The Cheshire Smile
by Quinn Tyler Jackson

He didn't know. As Farrah Donahue paced the waiting area of the airport, she continually chewed her bottom lip for need of a cigarette. The lipstick was gone now--she'd chewed it all away. She looked at her watch again. The flight was on schedule, but she had arrived early; it was the only way she could convince herself she would be able to make it to Heathrow Airport to meet him. But he didn't know.

Would he be angry with her? How could he be? She had every right to phone the University of Washington and find out that he would be in London that night, weather permitting. He did not have to tell her himself that he would be in London. He didn't even know she was in England for her graduate studies. They hadn't been in touch since before Christmas. She'd shown some initiative and discovered he would be there. What did it matter that he didn't know?

One of the dangers of getting close to someone because of a common misery was that the misery would heal and the bond would break. Anything for a smoke. If she ventured outside to relieve her need for a cigarette, it would take the edge off. She wanted that edge.

"Friend or family?" a middle-aged woman seated along the rows of preformed chairs asked Farrah.

"Pardon?"

"Are you waiting for a friend to arrive, or some family?" the women explained. She had been reading a paperback novel and folded the page she was on so she wouldn't lose her place.

He wasn't family, but was he even a friend? For all the time she had been right where she knew she was, he hadn't called. She'd called him once to wish him a Happy New Year, but not once had he shown any initiative. "Friend," she finally replied. The more she thought about the phone that never rang for her, the more she wanted to leave the airport, go back to her small apartment, and get on with her life without making a fool of herself by being at the airport when he arrived.

Twenty feet from her, on the other side of the large pane of glass looking out into the night, her doppelganger stood, starving for a smoke. Farrah laughed inside that her reflection in the glass was outside, and could light up if she wanted one that bad. She was dressed in the same new navy blue blouse and skirt, wearing the same new shoes, but stood in the looking-glass world, the land of the walrus and the carpenter. The world was that other Farrah's oyster, ready for the shucking. Run away! she shouted at her reflection in her mind. Beware the Jubjub bird!

"Coming from where?" the woman asked, bringing Farrah back to the other side of the looking-glass with a jolt.

"Seattle, New York, London," Farrah replied. "And how about you? Friend or family?"

"Family."

All mimsy were the borogoves, Farrah thought to herself, staring at that other her. Is there anyone in this place with a hookah?

When the flight finally landed, her heart starting pounding into her ears. She wasn't sure if it was her nicotine withdrawal or her anticipation about seeing him again. She wasn't itchy; it was anticipation. She played over in her mind the many possible expressions he might have on his face when he realized she'd shown up to greet him unannounced. She tried to rehearse with her mind's voice what she would say to him.

Long time no see. No, that wouldn't do. It was a silly thing to say.

Thought I would show you the town. Equally ridiculous, as was Fancy meeting you here.

Why didn't you phone me? Closer to what was on her mind, but he would be exhausted after such a long flight.

With each mental rehearsal, she tried to imagine the expression that would come over his face. He had an expressive face and would not be able to hide his true reaction after having spent so much time in a crowded, uncomfortable jet.

If I knew you were coming, I'd have baked a cake.

Come to my arms, my beamish boy! O frabjous day! Calloh! Callay! she would chortle in her joy.

Some others started to gather around where the passengers would arrive. Some held out signs. She had considered holding out a sign with Euler's famous equation, to catch his eye. The woman who had asked her the questions earlier was standing beside her. "Do you have a piece of paper by any chance?" Farrah asked.

The woman looked through her purse and produced a blank page from her day planner. It was half the size of a regular piece of paper, but would have to do.

Farrah took out her lipstick, reapplied some to her lips, and then, very carefully, in order to not break the stick, wrote, in bold red:

e^-iπ + 1 = 0

No mathematics professor on earth would miss that in a crowd of signs. But he didn't know she would be there, waiting for him, and wouldn't bother to look into the crowd.

When the passengers started to emerge, she scanned each one, trying to find him among them. After about twenty misses, she saw him. He had lost weight, and although he had always been handsome, seemed more so to her now. Maybe that was the passage of time, and not his face. He did look into the crowd, and started turning his head left and right, as if looking for someone, so she held up the lipsticked piece of paper and hoped for the best, her heart racing. She gathered up her courage to step toward him, to call out his name.

