by Liz Ziemska
I prayed this would never happen to me.
Right up to the time Uncle Szolem left for Texas, he had been trying to revive and continue Julia’s work. As far as I knew he had made little, if any, progress. After my success with the sefirot-powered Koch snowflake/Peano curve tessellated two-dimensional camouflage door skin, I believed I could do better than Szolem.
I began by thinking about the definition of an iterated rational function: an equation that uses its solution to plug back into itself, like a dog chasing its own tail (and the ancient symbol of the uroboros that the Zoroastrian magi used to cast their spells). I decided that I could use this sort of equation to build volume into a shape that was similar in its camouflaging abilities to the one I had made with the tessellated snowflakes, but this time I would pop it out not only to 2-D or 3-D, but into D, the Hausdorff dimension, a dimension that was bigger on the inside than the outside, so that I could hide not just our door, not just our apartment, but our entire neighborhood, which was shaped, much like the rest of Tulle, as I had noticed on my ride home from Brive, like cauliflower.
Or, more specifically, a single floret of cauliflower, which was exactly the same shape as the vegetable itself, a small part so like the whole, a fragment, a (what would you call a partial fraction?), a … fractal.
Yes, fractal was a good name. Now all I had to do was figure out the formula for the shape of a cauliflower floret and I would soon have a template for the tessellation that I would need to create my Hausdorff dimensional neighborhood-within-a-neighborhood infolding in which to hide my parents.
Going back to Gaston Julia, I found that his basic iterated equation was: Zn+1=Zn2+c, where “c” is a constant.
Well, I would have written that same function more simply as Z = Z2+c, and in order to mimic the shape of a cauliflower floret, thereby mimicking the shape of our little corner of Tulle, what better number to use than the universal constant that rules the architecture of the pinecone, the distributions of branches on a tree, the coiling of a seashell, a fern frond, and the spiral arrangement of seeds on the head of a sunflower: the golden ratio,
which can be described by the formula following:
*
That night, while my family slept (my mind always works best when all the other minds are switched off), I once again borrowed Father’s tape measure and began plotting out my Z variables.
The distance from the center of the kitchen table to the front door.
The distance from the front door to the lobby.
The lobby door to the sidewalk. The sidewalk to the bottom of the staircase that clung to the hill upon which our building was perched, like a cairn, teeming with life.
The bottom of the staircase to Vincent’s butcher shop, then a little farther, to the bakery owned by Madame Derrasse.
The distance from the front door to the little stream at the bottom of the hill, where my parents could get fresh water in the event of an aerial bombing and a disruption of the municipal water supplies. The distance from the bottom of the staircase to the end of the alley located between two factory buildings that connected our little hidden elbow of a neighborhood to the more desirable parts of Tulle.
I took all these measurements, and several incremental others, and plugged them into Z=Z2+c, and I did it over, and over, and over again, filling pages and pages of laborious calculations, in ever widening “spirals” of Hausdorff dimension, D, reciting the entire sefirot, just to be extra sure that it would work: Kether, Chochmah, Binah, Chesed, Din, Tiphereth, Nezach, Hod, Jesod, Malchuth.
As our last night in Tulle faded into salmon-pink dawn, I added the finishing touches to a semiporous (to allow for gas exchange) fractal canopy using my sefirot dimension generator (the way I would one day use a 3-D printer to create a scale model of the fjords of Norway). Inside their mathematically enclosed neighborhood (an invisible pocket sewn into the space/time continuum), my parents would be able to move freely, while remaining completely invisible to the outside world, because it was bigger on the inside than the outside by a factor of three, as the Hausdorff dimension is more capacious than our own and also more resilient. This should have been enough, given my parents’ limited mobility. I would have liked to make it bigger, but the construction of a fractal canopy roomy enough to cover all of Tulle would have required thousands if not millions more calculations, the kinds of calculations Emile Vallat could have cranked through in no time, but Emile was my enemy and the reason why I had to build that fractal canopy in the first place. Thus life iterates onward, one mistake following another, like a dog chasing its own tail. I did leave two holes for the river to run through, so as not to inadvertently flood my creation. Hadn’t I thought of everything?
