Circle Limit

Contributed by Rivka Jacobs on 13/06/11

“All right class, once you’ve finished your next loop of nine chain stitches on round ten, I want you to join this to the next group of four triple-crochets with a slip stitch, like so….” Mrs. Cerussi leaned over and worked her aluminum crochet hook, then held her nearly complete yarn doily up for the students to see.

The fifteen seventh-graders in the Introductory Art Studio class were seated in old-fashioned walnut chair-desks in a small, oak-lined classroom of St. Johns Jesuit Academy. To their fore, their instructor Mrs. Cerussi, sat on the edge of a large wooden table. To their right were three sets of extremely high triple-awning windows, the glass panes of which were cranked tightly closed.

Mrs. Cerussi’s bright red mouth stretched into a smile. She patted her short, asymmetrical bob and then pushed herself to her feet. She tossed her own crochet work behind her. She grasped her hands together at the small of her back and moved forward sensuously in her high-heels and tight, knee-length grey skirt. She craned her neck while strolling up and down the rows, observing her students, whose fingers worked slowly but with great determination. She paused. She inhaled loudly as she faced a petite, chubby girl in the last row. “Rosa Teresa Mendez, whatever are you doing?” she asked.

Rosa’s finger and hook were moving so quickly that it was hard to focus on them. She was slouched over her creation, her glasses sliding down her nose, her breath coming in raspy pants. Her dark, longish hair was frizzing and flying in all directions.

“Rosa!” Ms. Cerussi repeated.

The other students dropped their hooks and lowered their doilies. A combination of snickers and whispers filled the room.

On either side of Rosa, Mary Ann on her left and Tony on her right both leaned over at the same time to see. “What is that?” Mary Ann asked, pointing.

Rosa stopped her labors abruptly and lifted her chin. She pushed her glasses back into position, then slid upright as she realized everyone was staring at her. “Yes ma’am,” she said. “I’m … I’m … crocheting….”

Every boy in the class blushed and stared; Mrs. Cerussi’s pink sweater stretched tightly over her breasts as she bent over Rosa, to get a better view of what the girl was doing. “That doesn’t look anything like a doily,” she said. “I know you’d rather be in your mathematics class, Rosa, but your parents specifically wanted you to take this course. I don’t believe you’re trying very hard to….”

“But, this is a visualization of hyperbolic planes,” Rosa interrupted, her voice high and reedy.

Her classmates burst out laughing. “She’s crazy,” someone called out.

“What are you talking about, Rosa?” Mrs. Cerussi said, her voice cold. “You were supposed to be on row ten of a doily. What have you got there? It looks like a coral in a fish tank, or the underside of a mutant jelly-fish.”

“I told you, it’s the physical realization of a hyperbolic geometric model.”

“I see….”

“You know, like Escher’s Circles….”

“Who?”

“M.C. Escher….”

Mrs. Cerussi’s features tightened and she shook her head. She turned and squared her shoulders and began walking slowly, stiffly back towards her desk. The room became silent.

“Hyperbolic space is a kind of non-Euclidean geometry,” Rosa said, her voice growing in strength. “It has a negative curvature — every point is a saddle point. Hyperbolic n-space is the maximally symmetric….”

“Rosa!”

“Look, you can visualize it using different models, but I like the Poincaré model. By using an imaginary time coordinate, the Lorentz transformation can be possibly conceived of in four-dimensional Euclidean space, but this was developed further by Minkowski, who reformulated in four dimensions Einstein’s special relativity theory….”

“Rosa, ssshhhh,” Mrs. Cerussi said, remaining calm, bringing one finger to her lips as she faced forward once more, her rear end pressed against her desktop.

“But look, look how these crochet patterns can replicate the model of a unified four-dimensional space-time continuum!” She held up the mass of wavy, bunched wrinkles of wool stitching. It seemed to glow in iridescent colors. They’d all been working in brown.

The class went “ahhhh….” together.

“Isn’t it beautiful?” Rosa continued. “Minkowski didn’t use imaginary time coordinates but showed the four variables of space and time in physical form in a four-dimensional affine space. The Minkowski model is space-like, light-like, and time-like all at once. See I decided to use the Poincaré model of hyperbolic geometric space to represent Minkowski space and visually demonstrate Einstein’s conceptions of….”

“Rosa, shut your mouth,” Mrs. Cerussi said, then pursed her lips and glared at the girl, as if daring her to speak again. “One more word and you will go see Father Sullivan.”

