Mandelbrot the Magnificent

by Liz Ziemska

  “That’s what happens when you go around in a curve.” The rabbi stood up.

  “So what do we do now?” I said, trying hard to hide my disappointment.

  “Remove yourself and your brother, and your parents will not be touched,” said the rabbi.

  “Will you promise to protect them if I promise to work on the sefirot?” I said, unwilling to leave it on faith, though I had already decided to leave Tulle.

  The girl came back into the room before he could answer me. She was about twelve, I guessed, with delicate features positioned in the center of a round pugilist’s head. She glanced at the diminished sugar bowl.

  “Say hello to your father for me,” said the rabbi as he delivered me to his daughter.

  *

  As I rode out of Brive into the jasmine-scented, cricket-mad dusk, I felt the suffocation of the rabbi’s study leave me. And though the realization that he had simply been humoring me still stung (he had no right to treat me like a child, distracting me with pretty mathematical toys), that thrumming presence I had “seen” in the floor of his study was real. It was not a hallucination. In fact, I could still feel a residual quiver coming from inside the breast pocket of my jacket, where the sefirot was safely tucked, oscillating harmoniously with the beating of my heart.

  I didn’t need the rabbi’s praise of my “mathematical abilities” for them to be real, either. I had my own confirmation. But why hadn’t Uncle Szolem told me that he had been working on the sefirot? Had he wanted to keep it a secret, or had he thought that I wouldn’t be able to understand it?

  How wonderful it would be if I could solve it! How old had Kepler been when he solved the problem of planetary motion? I could certainly try, as I would have even fewer friends in that school in Lyon (fewer than zero?). At least I would have no enemies and therefore no distractions. If nothing else, the sefirot and its infinite curves would make a fine new shape for my Book of Monsters.

  And shapes were everywhere that evening: in the parabolic swoops of insect-chasing swallows, in the roiling buttermilk curdles of the clouds, in the bobbing, windswept heads of chamomile daisies, as dense and golden as a teeming beehive. And as I crested the hill above the town I had gotten used to calling home, I could not help noticing that the topography of Tulle, with its brainlike infoldings and outcroppings, resembled the Romanesco cauliflower Mother had served for dinner.

  Garments of Concealment

  WHEN I FINALLY ARRIVED home that night, Mother was sorting through our clothes, picking out the most essential items to bring to our new school. Léon was in the living room composing a good-bye letter to some girl he had met in the village (where did he find the time for girls?). Father was at the kitchen table sewing buttons onto matching plaid suits that were supposed to transform us from a pair of Polish immigrants fleeing deportation to young noblemen traveling for pleasure. No one bothered to ask me how it went with the rabbi, which offended me.

  I sat down on the couch next to Léon and pulled out my sefirot. I began to stare at the ten nodes and twenty-two interlacing pathways, willing them to unspool a puddle of magical magma onto the floor of our humble apartment, hoping to show them all, but the incident that had occurred at the rabbi’s house failed to repeat itself.

  “What are you doing?” Léon nudged me. “Shouldn’t you be helping Mother?”

  “Shouldn’t you?” I said, turning away from him.

  Why hadn’t it worked? Had it been the rabbi who had performed the miracle of the curves and splashes? Had he tricked me into thinking I was special and therefore “worth saving”? And then I realized what was missing: In the beginning was the Word. Sound was missing, the rabbi’s sonorous recitation of the names of the sefirot, but on my copy they were written in Hebrew and I could not read them.

  I glanced up at Mother, who was now standing in the kitchen pulling together another meal out of last night’s leftovers. She spoke French, Polish, Russian, and a little German, but she was a scientist, a secular intellectual, and despite the Sabbath prayer that had rolled so easily off her tongue, I doubted that she could help me. Then I looked at Father, who had just bitten the thread attaching the last horn button to my coat of many colors. Maybe he had picked up some Hebrew during his merchant days in the Warsaw Ghetto.

  I went over and showed him the paper. “Rabbi Feuerwerker gave me this diagram, but I can’t read it.”

