Mandelbrot the Magnificent

by Liz Ziemska

  But Mrs. Vallat was clearly delighted by this news, because soon Emile showed up at school with the first set of dental braces ever seen in the Corrèze region. With his vulpine face and silver-clad teeth, he looked positively carnivorous. He liked to flash that metallic grin at me during class, a warning to keep silent.

  I have often wondered why Emile waited so long to denounce me. Perhaps it gave him great pleasure to keep me close, yet muzzled (a trained bear on a delicate leash), so that he could savor the torture he was certain I felt, knowing the solutions to all those math problems in class but not being able to say them.

  After the Gestapo arrived, my parents no longer left the apartment. Patients started coming to us, quietly, despite the increased risk. Quotas were going up; the local police were looking closer into people’s backgrounds, searching for hidden Jewish ancestors. Families were taken away every day; sympathizers didn’t fare much better. Nevertheless, they came: a farmer’s wife, Vincent the butcher, an apprentice who worked for Madame Derrasse at the bakery, a maid at the Hotel St. Michel, they all slipped in after dusk and sat under the kitchen light, heads tilted back, mouths open, as Mother stood above them wielding her glinting instruments. In payment they would leave whatever food and supplies they could spare. I was reminded of Warsaw during the Depression, except for the fact that at any moment one of these villagers could decide to report us to the authorities. It was a risk, but Mother insisted that it was worth it.

  The food we received we shared with the dwindling occupants of our apartment building (many had died or moved away to a more desirable part of town). During the day, Father and Monsieur Hubert would clean, sharpen, and sterilize the instruments. From donated silver teaspoons and various bits of jewelry they would fashion crowns and bridges so that our friends would also have glittering mouths.

  While my parents were busy building a wall of neighborly support around us, I was too preoccupied with preparations for our year-end examinations to pay much attention to the Germans in our midst. For those students who managed to pass the notoriously difficult exams, there would be two additional years of school that would then qualify them to apply to the university. I had to pass those exams. Uncle Szolem, wherever he was, would expect nothing less from me. Why else had Mother and Father sacrificed everything to bring me here, a place of relative safety? All winter long, while Mother practiced her clandestine dentistry and Father devoted himself to the custom tailoring business he ran from the kitchen table, I studied Latin, philosophy, physics, geography, and history. The Book of Monsters would have to wait until after the exams.

  One night when I was up late studying, Mother came to me with a slice of mandelbrot, still warm from the oven. I hadn’t seen one of these cookies since Warsaw.

  “You will do well on your exams”—Mother set the plate down next to my books—“but not too well. Do you understand?”

  Where had she gotten the butter, the almonds? Madame Derrasse must have needed some fillings. “I understand,” I said.

  Not only Emile Vallat but also my mother wanted me to hide my talents. Maybe it would be better if I stopped studying altogether. Maybe I should go out and fight instead.

  There were rumors going around about hooligans hiding in the countryside. They were delinquents, gangsters, escaped convicts, Communists sent over from Spain, but also French citizens who were tired of living under German occupation. They were supplied by the Allies in a series of nighttime parachute drops over the French countryside. Well armed, the hooligans were coming out at night to blow up bridges, burn down factories, demolish train tracks, ambush supply convoys—anything they could do to sabotage the German war machine. Silently we cheered for them, these maquisards (named after the maquis, or “brushland,” in which they lived), but we also trembled. There were posters all over town threatening severe reprisals for collaboration with the “terrorists.”

  Several boys from my class had gone missing, though Monsieur Leguay never acknowledged their absence, simply skipped over their names during roll call. Which of them had run away to join the freedom fighters (or “rebels,” as the Vichy government and the Germans called them) and which had been denounced?

  How wonderful it would be to do something rather than pretend to be invisible. Maybe I should join them instead of memorizing Latin verbs.