Rosa remained next to her desk, the toes of her oxford-clad feet turned towards one another, her blue skirt rumpled and white shirt coming untucked. Her blue blazer seemed to pull and drape in all the wrong ways under her arms and across her back. She held her construct in the palm of her right hand. It was glowing like a rainbow. “I made it, I created a physical model of Minkowski’s and Einsteins theories. There is no more parallel postulate. Euclidean space has only strictures and rules and regular dimensions, but Minkowski took the Lorentz transformation and shaped it all over again in four dimensions using Dr. Einstein’s theory; he tried to make a physical model that showed the equality of time and space. And I’ve done it. Look!” The wool thing in her hand seemed to be growing, pulsating. She was holding it now with both palms side by side.

Tony and Mary Ann maneuvered from their seats and stood. They looked at one another and then suddenly took off, sprinting between desks to the blackboard on the far side of the room.

“What the…?” Mrs. Cerussi tried to say as she straightened and raised her hands, attempting to get everyone’s attention. “Ladies and gentlemen, please, calm down. Stay in your seats. You, young lady, Rosa Teresa Mendez, you get yourself immediately to the director’s office,” she said loudly.

“Oh, it’s so beautiful,” Rosa cried out. There was a humming sound now. The weaving, unfurling, coiling accretion of planes and curves was making a noise like the lowest string of a bass viol. “It’s rotating, expanding, like Escher’s fishes and bats. It’s sitting still on my hands and it’s also moving throughout infinity….” A pinkish-orange glare reflected off her lenses; the skin of her face looked gold, her hair seemed to shine white. The deep vibrato of one base note expanded; the room was filled with the sound of a hundred double basses playing at random.

“Students, I want you to leave this room quietly, in an orderly….” But before she could finish the sentence, a high-pitched screaming began, and the stampede of heavy shoes and blue uniforms pushed and shoved and stumbled out the door. Mrs. Cerussi sidled along the edge of her desktop, held onto it for a moment as she watched her pencil holder begin to vibrate, then slide in Rosa’s direction. Pens and paperclips on desks flew into the air and floated. The pencil sharpener beside the chalkboard groaned and the handle began spinning. The metal frames of the windows bulged and moaned and the glass rattled. “My God, Jesus help us,” she wailed.

Mr. Nagy from the next room appeared in the doorway. “What in the blazes is going on?” he shouted.

“Call the police, call Father Sullivan,” Mrs. Cerussi screamed as she pulled her way towards him. Her last sight of Rosa Mendez was of a swirl of clouds and darkness and brilliant colors, and then a kind of flahing shine orbiting the girl’s body like electrons around an atomic core. She reached the hallway and Mr. Nagy grabbed her around the waist and yanked her out.

The two teachers tugged the door with all their strength; they got it closed and locked. The walls of the hallway creaked. Brightness leaked from around the hinges and jamb. The shiny hardwood floor beneath them seemed to vibrate in tune with the tonal chaos on the other side of the painted plaster.

The hall was filling, staff and students spilled from classrooms. The Reverend Sullivan, S.J. now appeared running as fast as he could toward them from around the corner of the north wing. “Evacuate,” he shouted as he moved. “Get those kids out of here!” He came to a stop some fifteen feet away as everyone raced in the direction he had come from. “Mrs. Cerussi, Mrs. Cerussi, tell me what is going on?” he shouted at her as she dashed by. He watched her recede, then briefly faced her old classroom again. It seemed to him that the entire barrier between that room and the classrooms on either side of it was dissolving, along with cork display boards and posters and paper announcements hanging from disappearing shreds of masking tape. He made the sign of the cross. Fear seized him and began to back away, then spun around and loped after the others.

The beautiful but stocky middle-aged women pivoted. “Excuse me, could you repeat the question,” she said. She smiled sweetly and gazed up at the young man in the top row of the auditorium’s tiered seats. He stood. The space was so vast that the place was amped so everyone could be heard. “I apologize. I never can get used to that sight. It’s so amazing….” She waved her hand at the immense, curved globular windows behind her, through which they could view the stunning sight of a blue and white planet earth framed by fields of stars. ”But please, go on,” she added, nodding in the freshman’s direction.

“Uh, well, I was wondering, Dr. Mendez, why you called your work, your theory, ‘Crochet-1960.’ It seems kind of strange. Like the name of a painting or a band, or something. Does the date 1960 have any significance for you? Is it a random number?”

Dr. Rosa Mendez smiled again. “Since 1960 was almost one-hundred-and-fifty years ago, I would say my use of those numbers has no intrinsic meaning,” she said. “And I’ll explain the part about ‘crochet’ some other time. Now, let’s return to our physics lesson, please. Could you all move to alter-real, block six, and will someone please tell me about Albert Einstein….”