  “Let me see.” Father put down the needle and picked up a pencil. Without hesitation, he drew a fresh sefirot on the back of the page and began to fill in the name of each sefira, translating from the Hebrew as he wrote each one: “Kether (the Divine Crown), Chochmah (Wisdom), Binah (Understanding), Chesed (Mercy).”

  I could feel the vibrations through the table as the pencil scrawled over the page. Couldn’t he also feel them? I was too afraid to ask, in case I got sent to bed with a hot-water bottle. “How do you know all this?” I asked instead.

  “Seven generations of Mandelbrots were Talmudic scholars in Vilnius,” Father said as he continued to fill in the nodes with Gewurah (Justice), Tiphereth (Beauty), Nezach (Eternity), “but when the family moved to Poland, we lost some things. Or let them go.”

  “Who taught you?” I asked.

  “My father,” he said as he kept writing and the table kept shivering: Hod (Glory), Jesod (Foundation).

  “Why didn’t you teach me?” I asked.

  “It didn’t seem like useful knowledge for the modern world,” Father said as he filled in the bottom-most node, Malchuth (G-d’s presence in the World). “You should learn how to sew. I can teach you.”

  Ignoring his kind offer, I kept my focus on the drawing. “So what is this thing?” I asked, wondering if he knew more than the rabbi.

  Father picked up the pencil again and began drawing curves. “That’s a very difficult question to answer.” Dozens and dozens of curves spun out of his pencil, overlapping, interlacing, so many curves that I began to worry that the tabletop might dissolve at any moment and we would fall through. “Some people believe that the sefirot is an instrument by which the ten aspects of the Divine can be perceived in the physical plane,” Father began, “but my father taught me that these”—he tapped at the tangle of curves with the point of his pencil—“are actually Garments of Concealment.”

  “Concealing what?” I asked.

  “The Most Hidden of the Hidden.” Father looked at me as if I were slow.

  “G-d?” I said.

  Father shrugged. “That’s a simple word used by those who would be terrified if they knew the whole story.”

  I nodded, having experienced a tiny bit of that terror in the rabbi’s study. And then I had a vision of a delicate moth concealed by the clever pattern of its own wings, hidden in plain sight against the mottled tree bark by layers and layers of overlapping color, each tiny spot of color so like the whole.

  I looked closer at the sefirot. Though it was obviously just a diagram of an esoteric symbol, could it be that these ten nodes that had sprouted ten to the tenth to the tenth (ad infinitum) number of curves, like a pond in a rainstorm, were just a shape monster, another version of the Koch snowflake or Sierpiński triangle, a geometric creature that depended on the Hausdorff dimension for its prodigious proportions? Is that what I had seen in the floor of the rabbi’s study, the Hausdorff dimension in action? Then what was the sefirot itself? A diagram for the structure of the universe? A map that showed the way back to Paradise? A puzzle box containing G-d?

  “If this drawing was the blueprint for an engine,” I asked, “which of these sefirot would be the switch that turned it on—Kether, the Divine Crown? Chochmah, Wisdom?”

  Father pointed to the very center. “Tiphereth, Beauty.”

  “Why beauty?” I asked.

  “Because beauty combines compassion with strength,” said Father. “It is the sefira of transmutation.”

  “Like the transmutation of a two-dimensional object”—I tapped the drawing—“into a multidimensional Garment of Concealment?
”

  Father reached up and stroked my face. “You’re a good son.”

  Not as good a son as you deserve, I wanted to say to him, but if I said it, I would cry, and there was too much work to do for that sort of thing. Instead I asked, “May I borrow your tape measure?”

  *

  That night, while everyone slept, I went out into the hallway and contemplated the entrance to our apartment from the point of view of someone standing on the landing. I ran my hand over the door, collecting dark green paint flecks on my fingertips. I memorized the patterns of the water-stained plaster of the wall beside it, noting the traces of countless grimy hands, the bits of straw sticking out from the wattle. I could see the pattern in the pseudo-randomness of the stains and chips. Any moth could mimic it.