As her lips parted to call for him, she saw the other woman approach him, and her mouth shut and arms dropped beside her. He stepped toward the other woman, a brunette not much older than she was, and held out his arms to greet her with a hug.

Farrah could not move. She had rested by the Tumtum tree, in uffish thought, and had made a fool of herself. He put his arm around the woman's back and started with her for the baggage collection area.

"There are my children," the woman who had given her the paper said. "I must be off. Nice meeting you. Seen your friend yet?"

"Maybe he missed his flight," she lied as the woman wandered over to meet her son and daughter.

Shun the frumious Bandersnatch. Farrah slowly wandered over to the glass, coming ever closer to her reflection with every step, as the crown around her started to dissipate. She hadn't known.

As Farrah tapped her boiled egg with the side of the spoon she attempted to push the image of Anders' arm around the woman's back, but could not. Tap, tap. She peeled the shell from the egg, sprinkled some salt on it, and took a small bite.

She tried to imagine a reason he would have for putting his arm around the woman if they were not involved, but always came to the same conclusion. The quandary was like the Bridges of Königsburg--no matter how she lines of reasoning she crossed, she always had to double back over the most uncomfortable bridge of the truth.

She knew with her mind that she and Anders had never been anything to one another but Misery-Loves-Company, but her mind was not in control of how it made her feel to have been left standing at the window at the airport. The idea of showing up for graduate studies at Cambridge, in the same Department of Mathematics he would be teaching, even though she would not have to cross his path a second time if she was careful, made the egg hard to swallow.

When her phone rang, she wiped her mouth with a napkin, stood up, and went to the wall to get the receiver. It was Clark Kilkenny.

"Say, love," he began, "some of my friends and I are going out to the movies tonight. Would you like to join us?"

"How many will there be?" she asked.

"Three of my friends. You know Jenny, and her boyfriend Micah. Also, Raj. Nice fellow, you'd like him."

"I'm not up for a lot of company, I'm afraid," she replied, sighing.

"I can skip them if you'd like to go out and do something alone," he suggested. "Raj might feel left out in the cold, but I'm sure they won't mind. If you want to, that is." He'd been trying to get her to go on a date for two weeks, and she'd given him an opening she normally wouldn't have, but felt a certain sense of satisfaction at the idea.

"I'm not all that keen on going out into the open," she replied.

"What's up, Farrah? Anything you'd like to talk about?"

"Just not feeling outside-worldly," she answered. "If you pick up some makings for a salad, I could make a nice dinner at my place."

"Your flat at seven?" Clark suggested.

"Why not?"

Clark Kilkenny was asleep beside her in bed. He hadn't been all that artistic a lover, but she felt good about having had sex with him after dinner. Though they hadn't talked about what was bothering her, they'd had some pleasant conversation and she needed the company. It was almost enough to put Anders out of her mind, but not quite.

Why does it matter? she asked herself more than once as she lay under the sheets. It shouldn't matter.

"What's that?" Clark asked, coming out of his sleep.

Did I say that aloud? "Oh, nothing," Farrah replied.

Clark leaned over and kissed her arm. "I know it will sound odd at this hour, but do you have any tea? I've got this terrible craving for some tea, if you wouldn't mind making some."

Farrah got up, put on a robe, and went to the kitchen. She started a pot boiling and cleared the table for tea. The clock on the wall showed eleven twenty-three.

Clark came into the kitchen, dressed in his boxer shorts and a pair of socks. He was handsome even when disheveled, and Farrah asked herself why she had resisted his attempts to get closer to her before. There was nothing at all wrong with Clark Kilkenny. He was a good conversationalist, accommodating without being a pushover, thoughtful, persistent, and ambitious. His breath did not stink. Even though he was not a smoker, he hadn't once so much as mentioned Farrah's habit, or even made a sour face when she lit up in front of him. He also didn't act as if he was just putting up with her to get his way with her; he seemed genuine in all his attributes.

Any woman in England would be happy serving the man tea at eleven thirty at night after a good dinner, good conversation, and adequate night in bed. Even so, as he sipped his tea, smiling over the rim of his cup, Farrah wished he would decide to not stay the night. "What's on your mind?" he asked her.

