by Liz Ziemska
One weekend, not long after we arrived in Tulle, Uncle Szolem brought his family to our house, and before I could drag him off to share my latest perfect score, he took my parents into the kitchen and closed the door. When they came out several minutes later, Father looked stricken.
“Your uncle is leaving for America,” Mother announced, her expression unreadable (which was odd, because her mobile face never hid how she felt).
“We’re going to America?” Of course I assumed we would follow him, as we had done before.
“Not us.” Mother shook her head.
“But why?” Who would talk to me about mathematics, my favorite subject, if Szolem left? Not Father, not Léon, certainly not anyone at school. Mother cared only about practical calculations, like the number of molars in an adult mouth or the exact volume of buttermilk necessary to make one pound of farmer cheese.
“Tell the boy,” Father said to Szolem, an edge to his voice I had never before heard.
“It’s just a matter of time before they push me out of the university,” said Szolem. “The war isn’t ending as quickly as we thought. Things are about to change for us.”
“Us,” I repeated, but no one bothered to explain. They didn’t have to. We Jews were in the way again, just as we had been in Warsaw. No matter how hard we tried to fit in, we took up too much space, attracted too much attention.
“We will see each other soon,” said Aunt Gladys through a smile that did not hide her true emotions: she felt guilty. Cousin Jacques clutched his mother’s skirt, his eyes round, couldn’t understand why the grown-ups, normally so cheerful when they were together, were now so unhappy with one another.
“Why can’t we go with you?” I insisted.
“I have been offered a teaching position in the Department of Mathematics at the Rice Institute, in Texas,” Uncle Szolem said calmly. “They are sponsoring my visa.”
I looked at Father: Say something.
“I have a son,” Szolem added, looking away. “I have to think of his safety.”
“What about my sons?” Father snapped back before Mother could shush him.
Sorting out my feelings, I stood there comparing Father and Uncle Szolem, two brothers so alike, with their high foreheads and small, regular features, except that Szolem was free from Father’s stigma of perpetual worry. A freedom that came from his superior education and had been made possible, I reminded myself, by Father’s sacrifice.
A circle is a curve with one focus. An ellipse is a curve with two foci. Torn between the twin foci of Father and Szolem, I wobbled.
Dinner that night was spoiled. Hovering over the table was the growing sense that we might never see one another again. Yet around Uncle Szolem there was a flickering flame-colored aura, as if he were pretending to be depressed about his imminent departure, but in truth he was bubbling over with excitement. (Had anyone else seen it? Was I alone?)
As they prepared to leave, Uncle Szolem took me aside for a little pep talk. “Poincaré said that in most fields a person can be trained to become an expert, but mathematicians must be born,” he said, beaming conviction. “Never let anything, or anyone, get in the way of your genius.”
It was unsettling to be in his presence now, though I should have been pleased. How far had I traveled in Szolem’s eyes, to arrive all the way to “genius” from “childish”? Had I really changed that much, or was it Szolem who had changed? At this close range, I could see for the first time something missing from his shiny, unlined face, something that Father had but my uncle did not: dignity. It was obvious to me now that Szolem was leaving France not for his son, but for himself, because an American university had wanted him enough to rescue him from Nazi-occupied France. He was more than just flattered. It was exactly this consciousness of his impending fame that set his eyes ablaze.
“I won’t forget.” I turned away so as not to let him see the pain I felt at the fall of my idol.
The Book of Monsters
IN LOSING UNCLE SZOLEM, egotistical as I now knew him to be, I had also lost my mathematical sparring partner, so I decided to brave a visit to the public library to see if there were any books that would help me build my knowledge on my own. This was the first time I had ventured out alone, as my accent was still terrible. (Mother had informed the headmaster on the first day of school that I had a debilitating speech impediment, but she couldn’t announce that to the whole village.) As the library’s original building had been turned into a barracks for the small contingent of German soldiers stationed in our town to “keep the peace,” all the books had been moved to the top floor of an apartment building. When I got there I found Madame Vallat, the librarian, sitting behind an old farm table with Monsieur Ricard, the owner of the best hotel in town. They were deep in conversation and hardly noticed me when I came in. I went directly to the stacks and quickly found, to my delight, two entire shelves of books on various mathematical matters. I should not have been surprised given that Emile Vallat, the human calculator, my classroom nemesis, was the librarian’s son. I chose three books and brought them to the front desk. As I stood there patiently waiting to be checked out, the librarian and the hotelier continued their argument.