  For weeks I stayed in the house like a dutiful son, went straight home after school, even though I was growing restless from the boredom and isolation. Books and homework assignments were my only companions, besides Léon, who was fifteen months younger and therefore didn’t count. There was no one my age I could talk to, no one who shared my interests (mathematics was Léon’s worst subject). Finally, one Sunday in early spring, when my family was still sleeping and the rest of Tulle was in church, I packed a rucksack with my handwritten copy of The Book of Monsters, a wedge of bread, and some cheese and slipped away.

  The note I had left on the kitchen table said that I had gone out to forage for mushrooms, which was my original plan. As soon as I hiked up into the hills above Tulle, I was overcome by the beauty of the sky, which was as clean and blue as the Virgin Mary’s mantle (I say this in honor of France’s fabled Catholicism, as this is a French story, set in a French meadow, during a beautiful spring day in France). Birdsong sifted through the fragrant air as crickets harmonized in the underbrush. Wildflowers speckled the lush damp meadow: honeysuckle heavy with bees, rhododendron, ragged mallow, periwinkle, primrose, peony, and above it all, the mauve blur of almond blossoms.

  Everywhere that I saw flowers, Uncle Szolem would have seen numbers. Archimedes ruled the architecture of the pinecone, the unfurling of the fern frond, the distribution of florets in the head of a sunflower, he would have said. I could write the formula from memory:

  Φ being the symbol for the golden ratio, the sacred number that links the Archimedean spiral with Fibonacci’s integers, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …

  “God is a mathematician,” Uncle Szolem had said to me on our long country rambles (which seemed so long ago). I had shown him that I could do the numbers, that I could be his equal, given time and some additional tutoring—so why hadn’t he taken me with him? As soon as these traitorous thoughts invaded my mind, I could picture Father bending into the light, threading the needle, straining to make perfect stitches. Shame on me. Ungrateful, is what I was, and selfish.

  And furthermore, G-d is not a mathematician! Look at the clouds—they are not spheres. Mountains are not cones, light does not travel in a straight line, and bark is not smooth. Nature is rough and beautiful, not rigid and symmetrical, like the world of numbers in which Uncle Szolem lived.

  As I ran my hand over the trunk of a chestnut tree, I disturbed a moth I had not noticed. It settled onto a nearby branch, folded back its wings, and became invisible once again. Holding my breath so as not to disturb the creature, I leaned closer to examine the ragged distribution of color on its powdery wings: beige-blush-brown, beige-blush-brown, all over the moth, all over the tree. How clever of the moth to color itself to match the sun-dappled tree bark. What keen pleasure I felt in unmasking Nature, the archdeceiver!

  I looked back at the clouds, mountains, and trees: perhaps there was an order to this apparent chaos, one that Uncle Szolem and his mathematicians could not even begin to understand. Sitting in the shade of that moth-sheltering tree, I took out my notebook and began filling page after page with equations (not the ones Uncle Szolem had shown me, but the ones in my Book of Monsters), searching for a pattern in the weave. Until my eyes grew heavy and I decided to take a nap. A quick one.

  When the drone of a large insect finally woke me, it was full dark. The moon was just a crescent sliver in an inky sky seeded with stars, more stars than I had ever seen. It was late. I cursed myself for the worry that I was causing Mother. A shadow fell across the stars, diminishing their brightness, the silhouette of an ice-cream cone descending diagonally toward a nearby hillock. I grabbed my things and started running towar
d what I thought would be its landing spot.

  The parachute came down into a raspberry thicket, its payload a large wooden crate. I was just beginning to untangle the chute from the thorns (what a pretty dress Father could make for Mother from the silk) when a circle of men stepped out of the shadows. I saw the dirt-smeared ovals of their faces, heard the clinking of rounds chambered into guns, pointed at me. But I wasn’t scared. Somehow it all seemed as if I were still sleeping beneath that tree, dreaming of mottled wings.

  “Who are you?” said a dark-haired man. He was older, possibly forty, with a Spanish accent, possibly a Communist. “What are you doing here?” said a boy holding a gun with shaky hands. I recognized him from school, the son of a shoemaker. A third one, somewhere between a boy and a man (eager to prove he was the latter), cried, “He’s a spy!” A fourth, bespectacled, spit into the grass. “He’s a Jew!” I knew what he meant: Jews are not supposed to have nationalistic feelings. They weren’t fighters, because they had no state to fight for and therefore could not be trusted.