From our top-floor kitchen window, I gazed out onto the courtyard below. The same crows nested in the linden tree across the street, because they were real; but the mist curling into the creases of the foothills across the valley had more in common with a painted scrim than an actual product of nature. That said, I couldn’t tell the difference, and hopefully, neither would my parents. I only prayed that it continued to appear beautiful and rough, like nature itself. Would my construction hold until I could come home and dismantle it? I had no idea, but I put my faith in mathematics, because in mathematics there were no Jews, no Muslims, Hindus, Buddhists, Freemasons, or Zoroastrians. There were only numbers, real and imaginary, predictable and reliable, and, above all, dispassionate. Numbers did not separate mankind into artificial problem sets.
The Tulle Massacre
FOR SIX LONG WEEKS, our past and heritage obscured, my brother and I lived at a lycée located in the suburbs of Lyon, which was a beautiful city, a real metropolis, unlike Tulle, with ancient cathedrals and elaborate villas built by Renaissance silk weavers. Recently it had become infested with Nazis, so we hardly saw it. Topographically, Lyon consisted of two large hills guarding a flat plateau created by the confluence of the Rhône and the Saône Rivers: almost impossible to fractalize, it lacked the uniformly convoluted terrain that was so critical to my work in Tulle. We never went anywhere, Léon and I, except to our classes, the dining hall, the library, and the dormitory. Every day was exactly like the one before. We did well at school, but not too well. We never spoke anything but French, though I rarely spoke at all, on account of my atrocious accent. We existed in a cocoon of our own making, which turned out to be a supreme luxury, because not two kilometers away, we would later learn, at the Hotel Terminus, Hauptsturmführer Klaus Barbie was skinning and impaling prisoners (what he did if he discovered they were Jews is unprintable). Compared to the “Butcher of Lyon,” Emile Vallat was a kitten.
In late May 1944, the Allies bombed Lyon before breakfast. Our school was safe, though classes were canceled for the afternoon. Nothing unusual, seeing as we were at war, nevertheless there was a change in the air. We all could feel it, a slight but significant shift in the barometric pressure of the spirit. A little over a week later, on June 6, all students and faculty were called to an assembly in the chapel. The headmaster appeared before us and proclaimed that the Lord had answered our prayers: the liberation of France had begun. Boats laden with Allied troops had already landed on the beaches of Normandy. Soon they would purge all invaders from our soil. (Were Léon and I invaders?) Already the local Gestapo unit had decamped north to provide support for their countrymen. The war was over, said the headmaster. All classes were dismissed indefinitely. We were free to go home.
One by one, two by two, parents and other relatives arrived through the many traboules (secret underground passages built by the silk merchants) to pick up their boys, but no one came for us. Not that we expected anyone. According to the school record, we were orphans. So we packed our few belongings, changed from our uniforms back into the plaid traveling suits Father had sewn for us, and walked off campus to discover that the war was far from over.
All pretense of collaboration finally dropped, every Resistance fighter in France now crawled out of the maquis and started pi
cking off German soldiers wherever they could. The night sky filled up with fireworks, the hills reverberated with shell concussions. Under cover of all this chaos, it took us four days to walk from Lyon to Tulle. The whole time I kept thinking about the journey our parents had made at the start of the war, the distance from Paris to Tulle twice as long. We kept to the side roads and hid in the thicket during the busiest hours of the day. Panzer divisions tore up the country roads, belching diesel fumes like some Iron Age dragons. The traveling suits Father had made us provided excellent cover, as the fabric mimicked the leaf-twig-shadow pattern of the typical scrubland biome of southeastern France. Had he chosen that hideous brown-and-green plaid on purpose?
Most of the divisions were heading north in the direction of Normandy, but some, disturbingly, were heading west, in the direction of Tulle.