“The Allies only want our interests in Syria and North Africa,” said Madame Vallat. She was fair and plump, despite the food rationing. A widow of the last war, somewhere in her early forties, she looked nothing like her son. “Better to stick with Germany.”
“If you think Hitler wants France for anything but her resources, you are delusional, Violette,” said Monsieur Ricard. He was a fine-boned man in his early sixties, wiry and dark.
“Things will be better when we are allowed to do what Germany has done,” said Madame Vallat. “If we don’t cleanse the country of all foreigners, Freemasons, and Jews, we French will disappear from France like the Gauls. All that will be left from our culture is the word merde.”
Heart pounding, I placed the books on the table. Madame Vallat looked me over, visually measuring the width of my temples, the length and slope of my nose. My skin began to itch, as if there were a spider crawling over it.
“How may I help you?” she asked in deliberately archaic French. I was afraid to open my mouth and give her a blast of my Cockney French. Would she denounce me, the way her son had done in class?
“Leave the boy alone.” Monsieur Ricard pushed my stack of books closer to Madame Vallat.
Never taking her eyes off me, she stamped all three and pushed them back.
“Shoo,” said Monsieur Ricard.
“Merci,” I mumbled, tucking the books under my arm and forcing myself to walk slowly down the stairs. Out on the sidewalk I felt nauseated from the knots in my stomach. Was this how Uncle Szolem had felt every day at the university? Did he also encounter people who did not like how he looked? I felt pity for him now, and shame at the cold and formal way I had treated him when we said good-bye.
*
Later that night after dinner, I sat down and started looking over my books, so perilously acquired. There was one volume on advanced calculus, another on infinite sets, and a third entitled The Book of Monsters. I opened it and on the flyleaf encountered a quote from Uncle Szolem’s favorite mathematician:
These functions are an outrage against common sense, an arrogant distraction. But logic can sometimes make monsters, and it is the beginner that would have to be set grappling with this teratologic museum.
Henri Poincaré, 1899
Teratology, I knew that word from Mother: it was the study of congenital abnormalities, like the two-headed calf. Terato, Greek for “monster” or “marvel.” I flipped through the book and encountered not monsters, but many beautiful shapes. Like the snail coil of the Archimedean spiral:
The crystalline elegance of the Platonic solids:
But then things got really strange with Cantor dust, which looked like the tallit prayer shawl worn by Father on Yom Kippur, with its twined and knotted tzitzit, a reminder of our obligations as Jews to heed G
-d’s commandments:
And the Koch snowflake, which looked exactly like a snowflake in its middle stages, but in its early and late stages began sprouting triangles, which sprouted more triangles, and more triangles, until the triangles became movable arms that could manipulate space and time:
Then there was the Sierpiński triangle, which looked like a geometric Cronos that had eaten up its tiny offspring and now had a stomachache from all those pointy apexes sticking into its intestines:
I stayed up all night thinking about those infinitely nesting triangles (like those Russian dolls Mother used to have in her dental office to amuse her younger patients while she drilled their teeth). I vowed to master The Book of Monsters, no matter how long it took me. It would be my atonement to Szolem for accusing him, in my heart, of selfishness and vanity.
The following morning I went off to school feeling light-headed from my sleepless night, but not tired. Everything I saw—the trees, the buildings, the hills surrounding the town—had an extra shimmer of possibility around it. How wonderful the world was, how full of mystery! Maybe today I would stop cowering in the back of the classroom. Maybe if I said something brilliant, Emile Vallat and his friends would overlook the shape of my face. Perhaps we could even start doing homework together.