  You can trust me, I wanted to say to them, it doesn’t matter where I’m from, but the men and boys kept inching closer with their guns, and all I could do was to keep staring at the girl who stood behind them, a beautiful dark moth with liquid brown eyes and mahogany curls. I had seen that girl in the village, walking on the arm of a German officer. Tonight on this desolate hillside she was unescorted, unarmed, and even more luminous in her camouflage of brown wool and muddy corduroys than in the printed dresses and genuine silk stockings she had worn on her “dates.” She winked at me, and then the one with the glasses took me roughly by the arm and began to lead me away.

  “Let him go.” Another man stepped into the clearing, and at first I had trouble recognizing him, so out of place was he in this setting. It was Monsieur Leguay, and he looked as etiolated as ever, though no longer frail, now that he had shrugged off his habitual stoop.

  He looked at me as if trying to decide my fate, and I became aware of the sheer bulk of my shoulders, my plowman’s hands, how much taller I stood than most of the boys holding me at gunpoint. What a good addition I would have made to this band of outsiders, if only I had been the son of a different mother. I shrank myself smaller, ashamed at being able-bodied but unwilling to put myself at risk.

  “Go home, Mandelbrot,” Leguay finally said, shouldering his rifle. “You have a lot of studying to do if you’re going to beat Vallat next week.”

  And just like that, my life became interesting.

  My Keplerian Moment

  LATER THAT NIGHT WHEN I got home and explained to my frantic parents that nothing had happened to me, that I had merely fallen asleep, and that I would never go out again, I savored all that I had learned that day: Monsieur Leguay had failed to defend me against Emile Vallat not because he agreed with that little anti-Semite, but because he had not wanted to draw attention to himself and blow his cover! And furthermore, by mentioning the upcoming exams in front of his maquisards, Leguay had instructed them to understand that I was not a coward for not joining them, but that I was a scholar, and that my talents were best utilized elsewhere. One day when the war ended, this country would need mathematicians. And finally, by mentioning Emile specifically, Leguay showed me not only that he had been tracking my academic progress (or lack thereof), but that he was encouraging me to work harder, to surpass Vallat and take my rightful place at the head of the class. Satisfied that I had figured everything out, I opened my books and set joyfully to the task of becoming excellent.

  The dreaded exams, when they finally arrived, were like a particularly satisfying military skirmish. I pulverized Latin, destroyed philosophy, annihilated physics, buried geography. Biology was no match for my knowledge of the physical world. History was but a fairy tale I had memorized in childhood. At last there was mathematics, the final exam of the week. I had consumed the textbook, rewritten all my notes, particularly the ones I had scribbled down while rambling the countryside with Uncle Szolem, listening to his treatises on complex polynomials. I was ready.

  The morning of the math test I put on a clean white shirt, combed my unruly thatch of hair, and walked slowly to school, savoring the gentle warmth of the late-spring sun. When I entered the classroom, Emile Vallat was already seated at the front, reeking of lavender soap. I went to my customary place at the back and began ranking freshly sharpened pencils into ascending order of length.

  Monsieur Leguay strode into the room in a swirl of black robes. Without saying a word, or even glancing at us, he rolled up his sleeves and began cleaning the blackboard. My classmates fidgeted in their seats, scratching at mosquito bites, cracking knuckles. Almond petals floated in through the open window.

  Monsieur Leguay turned around and addressed us. “Today’s examination will consist of as many problems as the class can solve in one hour.” We groaned in anticipation of some impossible task. “You may use your notebooks to work on them, but the answers must be given orally.”

  Leguay turned to the board and began filling it with mathematical symbols.

  I told myself not to be fooled by the pleasing trinity of those integral signs, which recalled to me the supple backs of girls strolling down the street on the arms of German officers, laughing over their shoulders at the tall, galumphing Warsaw Jew clutching an armful of books, his only friends in this village of monsters those same books attained at great peril from the library, which was run by a gorgon, hurrying home (along the opposite side of the street) so that his mother wouldn’t worry.