We scavenged food wherever we could—stolen eggs, dug-up fennel bulbs, wild cherries, edible flowers, mushrooms, and one absolutely regrettable frog (we were very hungry).
Roving bands of maquisards traversed the countryside, looking for fresh recruits. We hid from them as well, not wanting to be conscripted into their cause. We needed to get home as quickly as possible.
On the fourth day, not seven miles out of Tulle we spotted a farmer heading east with an oxcart loaded with whatever possessions he could gather and one tiny girl clutching a filthy doll. I remembered him from our family kitchen. He’d had an abscessed molar that was so firmly rooted in his jaw, Mother had to brace her foot against his chest to pull it out. We stepped out of the bushes and yelled for him to stop.
“Turn back,” said the farmer. “There’s nothing left for you in Tulle.”
Five days ago there had been an uprising by the local Resistance, he explained. They had taken over the town, taken some prisoners; there were casualties among the German POWs, perhaps too many. Two days after that, a Panzer division pulled into town. In less than three hours of heavy shelling, Tulle had fallen once again. When the Germans learned what had happened to their comrades, they went door-to-door, taking every man aged sixteen to sixty (Father had just turned sixty that year!). They were rounded up in the armory, questioned, and tortured. The Germans said bullets were too good for us, shooting too honorable a death. They stopped the hangings only when they ran out of rope. Then they packed up their casualties and headed north to join their countrymen. May they all be driven into the ocean, the farmer spit into the dirt.
“What about our parents?” I said.
The farmer shook his head. “Haven’t seen them in weeks.”
Silently I prayed that my fractal had worked.
“You were right,” Léon said to me. “We should have stayed.”
“To be slaughtered like the rest?” The farmer shook his head. “Head east to Clermont-Ferrand, there’s a train still running to Paris. The Red Cross will help you.”
At this point Léon started sobbing, and so I was able to convince the farmer to take us at least to the train station at the outskirts of Tulle. As we crested the hill, it looked as if the streets of Tulle were filled with ants scurrying around the ruins of an anthill, hurrying to save their larvae and eggs. We climbed down and the farmer turned around and rolled away without looking back.
As we walked farther into town we could see that many of the buildings had lost a wall, exposing beds, bathtubs, kitchens, entire lives to the streets below, which were covered with rubble and giant craters. Cobblestones had been crushed under the tank treads into jagged shards. The air was hazy with the stench of burning hair. Hanging above the ruins were what appeared to be dripping tendrils of rot, as if a pernicious fungus had invaded the town, covering every light post, balcony, and tree limb. Moments later, our eyes adjusted and the tendrils resolved themselves into lengths of shredded rope. I had a savage vision of Father hanging from one of these makeshift nooses, his neck bent at a sickening angle, the skin of his face stretched tight and shiny over his purple cheeks.
Someone grabbed me by the sleeve and jerked me out of that awful reverie. It was Monsieur Hubert, our neighbor, who was sixty-eight but today looked eighty.
“What are you doing here?” I asked him. He should have been inside the fractal. Monsieur Hubert never left our neighborhood.
“I left the house a week ago,” said Monsieur Hubert, “to mail a postcard to my daughter, on account of her birth.”
“I didn’t know you had a daughter,” I said, my mind spinning with the possibility that my parents had simply walked out of the protective dome I had made for them.
“The house is gone,” Monsieur Hubert said slowly, so as not to frighten me.
“What do you mean?” I stepped closer.
“The entire neighborhood has disappeared,” Monsieur Hubert whispered. “The Germans must have developed some new kind of bomb.”
“Did you find debris?” I insisted. Bombs fall short of targets all the time. My fractal wasn’t strong enough; it was more an optical illusion than a pocket universe. I hadn’t had the time to make more calculations, to bury my parents more deeply inside the Hausdorff dimension!
“No debris,” Monsieur Hubert said vaguely. “I just couldn’t find our street… .” He trailed off.
A man ran over and handed me a shovel.
“Come help us bury the dead,” he said, “before the vultures get to them.”
