through Cockney.) I was kept back two grades at school, but my good visual memory served me well and I was soon able to master the French spelling and grammar.

  In the fall of 1939, Uncle Szolem received a tenured professorship in Clermont-Ferrand, in the Auvergne region of France, and departed Paris with his family for a small town called Tulle. Father and Szolem seemed to be in agreement about this move, but I was surprised—did my uncle not want to live with us in the same city?

  The following spring, my parents took Léon and me out of school and sent us to stay with Uncle Szolem, telling us that there was a meningitis epidemic running through Paris and that the fresh country air would do us good. His new house in Tulle was a simple wooden box built on scrubland near the train station, but it seemed like a palace to slum dwellers like us. Aunt Gladys pampered us and taught us French table manners. My brother and I shared a room, which wasn’t ideal, but there was modern indoor plumbing, and out the kitchen door was a small garden. Léon revealed his natural babysitting skills, and he and our new cousin, Jacques, played for hours, which freed me up to pursue my own agenda. I kept hidden my Keplerian dreams and wooed my uncle patiently, incrementally convincing him that I was worth his time. It worked. He became interested in me again and spent many hours talking to me about mathematics and the natural world.

  Uncle Szolem began with simple exercises culled from the lycée curriculum: “If Étienne puts a rectangular fence around his cabbage patch, and the patch has a length that’s nine meters less than three times its width, what is the perimeter of Étienne’s fence if the area of his cabbage patch is five thousand six hundred and seventy meters?”

  I found this pedagogical exercise almost insulting in its simplicity, as my uncle must have intuited because soon we moved on to more interesting thought experiments adapted from one of Zeno’s paradoxes:

  “Étienne tries to walk to the end of his sitting room, but before he can get there, he must walk half the distance, then a quarter, then an eighth, then a sixteenth, and so on. Will he be able to leave this room and join his wife in the kitchen, where the chicken needs plucking for dinner?”

  I didn’t have to think about it too long, for the answer seemed obvious to me. “Étienne may never get to the end of his original sitting room, but as he comes incrementally closer to the middle, he kicks up the carpet of our world and creates a space between it and the floor, and as that space grows larger, he will create a parallel farmhouse in which he can live and never have to worry about plucking his wife’s chickens again.”

  Uncle Szolem eyed me uneasily and said, “It’s time for you to grow up, Benoît.”

  *

  One morning, a few weeks into our stay, a telegram arrived from my parents. I remember the breakfast Aunt Gladys had served that morning: toasted buckwheat groats with fresh milk and raisins, the aroma so distinct, so warm and nutty. But the look on Uncle Szolem’s face set a frozen stone in the middle of my stomach.

  “Germany has invaded France,” Szolem announced. Aunt Gladys pulled little Jacques out of his high chair and cradled him in her lap.

  “Your parents sent this from the train station,” Szolem said to me and Léon, a hopeful smile on his normally somber face. “They will arrive in Tulle by nightfall.”

  What followed was a week of terror and anguish, during which time my brother and I became convinced that we were orphans. I let Léon climb under the covers with me at night, though he kicked like a mule and sometimes wet his bed. Finally, toward nightfall of the eighth day, my parents arrived. There had been no trains; all roads heading south had been clogged with cars and trucks piled high with household goods. My parents had left everything behind in the Paris apartment, except for Mother’s precious dental equipment, and walked over four hundred kilometers to get to Tulle, crossing unplowed fields, avoiding main roads, sleeping in abandoned farmhouses.

  My parents looked tired and bedraggled when they finally arrived, much older than I remembered them. Mother’s lips trembled as she crushed me to her breast. There were crescents of dirt under her fingernails. The stench of her unwashed body brought tears to my eyes, and Father looked like a golem that had risen out of a dried-up riverbank. At that moment there was a tiny shameful part of me that wished I could go on living with Uncle Szolem and Aunt Gladys.

  Tulle

  TULLE, THE CITY OF SEVEN HILLS, stretches three kilometers over a deep, winding hollow created by the confluence of the Corrèze River and two of its tributaries. Many streets go straight uphill, with long staircases in stone or concrete. One benefit of this to me was that the girls in Tulle had much better legs than their sisters in Paris. I was fifteen then and quite interested in such things, though unable to act on my interests, due to Mother’s strict instructions that we remain inconspicuous. She had recovered, my stalwart mother, as she had done in Paris. Her word was once again law.