  Or perhaps I was the monster, the impossible being that should not exist, not in this world in which I would never be allowed to be good enough, never reach the brilliance (because Mother told me not to) of Johannes Kepler, or Uncle Szolem, or even Emile Vallat.

  But truthfully, ruminations aside, what Monsieur Leguay had written on the board was the worst equation I had ever seen! Nothing that Uncle Szolem and I had talked about on our rambles through the Tulle countryside had prepared me for this. Nothing that Leguay had taught us in the past year came even close. Was it a trick? I tried to talk it out to myself silently, maybe a verbal recitation would penetrate a hidden corner of my mind, but it sounded even worse than it looked: “an integral over the variables theta (from zero to pi), phi (from zero to two pi), and rho (from zero to one), with an integrand of rho squared times the sine of theta.”

  I glanced around the room to see how everyone else was doing. None of the other students seemed to know where to begin; they just sat at their desks like dazed chickens. Except Emile Vallat. His head was down, the sinews of his skinny neck strung tight as bowstrings, scribbling steadily in his little blue notebook.

  I glanced at Monsieur Leguay, who was seated behind his desk reading a local newspaper, swatting flies. This was the end for me. How would I survive the humiliation of disappointing not only my parents and Uncle Szolem, but also this man who had saved me from the maquisards?

  Just as I was beginning to ring the death knell of my career in mathematics, something strange happened to those integrals, sines, cosines, pis, alphas, thetas, and phis: they floated away from the blackboard and swam around in the air above my desk, lit by the rays of the sun, like fireflies dancing above a flower-filled meadow. I began to feel dizzy and forced myself to look away, to think of something else or, better yet, nothing, to empty my mind entirely. Quiet surrounded me like a gentle mist. I felt calm in this thought-void, in the stilling of normal perception. Then, on the very edges of consciousness, I began to detect a little flutter. Soon it invaded the edges of the non-space in my empty mind, kicking up density, curving inward, taking shape, until I finally saw something, and once I had seen it (4*pi/3, of course!), I could never unsee it.

  My hand flew up.

  Monsieur Leguay looked surprised. “What is it, Mandelbrot?”

  “Monsieur,” I began, unable to keep the grin from my face, “I see an obvious geometric solution to the problem. What you have written on the board is nothing more than
an overly complicated way of stating the volume of a sphere!”

  My fellow students held their breath. Emile put down his pencil but did not turn around to look at me.

  Leguay glanced over his shoulder at the board, then back at me. “Correct.”

  The room exploded in whispers. Emile closed his notebook.

  Monsieur Leguay erased the board and wrote out another, even more elaborate equation. It was no match for me. At this point the numbers were yielding up to me their three-dimensional counterparts as readily as foam bubbling out of a pot of boiling milk.

  “The volume of a three-dimensional shape created by a plane bisecting a cone at an angle of thirty degrees,” I announced, remembering Apollonius of Perga.

  Leguay wrote another equation.

  I solved that one, and the one after that, as my classmates stared at me, eyes wide, lips parted, marveling at the magnificent exploits of their once mute Mandelbrot. Surprisingly, they all seemed happy for me. Except Emile Vallat, who looked at me with eyes as cold and lifeless as a pair of amber nuggets stuck into the sand at the bottom of the North Sea.

  It’s just a trick, I wanted to say to him. Absurdly easy, anyone can do it.

  But I knew that it wasn’t a trick. This was it, the event that I had been waiting for: my Keplerian moment had arrived. If only Uncle Szolem had been there to witness it.

  *

  “I have something amazing to tell you!” I said as I burst through the door.

  “Go wash your hands,” said Mother as she leaned over the kitchen table. “We’ll be eating soon.”

  Only when she straightened did I realize that she had been lighting candles. Four, to be precise: one for each of us. Her gray-streaked hair, which she normally wore in a twist at the nape of her neck, was covered by a white lace shawl.