I handed the shovel to Monsieur Hubert, who looked at it as if I had given him a snake.
“We have to go find my parents,” I said by way of apology, and pulled my grief-stung brother after me.
I led us behind the ruins of the Hotel St. Michel (yes, there was a twinge, a secret shameful desire) and took a series of shortcuts through alleys and courtyards until we got to the factory part of town. Here in the outskirts of Tulle, where it dropped off into the steep valley of the Corrèze River, all was quiet. I counted buildings arrayed around a square in the middle of which stood a statue of the Roman goddess Tutela, protector of property and persons: one, the accordion factory; two, the tire factory; three, the lace factory; four, the glass factory … there was the alley that led to our section of town. It was clear, no bombs; no debris.
We stepped into the entrance to the alley, and the light changed register. Up ahead, where there should have been an opening, was a dead end. But this was part of the illusion, wasn’t it?
I ran to the end of the alley and put my hands on the wall. It was solid, made of the same old bricks as the factory buildings on either side. I hadn’t made those bricks. But then again, I never did look down that alley on the morning we left Tulle. Had the fractal become self-propagating?
“Where’s our street?” Léon wailed. “Wasn’t this open before?”
A dusty rag stirred at my feet and Madame Popova’s Pomeranian raised his head off his paws. He cocked his chin and blinked twice. Then he jumped to his stumpy legs and started barking. We had been friends ever since the pig snout, so this was unusual. I picked him up and immediately he stopped barking. He wriggled in my arms and I could feel that he was well fed and that his fur was clean. His eyes were bright in the dim light, and they were focused on me with an expression that said, What took you so long?
“Please tell me what’s happening here,” said Léon.
“Don’t worry, he’ll show us the way.” I set down the dog and he started scratching at the wall. I got down on my knees beside him and put my ear to the bricks.
There was something stirring inside, like a small animal trapped behind masonry. Now all I needed to do was peel it apart, like an onion. But how to unravel a fractal? I reminded myself that fractals do not glorify complexity. They are simple structures consisting of initiator and generator, iterated over and over again.
I took a deep breath and began unraveling my spell, calculating backward, reciting all ten sefirot in reverse order. The Pomeranian kept digging at the wall as if he were helping (maybe he was), and soon we were able to crawl through and come out the other side.
*
The sky i
nside my artificial Tulle was a clear, cloudless blue—no smoke, no vultures. The air felt warm and damp, like the inside of a greenhouse, and smelled a little bit like compost. Not one person walked the empty streets. Cherries fallen from the trees lay rotting, undisturbed by birds, for there were none. Forgotten laundry flapped outside closed windows. The door of the bakery stood open. Some bread had been taken, but many loaves remained, most of them covered in blue-green mold. The door to the butcher shop was closed, but it was dark inside. We pressed our faces to the window and saw a whole precious chicken deliquescing into its bed of wilted parsley. Strings of sausage hung from the ceiling, writhing with maggots.
We kept walking through the deafening silence, only the sounds of our footsteps to keep us company, too afraid to speak in case we said what we both feared: something had gone wrong here. But as we reached the bottom of the staircase, we began to hear some sounds. Could it be music? As we reached the top of the hill upon which the apartment building was perched, the music grew louder. We looked up at our kitchen window and saw that it was open, with clean white muslin curtains flapping in a subtle breeze. The mellifluous strains of Chopin’s Grande Polonaise unfolded into the air around us, Mother’s favorite piece of music.
In Memoriam
ALIETTE TAKES AWAY THE cauliflower dish and sets down a plate of freshly baked mandelbrot.
“That’s enough writing for tonight.” She places her warm hand on my bald head.
“Almost finished.” I remove her hand and kiss her palm, then take a bite of the cookie and notice the sharp aroma of anise among the almond and butter flavors. It’s my wife’s French variation, very different from how Mother used to make them. The seeds get stuck in my crowns, but I never tell her. There are many things I don’t talk about with my wife. She will know everything once I’m gone. This document will reveal to her the kind of man she married.
