The Vanishing Point
“The universe is passive aggressive” were the last words Triglyf Loench uttered before collapsing to cries of “Oh, my God!” from the horrified students filling the lecture amphitheater. For nearly three hours, he had scribbled equations across the huge blackboard, beginning slowly like a jazz musician and building to a crescendo of wild ecstatic improvisation, erasing the cascading numbers to make room for more, while everyone from math major to PhD candidate took pictures with their cells and tried to follow the most amazing mathematical performance since the publication of the Baklanov Equations in 1937.
As a dozen cells called 911, Carolyn Seepers, a thin, driven woman who had transferred from the College of Nursing after sitting in on one of Loench’s lectures, held his mouth open to keep him from swallowing his tongue. It took the emergency squad forever to arrive. Many were weeping, many were nearly mad with frustration that he had collapsed before completing the most provocative equations ever written.
His hand still gripped the chalk stub, his lips quivered, but he could not speak. The paramedics lifted him gently onto a stretcher and carried him away. No one could complete his work; no one would ever see the vision that struck him dumb. Like Shakespeare, like Freud, like Baklanov himself, Triglyf Loench had no successor.
Until now. I am Walter Ecriveur, PhD, one of the last of those golden acolytes who were with him when he achieved enlightenment. Except for Carolyn, the others left their novitiates to write algorithms for stock traders or to develop killer apps for the selfie public.
It is difficult to visit a man in skilled nursing who once embraced the universe but now can’t press a call button. Yes, Professor Loench had survived his encounter with the act of creation, but like Fermat, who died before he could write his famous proof, was unable to share his vision.
Just a few days before he collapsed, he had agreed to supervise my dissertation on the infuriating Baklanov Equations. I was intrigued by Baklanov’s Seventh Equation, which, as I read it, suggested that Pi was not an infinite number. The problem with the equations was Soviet mathematician Dmitri Sergeiovich Baklanov (1892-1938?) died before he could demonstrate they had any application to the real world. During the Great Purge Stalin had him shot for writing a mathematical parody of the Soviet Union. Like his equations, however, even Baklanov’s death is not well understood. After the collapse of the Soviet Union, documents were uncovered in a secret archive suggesting Stalin had kept him alive until 1947, when he was shot for refusing to work on the Russian atomic bomb.
So after Professor Loench’s stroke (if it was a stroke), I changed my dissertation topic to solving Loench’s Blackboard, what the mathematical world now called the overwhelming series of equations he had spun in his last wild improvisation. Even if I failed, I would join a group of dedicated theorists who would not rest until it was understood. In mathematics as in medicine, it is sometimes necessary to rule out incorrect solutions to determine the correct one. During that dazzling and terrifying last class, I had detected threads of Baklanov’s Seventh and Ninth Equations intertwined on Loench’s Blackboard.
Efforts to reach him through whatever neurological event had robbed him of speech and motion were hopeless. The best neurologists at City Hospital followed by the best at the Mayo Clinic were unable to reach a diagnosis, much less develop a treatment plan. The closest was a psychiatrist at the Mayo Clinic, who opined it was an extreme case of Baklanov’s Syndrome, an amalgamation of bizarre symptoms that afflicted mathematicians who succumbed to the lure of the Baklanov Equations. There was no known cure.
I was the only one in the department who had the patience and could endure the anguish of sitting beside a man who had scoured the universe for its secrets only to be struck dumb before he could reveal them. So as faculty and students tried to decipher Loench’s Blackboard, I would take their work to the Professor, place a lap top on his tray table, and let him watch the equations stream across the screen. When his eyes grew dull I knew the writer had failed, much as generations of mathematicians failed to find a proof for Fermat’s Last Theorem until nearly 400 years after his death. Many, like Carolyn Seepers, were dejected by what they perceived as a personal rejection when he did not affirm their work.
“For a second he brightened, but then it was like he flared out,” I told her. Nevertheless, this was the most positive response anyone had received from him.
But Carolyn was more than dejected. She was devastated. I tried to console her at Zenos’s, the bar favored by math majors in the strip of second hand bookstores and Chinese restaurants across from City University. She thought she recognized a symbol from Baklanov’s seemingly impenetrable Third Equation in the last section he erased from the blackboard before he collapsed. Like a great poet inventing a word, Baklanov invented symbols to express concepts necessary to understand his equations.
In 1967 Milton Deupree at the University of Chicago had speculated this particular symbol might represent an algorithm to reduce incomprehensibly large numbers to manipulable comprehensible segments much as travel to the stars is measured in light years, so that one might speak of a number to the power of so many Baklanovs. Until Carolyn, however, no one had been able to confirm this because no one could conceive of numbers so large they called for a Baklanov Compression.
“He lit up when he saw it, he really did,” I said to console her.
She looked so sad and helpless I wanted to touch her, but graduate students rarely ask each other the probing questions such actions now require.
“But then he drifted off,” I lied. In fact his features relapsed into hopeless despair. “At least for a moment you spoke to him.”
I felt like a long term care nurse telling a patient’s daughter her father lit up when he heard “I’m Nothin’ but a Hound Dog” on an iPhone but slipped back into senility the instant the song ended. She sniffed and tried courageously to sip her beer. This was as much encouragement as mathematicians ever give or get from one another.