The school day went by slowly, medieval history, Latin, and zoology of no interest to me whatsoever anymore. Finally Monsieur Leguay came into the room. He lectured for a bit on how to find the volume under a curve, all very basic, reasonable stuff. I didn’t even bother to take notes, but when he finally put down his chalk and asked if there were any questions, I quickly raised my hand.
“Could you please explain the Sierpiński triangle?” I was showing off, I won’t deny it.
Monsieur Leguay looked surprised and thought for a bit. “No,” he finally said. “That subject is beyond the scope of this class.”
I was disappointed but knew better than to insist. In those days, students did not challenge their teachers as they do now.
“Any further questions?” Monsieur Leguay looked around the room.
Emile Vallat, who had been staring at me the entire time with his little amber fox eyes, now raised his hand. “Could you please explain, Monsieur Leguay, why it is that Jews in the Free Zone are not required to wear yellow stars on their clothes?”
The entire room went silent, so silent that I was certain they could hear my heart beating through the thick wool of my jumper. I kept my eyes on Monsieur Leguay, wondering what he would do. Would he defend me? Would he order Emile out of the room? Monsieur Leguay kept staring down at his desk. His face grew pink, then red, and then completely white. The bell rang. All the boys grabbed their books and ran out the door into the courtyard, where the ordinary world, as I could see through the classroom window, had not lost its radiance.
I remained at my desk, hoping for a word from Monsieur Leguay, any word at all, but he just packed up his bag and walked out of the room. I was crushed. Another door had slammed. What had I hoped for? An apology? More praise? A replacement for Uncle Szolem? I could almost hear his voice in my head: That is a childish dream.
The Hausdorff Dimension
AFTER THE INCIDENT WITH the yellow stars, I vowed never to speak in class again, though I continued to turn in my homework, as I had always done. If Monsieur Leguay was one of the poisson pourri, as Mother called the French collaborators, then I did not want to attract any more attention in his class. I stayed home after school with The Book of Monsters, which I had copied out by hand, like those Talmudic scholars in Vilnius, before returning it to the library. Deeply absorbed with my pathological shapes, I paid very little attention to the pathological events of the “real” world.
When Germany invaded the Soviet Union in the summer of 1941, Mother became almost cheerful. “Russia kicked Napoleon in the tuches,” she said, abandoning her proper French for one choice Yiddish word, “now let’s see what she will do to Hitler.”
“Things will be easier for us once German troops are pulled out of France and sent east,” Father agreed.
We had heard about the Jewish doctor living in a nearby village, the one who had been denounced by a jealous rival and deported to a concentration camp. For this reason, Mother had been hesitant to resume her practice. Emboldened by the news of the Soviet invasion, she took out her dental equipment for the first time since we came to France, crossed the landing, and knocked on the door of the only other apartment on our floor. Monsieur Hubert, a retired engineer, lived there all alone. He had been complaining of toothache to his downstairs neighbor, Madame Popova, a retired dance instructor. We could hear everything through those flimsy walls, every marital squabble, every sneeze (and many other things, too). Mother pulled Monsieur Hubert’s tooth free of charge, then she went downstairs and pulled one from the odoriferous mouth of Madame Popova’s ancient Pomeranian.
Word spread of her skills, and soon Mother was on a first-name basis with Vincent, the butcher, who had terrible gums. He set aside special cuts of meat for us. Chickens were out of our reach, but a rooster could be made palatable if cooked long enough with tarragon, peppercorns, and onion.
Madame Derrasse, who owned the bakery, received a new bicuspid made of gold melted down (by Father, with a desktop smelting forge fashioned by Monsieur Hubert) from her wedding band (her husband had passed away during the middle years of the Third Republic). From her we received every Friday a loaf of braided brioche studded with brandy-soaked raisins.
As my parents grew more comfortable going out into the village, I was drawn inward, to my little corner of the living room and the book of mathematical shapes that I had copied out by hand. One night, after several weeks of fruitless puzzling, I began to sense that I was on the verge of a breakthrough.