Farrah knew that she couldn't ask him to leave. He had been too perfect a gentleman from the time he'd arrived. She searched her mind for the least offensive thing to say, and replied, "I was just thinking about Gödel."

"Girdle?" he asked.

Maybe that was it. Clark Kilkenny, a history major, appeared to know nothing whatsoever of mathematics.

"Kurt Gödel."

"Ah. Cute girdle! Anyone I should be worried about?" he asked, half grinning.

Good grief! "He is a dead mathematician," she explained.

"Anything particularly interesting about him?" Clark asked, still sipping at his tea.

"In 1931, Gödel wrote a paper that shook all of mathematics. For centuries mathematicians had been debating as to whether or not math could be a perfect system. His paper showed that a sufficiently complex axiomatic system could never be both consistent and complete at once."

Clark tapped his cup with his index finger. "I'm afraid you've lost me, but I'd really like to know what the means, if you don't mind explaining it in terms a historian specializing in the Byzantine Empire could understand."

Farrah wanted to scream. Clark's perfection had shown itself to be incomplete, just as Gödel had proven it must be.

"Well, it won't be easy, but I shall try," she replied. "Do you know what an axiomatic system is?"

"A system of self-apparent truths?"

"Good enough. An axiomatic system is a collection of axioms, yes, and axioms are self-apparent truths. For instance, one is the integer that follows zero. The successor of any integer plus one can be expressed as the successor of the successor of that integer. And so on. All of these rules, together, make up an axiomatic system."

After a long, blank stare, he said, "OK, I'm with you so far. What did Kurt do?"

Farrah stirred her spoon, trying to find a way to put the importance of Gödel's work to Clark. "Well, there was a school of thought, before 'Kurt' that believed that math was, at heart, perfect, and a potentially complete set of axioms could exist that would wholly and truly represent all mathematical truths. The perfect system would be consistent--that is, could never say A was true, while A wasn't true--and would be complete, in that it could be used to prove every mathematical notion that was true, and weed out the untruths."

"OK."

"And then Kurt Gödel came along and proved mathematically that any system of sufficient complexity that was entirely consistent could not be complete, and that any system that was complete would contain at least one statement that was patently false, but which could not be proven-- in the rigorous mathematical sense of a proof--to be false. In other words, a sufficiently complex axiomatic system could not be both consistent and complete at the same time. The proof Gödel supplied toppled the faith of pure mathematicians everywhere, since mathematics was shown to be less than perfect." Less than perfect. Wasn't Anders also less than perfect? If so, why was she still thinking about him and the woman, and his arm around her? She was inconsistent.

"You're very sexy when you talk maths," Clark kidded.

Could such a relationship go any further than this? Clark was a complete ignoramus when it came to mathematics. What would she have to do to get rid of him for the night and have him never come back, without making herself look bad -lie and say she had pubic lice? The thought was tempting. Had this been a conversation with Anders, she could have slipped a mention of Gödel's Incompleteness Theorem into a conversation with such nonchalant ease. Gödel shows that is impossible. Five words to the wise was sufficient. She knew why she wasn't happy with perfect Mr. Kilkenny--he and she had nothing in common whatsoever.

"Thanks for the dinner, the tea, and everything else," Clark finally said. "I don't want to seem a cad, but I've got an interview tomorrow morning, and really have to be going home for the night so I can shower and put on my pressed suit before heading out in the morning. Would it bother you if I headed home?" He leaned over and kissed her forehead.

"I had hoped you would stay the night," she said, "but I understand." Thank God! He's going to leave!

By midnight, he was gone.

Even if Gödel had pulled a Fermat's-Lost-Proof in the margin of an address book, the conversation Farrah had had with Clark Kilkenny that night would have proven to her beyond doubt that he'd been correct. QED