  I went back inside and pulled out my Book of Monsters. Flipping through the pages, I found the chapter on tessellation, the process by which a shape is repeated over and over again (iterated) to create a plane without any gaps or overlap, sort of like laying tile. In order to mimic the dirt patches of the walls in the hallway, I thought it would be best to use Koch snowflakes of two sizes and two degrees of complexity as my “fundamental region.”

  Then I could fill each snowflake, at random intervals, with five different degrees of the Peano curve.

  If I iterated these shapes over the entire surface of the door, it should blend perfectly with the surrounding wall. In theory.

  No, there was no time for self-doubt!

  I measured our door and plugged the dimensions into my perimeter equation (the equation that told the Koch snowflakes when to quit making more snowflakes). Then I performed three straight hours of calculations, making sure that the tessellations fitted together perfectly, filling them in with variegated Peano curves. Then I rolled up the tape measure, gathered my pencils and papers, stepped back into the apartment, closed the door, and whispered one magical word: Tiphereth.

  Nothing happened. The heavens did not open up, lightning did not strike, G-d, “the Most Hidden of the Hidden,” did not speak to me from His primordial broth. And yet, I was too superstitious to open the door and see whether or not my mathematical spell had actually worked. I forced myself to trust myself, the way Father seemed to trust me now, which was amazing to me, because I could count the number of our significant interactions during my entire childhood on one hand. Father had always been too busy making money to pay attention to me. He had done it for us, so that even within the confines of the Warsaw Ghetto, we had everything we needed. Now it was time for me to do something for him.

  *

  They arrived first thing Sunday morning, while the rest of Tulle was still at church. Mother saw them from the kitchen window: four German soldiers climbing the hillside to our building, led by a small dark boy with the narrow face of a fox.

  Was it perverse kindness that had inspired Emile Vallat to give us the Sabbath? For he certainly could have brought them to our house on Friday, right after the math test. Or did he take the precaution of first going to church and receiving the sacrament before committing the sin of consigning innocent people to their deaths? Either way, we were trapped. Our apartment building was perched at the very top of a hill, at the very edge of the town. The only way out was the same steep and narrow path that was now blocked by men in black uniforms and swastika-emblazoned armbands.

  “I knew this day would come,” Mother said bitterly. The look of genuine despair on her face almost made me lose my nerve.

  “You did this to us!” Léon looked at me with hatred.

  “Hush!” said Father.

  “We have to stay calm,” I said, trying to believe myself.

  “We have to hide!” Léon cried as he started pushing the couch in front of the door.

  “Get in here!” Mother grabbed me by the sleeve and tried to stuff me into the broom closet. “We’ll tell them we have no children. Maybe they’ll just take us and go.”

  “Maybe we can climb out the window,” said Léon as he shook out his bedding and started tying sheets together.

  “It’s six stories, we’ll never make it.” Father took the sheets away and refolded them. “Your brother has a plan, don’t you?”

  I nodded at Father, pushed the couch back into place, and made them sit down.

  We heard a door slam below.

  Madame Popova’s Pomeranian began yapping hysterically.

  “They are coming up the stairs!” said Léon as he tried to get up.

  “We need to keep very still,” said Father as he grabbed Léon’s hand and pulled him down.

  My family watched me in terror as I got up and walked over to the door—the sound of military boots growing louder than the yapping—and opened it carefully. Monsieur Hubert hated to throw things away, so in addition to his rusty bicycle, I found a pair of old galoshes from the pile of debris by his door, also a shovel, an empty rabbit cage, and various other odd bits to drag in front of our door. Then I closed it, making sure once again not to look.

  “We’re hiding behind our neighbor’s trash?” Léon wailed.

  “Shhhh!” Father whispered.

  Mother’s lips were moving in prayer, but there was no sound coming out of her mouth. I sat down between my parents and took their hands. Mother’s fingers were trembling, but Father’s hand was warm and dry. He believed in me.