After gulping down the oxtail stew Mother had set in front of me, I began to etch the Archimedean spiral in the thin brown gravy with the tines of my fork, counting out the Fibonacci sequence as I drew the overlapping curves: 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13, 13+8=21. I could feel the brittle surface of the plate becoming spongy, the fork piercing through the flat plane of the tabletop, my fingers sinking down into an undiscovered realm of pure … until Mother took my plate away.
“You’re too old to play with your food!” she said as she dropped it into a pail of sudsy water.
The spell was broken, and yet the feeling lingered, that indescribably delicious sensation of penetrating a new realm. That chipped blue plate had been given to us by Aunt Gladys before she left for Texas. It had ordinary, real-world dimensions. How was it possible that I could stick practically my entire hand inside it (perhaps even my whole body)?
The kitchen was hot and stuffy. The backs of my legs stuck to the chair. I left the table and hurried back to the corner of the living room, pulled out The Book of Monsters, and flipped through the pages until I found what I was looking for: the Hausdorff dimension. It’s what all the monstrous shapes, including my dinner plate, had in common.
Hausdorff’s innovation was a simple one. Previously, the concept of dimension had been thought of as something that extrudes into space and could be described by a set of coordinates, x, y, z. A point (x) has the dimension of one; a line (x,y) has the dimension of two; and a cone, cube, or sphere (x,y,z) has the dimension of three. This is the ordinary kind of dimension, the Holy Trinity of Euclidean geometry. It is called topological dimension, or DT.
But when Felix Hausdorff encountered the geometric monsters, he quickly realized that DT would never work for them. For instance, if the Peano curve (another shape monster) is like a pot of soup to which you can keep adding handfuls of noodles, what happens to the hundredth handful? The hundred thousandth handful? All those noodles can’t go into the third dimension because the Peano curve is two-dimensional, and the noodles can’t go into infinity because the curve is bound at the top, sides, and bottom by the square (or whatever other shape) that it inhabits. It defies logic and all known laws of the universe. And yet the noodles must go so
mewhere… .
Frustrated by the limits of mathematics, Felix Hausdorff decided to create a new kind of dimension, one that went in instead of out. He wasn’t the first person to think about dimensions in this way. The Oxford mathematician Charles Dodgson, better known as Lewis Carroll, had the same ideas. Only after Alice drinks the potion and becomes small is she able to open the door and enter the garden.
The Hausdorff dimension, D, is that garden.
D is where I would one day hide my family.
Life Under the German Occupation
WHEN THE ALLIES GAINED a foothold just across the Mediterranean in North Africa, Germany abandoned the pretense of the Free Zone and moved south to occupy all of France.
To commemorate the day, Emile Vallat (at least I assumed it was him) left a pig’s snout on my chair. Did it hurt me? Oh, yes, it did, because I had still hoped we would become friends one day. We both loved math—wasn’t that enough to cancel out our differences? Apparently not. The pig snout made me feel queasy, but I tried not to show any sign of weakness, so I forced myself to stand there and contemplate the offering. Then I actually began to admire the tender symmetry of the fleshy circle with its delicately flared breathing holes. With a handful of dried peas and a yellow onion it would make a fine soup. But not for us.
I am impervious to your taunts, I wanted to say to Emile. You have your handful of cronies, but my friends are numbers and they are legion. I tore a page from my notebook, wrapped up the snout, and tucked it into my satchel. Later that day, I fed it to Madame Popova’s Pomeranian. The dog chewed the snout ecstatically with its remaining teeth, while the mistress (blond, berouged, seventy) fed me stale caramels and complimented me on the width of my shoulders.
A Kommandatura of veterans from the last war and wounded soldiers shipped back from Russia moved into Monsieur Ricard’s Hotel St. Michel, along with a small Gestapo division. As a gesture of goodwill, their senior officer, a dentist from Bonne, began offering free dental care. We heard about this from Madame Derrasse, who came to our house to deliver some plain country bread (no more braided brioche, too risky) and to assure us that no one but Dr. Mandelbrot would be allowed to touch the mouths of the inhabitants of our little corner of Tulle.