“I don’t drink much beer,” she said pushing the glass away.
“I have some Bach CDs at my apartment,” I said, as amazed at my temerity as she. Next only to Baklanov, Bach intoxicated mathematicians by his effortless exposition of every conceivable aspect of a theme with a mathematical precision almost beyond human capacity.
“That would be nice,” she said.
So we walked back across campus to my apartment and sat on the floor listening to a recording of The Art of Fugue. Carolyn was slowly reviving until it stopped abruptly with the unfinished Contrapunctus XIV. She broke into tears.
“He died before he finished it,” she sobbed. “Just like Professor Loench.”
“Loench isn’t dead,” I said and suddenly I understood.
“Listen, Carolyn, Bach knew how that was going to end the same as Fermat knew the proof for his Last Theorem before he died. Every mathematician is always several equations ahead of his writing. Loench knew exactly where he was going. Something stopped him from completing it.”
She stopped crying and looked at me as if I were a genius.
“So when he said ‘the universe is passive aggressive,’ he was talking about something he saw ahead but hadn’t written,” she said.
“Yes. And when he froze up looking at your work, it’s because he was afraid you were going to find out why the universe is passive aggressive.”
We just looked at each other, and then we made love. I think the only time either of us could think of anything besides the Baklanov Equations and Loench’s Blackboard was when we climaxed. And then she sat up and reached for her clothes.
“What’s happening?” I asked.
“I have to get home to Carla.”
“Your roommate?” I wondered.
I was as stunned as when Professor Loench collapsed. Mathematicians don’t usually share much of their lives, but I thought we had shared everything.
“Can I walk you back to your apartment?”
“Yes. Maybe we can talk some more.”
That’s how I learned about an unexpected pregnancy, a failed marriage, and four years trying to be a mother and a graduate student while working part time as a teaching assistant.
“This has really helped, Walt,” she said pecking me on the cheek at the door. “We’d been talking about perfect numbers, and then something set him off.”
“What could set him off about perfect numbers?”
“My thesis on Baklanov’s Ninth. No one has ever found an uneven perfect number, and no one has developed a proof to show one could or could not exist.”
“Something about the Ninth suggests it’s about perfect numbers. I told him I’d like to try.”
“Carolyn, are you alright?” a female voice called from behind the door.
“It’s the baby sitter,” she laughed. “Yes, Beth, I’m alright. Open up.”
“Go on. Don’t stop,” I pleaded as the door opened to reveal a startled undergraduate expecting the worst.
“Beth, this is Walt,” she introduced me and went on about her thesis. “We were going to do it with the Baklanov Equations.”
Beth froze. She must have thought she was saying that to her.
“The Seventh and the Ninth are related,” I gasped, suddenly realizing expressions or parts of Baklanov’s Seventh and Ninth recurred again and again on Loench’s Blackboard.
“Let’s talk tomorrow,” she said, pressing my hand and stepping inside.
I floated back to my apartment. We were beginning to untangle something as infuriating as the Baklanov Equations themselves.
“There’s only one way to test it,” I said the next day. “Let’s tell Professor Loench and see how he responds.”
When she found a babysitter, we took the bus to his nursing home. He was lying motionless on his back, staring at the ceiling as intently as Descartes watching the fly that inspired him to invent calculus to measure its progress.
“Professor Loench, you remember Carolyn Seepers,” I introduced her. His eyes glowed softly, but he kept staring at the ceiling.
“Do you remember our conversation about perfect numbers, Professor?” she began.
His eyes were brimming with tears.
“Were you working on a proof for an uneven perfect number?” she asked.
My breath nearly stopped. His eyes turned to her.
“Using Baklanov’s Ninth?” I asked.
Now he was looking directly at me.
“Is Baklanov’s Third part of it, too?” she followed up.
He looked at her as proudly as if he were hooding her for her degree.
“And his Seventh?” I continued.
His lips quivered.
“But what is the Seventh about?” I nearly screamed.
Loench’s eyes were burning, but he did not speak. Carolyn gripped my hand.
“My nerves are at the vanishing point,” she said.
“The vanishing point,” Loench repeated, a look of inexpressible anguish contorting his face.
We were both as surprised as if the statue in the Lincoln Memorial had spoken. Recovering, I ran down the hall for the nurse. When we returned to his room, his eyes were closed.
Triglyf Loench died that night. A teaching assistant called to tell me the next morning while Carolyn and I were sitting in her kitchen trying to keep Carla amused and talking about the Google definition of the vanishing point.
“My God,” she said. “I killed him.”
She broke down, and Carla started to cry along with her.
“Are you alright?” the TA asked over the phone.
“We’ll be OK,” I said and ended the call.
How could a term from art criticism cause a man’s death? I wondered. And then I saw it.
“Carolyn, listen,” I said gripping her hand. “The vanishing point is where two parallel lines converge in a painting.”
“Or in non-Euclidean geometry,” she sniffed.
“It must have something to do with Baklanov’s Seventh,” I said.
“So what’s Baklanov’s Seventh about?”