  France was cut into two regions after the invasion. Germany occupied the north, and Marshal Pétain controlled the “free” south with his (puppet) Vichy government. Tulle was technically in the “Free Zone,” though not free enough for foreign-born Jews like us, who were not protected by the new laws the way the French-born Jews were. For now. Nevertheless, Uncle Szolem had led us here just in time. It might have been a coincidence, or that he had friends in high places who watched over him, but to my young mind, Uncle Szolem had anticipated the fall of Warsaw, the fall of Paris, had crossed Europe just one step ahead of the Nazis. His omniscience was unassailable.

  Uncle Szolem even found an apartment on the top floor of a small tenement in a little elbow of a village on the very edge of town. As refugees (with his help we registered as Parisians, hiding our Jewishness entirely), we were eligible for welfare and received some furniture and a Franklin stove to cook our food and heat our home. The walls were made of plaster and straw. In winter there were icicles hanging from the window frames. The luxuries of Uncle Szolem’s house beckoned. Why hadn’t he found us anything nicer? Father had insisted that this was the best we could afford.

  The Tullistes, who had a reputation for being unfriendly to strangers, especially Parisians, were kind to us. It was a poor region back then, nicknamed Tulle-la-Paillarde, “the Poor One Who Sleeps on Straw.” We no longer had the sorts of possessions that would attract attention or envy, and thanks to Mother’s language efforts, we did not seem like strangers (except for my terrible accent, which kept me silent most of the time, though I was bursting with the desire to connect with others). It was from her that I first learned the subtle art of camouflage.

  My brother and I went to a school reached by one of Tulle’s endless staircases. It had once been the town house of a local landowner who had been killed in the previous war. The stately rooms of the nineteenth-century mansion had been stripped of their furnishings and crammed full of clumsy wooden desks, but the walls still held their decorative moldings. Sitting in those rooms, I felt like a young nobleman absorbing knowledge amid ancestral splendor.

  We had the best teachers from an old lycée in Alsace that had been closed after Hitler incorporated the region into the Reich. Math was taught by a slim, exceedingly pale man named Monsieur Leguay. In another time, he would have be
en supervising Ph.D. candidates at a university, but there was a war going on, so he was obliged to share his genius with us. He led us through a quick review of algebra, geometry, and trigonometry before scaling the heights of calculus.

  Thanks to Uncle Szolem’s tutoring, it all came easily to me. I was the best in class on some days, on others the crown belonged to a student named Emile Vallat, son of the town librarian. He was a small, dark-haired boy with an unnatural ability to grind through complicated equations at great speeds. How I envied his agility, his grace. How good it would be to have a friend to bring home and show off to my uncle. And that’s when I realized how lonely I had been all this time.

  Every morning while I ate my breakfast porridge, I devised in my head various ways to approach Emile, who was always surrounded in school by a circle of worshipful friends. But on my fifth day of school, when I had finally gathered the courage to approach my soon-to-be best friend, Emile Vallat watched me cross the room and said to his allies, loud enough for me to hear, “This one claims to be from Paris, but that face could only come from the ass-end of Europe.”

  I swerved away before Emile could say anything else, made it back to my desk on trembling legs, the laughter of those jeering boys like shrapnel penetrating the coarse blue wool of my school uniform. Eyes stinging with tears of humiliation, I bent my head and began to draw ellipses into a blank page in my exercise book. (I drew and drew, the lead point of my pencil growing dull and smudgy, until I had created a cool metallic trachea down which I slid and hid for the remainder of the class. Was there really something wrong with my face?)

  Friendless, lonely, I looked forward to the weekends, when my family and I would go to Uncle Szolem’s house or he would come to us. Food was becoming scarce, but between his extra rations from the university and Mother’s ingenuity, an excellent meal was always pulled together. After dinner, Uncle Szolem would quiz me on everything I had learned in class from Monsieur Leguay. I answered as best as I could, savoring his attention.