  The soldiers arrived at the landing just outside our door. We could hear the low murmur of their voices, the clink of their guns as they examined Monsieur Hubert’s door, checking the number against the one they had been told contained Jews. They knocked anyway. Thank God Madame Popova had recently convinced our neighbor to start attending mass, for appearance’s sake, as Monsieur Hubert was a devout atheist. The Nazis didn’t care for Catholics, but they hated Communists almost as much as they hated Jews. We waited for the pounding on our door, but nothing happened. Where was Emile? I could not sense him out there. No doubt he had run home after leading the Nazis to our building. Coward.

  The Pomeranian kept yapping. The clock above our kitchen sink ticked loudly. Would our breathing give us away? Would the enchanted tessellations hold? Had I missed anything in my calculations?

  The squeak of boots; the clink of metal against metal.

  “They’re leaving!” Léon whispered.

  “No, they’re just going down one flight,” I whispered back.

  Mother cried softly as they came back up and then climbed one flight above ours. The attic door opened with a screech, slammed shut. We could feel more than hear as they tore the place apart, searching for us. Then the soldiers came back, cursing, joking, “Der kleine Scheiße Kopf,” and clattered all the way down the stairs to the ground floor, out the door. Only then did Madame Popova’s Pomeranian stop yapping.

  We sat still for a moment, relearning how to breathe.

  “Stupid Germans.” Léon laughed with relief, tears in his eyes. “They don’t know how to count.”

  Oh, they do, numbers are their specialty, but I have disappeared our door.

  Mother regained her voice. “Get ready to leave tomorrow at dawn.”

  “Yes, Mother.”

  The Fractalist

  I HAVE OFTEN WONDERED why the soldiers did not simply break down our neighbors’ doors and search for us when they could not find our apartment number. My only guess is that they were acting on orders not to antagonize the civilian population, to keep them calm, complacent.

  That would all change in a matter of weeks.

  After the Gestapo left our building, I had renewed faith in my ability to protect my parents using my mathematical skills. Only now I had to build a different kind of mathematical camouflage, something that would protect them for weeks and months, if not years, and yet allow them to move around a bit in the neighborhood, as Léon and I would no longer be there to do the shopping for them.

  After the great starvation of the previous winter, our neighborhood gathered together and started making plans for a longer war. Vincent the butcher acquired a pregn
ant sow and now there were piglets, six of them, too adorable to contemplate eating, but this was something I could not think about now. Madame Derrasse kept five chickens under lock and key in a roomy mesh enclosure behind her bakery. They were good egg layers, secure in their avian knowledge that they would never be eaten. Madame Derrasse’s niece kept rabbits. Everybody had a vegetable garden these days, from the poorest farmer to the most exalted aristocrat. The basement of our apartment building was stocked with root vegetables, jars of preserved tomatoes, and several hefty blocks of burlap-wrapped cheese made from the milk of a cow my parents would no longer be able to access. Like the Tatars who were our distant ancestors, we had laid by several sacks of dried apples. The stream beside our house ran clear with cool water from which an agile fisherman could pull the occasional silver-mottled trout. In the few hours remaining before our dawn departure, I had to do all that I could to protect our fragile Eden.

  Once again I reached for my Book of Monsters and reread my favorite part of the Poincaré epigraph:

  Logic sometimes makes monsters.

  That Emile Vallat hated me so much for being better than him at math that he had denounced me to the Germans was logical, but monstrous. That the German people who suffered so much after losing the previous war would choose a charismatic leader like Hitler who promised to return them to their former international status was logical, but monstrous. That the Jews were blamed for everything that went wrong in the world … well, that’s where logic broke down and became doubly monstrous, or monstrous2. In any case, this was not why I had turned to my Book of Monsters. What I had been searching for were my notes on Gaston Julia.

  Gaston Julia, who was still alive at the time of these events, was a French mathematician whose nose had been blown off during the previous war. Despite the inconvenience of having to wear a mask over his mutilated face, he went on to write, at the tender age of twenty-five, a 199-page essay entitled Mémoire sur l’itération des fonctions rationelles, describing the iteration of rational functions. This had been Uncle Szolem’s favorite piece of writing. Julia became so famous for his essay that he was awarded the Grand Prix de l’Académie des sciences. But despite his early fame, his work was largely forgotten.