“If it has Baklanov Compressors in it, it must be the largest number in the universe.”
“Mommy, what’s happening?” Carla asked.
“The divisor of Pi!” Carolyn exclaimed. “It goes out to so many places everyone thinks it’s infinite.”
So while the math faculty planned a memorial service for Professor Loench, Carolyn and I untangled Baklanov Seven and Ninth Equations from Loench’s Blackboard using the Third to compress the numbers. If the faculty allowed her to use the same subject matter for her dissertation, we would receive our degrees together. We were so excited we failed to ask why Professor Loench had intertwined them like strands of DNA to make their solutions nearly impossible.
My new thesis advisor called to ask me to give a remembrance at the memorial. I said I would as long as Carolyn could speak with me. I didn’t think I could get through it alone. When he called back to say the memorial committee had agreed, we were confronted with one of the greatest problem mathematicians can face: giving a speech.
“So let’s not speak,” Carolyn said. We were in her kitchen with Carla, trying to complete Loench’s Blackboard while the little girl offered us wooden delicacies from her play oven. “We’ll write the solution on the blackboard and let them see for themselves who he was.”
The service was going to be in the largest lecture room on campus. She was so brilliant I teared up.
“Be careful,” she said softly.
My tears had dropped onto the page with the solution, and the last numbers ran together. And then I saw it.
“The vanishing point,” I whispered.
She saw it at the same time. Pi was not infinite, and there was an uneven perfect number. Both had the same number of integers. When they were expressed side by side on Loench’s Blackboard, they touched like parallel lines in a painting extending to the horizon. According to Baklanov’s non Euclidean proof, the horizon was the outermost point in the universe where creation was still bursting out of nothingness.
“Yes!” I exclaimed so loud that Carla put her hands over her ears.
Carolyn scribbled the segment Loench dared not disclose on the tear stained page. When these numbers touched, the Baklanov Equations showed the universe would stop expanding.
“But that’s impossible,” I said.
“Einstein’s Theory of General Relativity predicts it,” Carolyn said.
“Yes,” I said. “And the point the numbers converge is the same point where relativity and the uncertainty principle converge.”
Carolyn gasped. She looked like the world was ending.
“Mommy, what’s the matter?” Carla said.
Carolyn picked her up and held her on her lap.
“According to the uncertainty principle, we can’t know the position and momentum of an electron at the same time,” she began.
“Because the act of measuring alters its position,” I continued.
“So when we determined Pi and the uneven perfect number, we stopped the expansion of the universe. That’s why Professor Loench stopped writing.”
“I had never thought the universe was so sensitive to a human touch,” I said.
“Walt, what have we done?” she whispered horrified. “Does truth change when you reach it?”
She set Carla on the floor and went to the stove.
“Cookies, Mommy?” Carla asked as she turned on a burner.
“No, Carla. No cookies now.”
She moved her hand toward the flame and back quickly, turned off the burner, and collapsed in her chair. Carla handed her a wooden cookie, but she did not pretend to eat it.
“We’re wrong about everything,” she said. I had never seen her so dejected. “If the universe had stopped expanding, the Second Law of Thermodynamics would not exist anymore, because the same law of dispersion of energy on earth applies to the dispersion of energy everywhere in the universe. But heat still flows from hot to cold. So what did Loench see? That it’s all a ridiculous charade and we’re fools to chase it? That everything Baklanov and he had written was wrong?”
We did not know it, but she had just conducted what would become the most famous experiment of the 21st century. But I looked away from her, because she had the same expression of frustration and anguish as Professor Loench when he said, “The universe is passive aggressive” and collapsed. If Loench was right, it lures us to discover its secrets only to drive us mad.
My eyes dropped to the last segment of Loench’s Blackboard, stained by my tears. So this was the end. But no. I suddenly realized the Blackboard was incomplete. The grand equation of the universe was unbalanced. So I took the pencil and continued to write to see if anything lay beyond Triglyf Loench’s annihilating vision. And there it was. Because the Second Law of Thermodynamics still applied, the universe was continuing its expansion in another dimension predicted in the last seemingly impenetrable segment of Baklanov’s Seventh.
When we wrote the results on the blackboard at Loench’s memorial, it took the mourners several minutes to realize what we were doing. Then they took out their phones and took pictures of what would be known the Seepers-Ecriveur Blackboard. After we finished, I stepped to the microphone.
“Professor Loench did not leave us in despair,” I said. “He led us to the edge of another universe.”
People would not leave after the service until Carolyn and I returned to the podium. They were shouting questions about the new universe. Finally we held up our hands for quiet.
“The new universe is there for us to discover,” Carolyn said. “There are twelve Baklanov Equations. We only know three.”
That’s how we will spend our lives, reaching into a universe we helped to create.
Fred McGavran retired from practicing law in June 2010 and was ordained a deacon in the Diocese of Southern Ohio. Black Lawrence Press published The Butterfly Collector, his award-winning collection of short stories, and Glass Lyre Press published Recycled Glass and Other Stories, his second collection, in April 2017. He is a frequent contributor to Spank the Carp. Find more about him at www.fredmcgavran.com.
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