Their paths were simultaneously dissimilar and comparable. Charles Dodgson, although a mathematician and ordained deacon, was best known for his imaginary worlds created under his penname Lewis Carroll. Sir William Hamilton was best known for his hyper-imaginary work created under the name quaternions. Anders Byrne, although never ordained, had once studied towards the priesthood, and later went on to win the coveted Brückemann Prize for Best Dissertation in Mathematics. Hamilton found his royal road through mathematics, a path not unlike the bridges Euler crossed in Königsburg. Leaving a mark on the stone foundation of nineteenth century mathematics, Hamilton was eventually anointed with the oil of knighthood. Meanwhile, Dodgson frolicked at fanciful belletristic gambol, proving himself a transcendent court jester who carved his legacy at the eccentricities of Kings and Queens. "Alice's Adventures in Wonderland" was published the year of Hamilton's death. Anders and Farrah only crossed the bridge from their professor-student relationship after Anders' wife's fatal car crash. It all made perfect sense to Farrah Donahue as she manipulated equations and tried to supply proofs for conjectures that might one day become Donahue's Theorems. The whole system was consistently interwoven.

It had been over a week since the Mad Tea Party with Clark in her kitchen near midnight, and she was completely content that he hadn't bothered to call her to follow up. Poring over a math journal in the University Library made much more sense than bothering with the likes of Clark. She was deep into the system, where no one would find her, leaning with her back against the spines of books, trying to find some glint of religion in lemmas and theorems. She could not smoke in the library, but the reading was relaxing satisfaction enough.

It was then that she heard Anders Byrne's voice. It was muffled by the wall of books between her and his aisle, but she could tell without doubt that it was he who spoke on the other side of the shelves. His accent was American, like hers. She slowly turned, peeked over the top of the bottom row of volumes, and saw two sets of men's trouser legs. At least he wasn't with her.

On and on the two of them went about groups, rings, ideals, and occasionally, lattices. Cabbages and kings. Mathematician word salad. Utter jabberwocky, even to her. Her heart raced.

"I'm going to step out for a smoke," she heard Anders say.

She shoved the journal she'd been reading atop some books, quickly made her way to the exit she knew was closest, hastily lit up a cigarette, and leaned against the wall as if she'd been there for hours.

"Fancy meeting you here," she muttered almost nonchalantly when he stepped outside, smacking a pack of cigarettes to get at one. It wasn't Calloh! Callay!, but it would have to do. She had slain the Jabberwocky--snicker-snack!

Anders' eyes lit up when he saw her. He seemed pleased at the circumstance of fate that brought the two to a library door for a smoke outside the library. "Farrah!" he exclaimed, putting his pack into his jacket pocket. He stepped right up to her, put his arms around her, and held her longer than she'd expected he would. He had never hugged her in the past; except in her mind. Maybe it was the English air.

Because she hadn't expected the hug, she didn't know how to respond. She put her arms around him finally and returned it.

"What are you doing here?" he asked.

"I guess I settled on a place to do my doctorate," she replied.

"That's wonderful!" He stood back, took his pack out again, and started a cigarette going.

"You here on a visit?" she asked. She knew he was in England for a year long guest lectureship, but asked anyway, so that he would not know what she knew.

"One year of rest and relaxation as a guest lecturer. What's this about doctorate, by the way? No master's, then?"

"I was allowed to skip a level," she replied. "It will save me some money and time. Somebody up there must like me."

"Excellent!"

"You've lost some weight," she noticed. She had seen it at the airport, but now that he was close to her, it was even more apparent. He'd lost at least twenty-five pounds since she'd last seen him.

"Depression will do that to a man," he said, sucking at his smoke. "But you know what? Now that I've seen you again, I'm sure it will pass and I'll gain all the weight right back."

Farrah brushed her hair out of her eyes and smiled the first genuine smile that had come to her lips since the night at Heathrow.

"Staying with anyone while you're in England?" she fished.

"Oh, yes," he replied.

"That's nice," she lied, puffing at her own cigarette comfort.

"My cousin Gillian and her husband have me in a room at their house," he then explained. "Such a nice family. She works nights at a hospital, but took a night off to get me at the airport. Blood is thicker than water."

Cousin Gillian. The woman he'd been so friendly with as he arrived off his flight. Damn, I'm so stupid! she cursed herself.

"Listen, Farrah...," he began, and then stopped himself.

"Yes?" Her heart was beating again.

"I'm sorry I didn't get in touch with you after you left University of Washington. I was in terrible shape. Took me some time to get over Joan's death. Still not quite over it."

"I understand," she lied. No, it wasn't a lie. She did understand. It had just hurt so damned much to not hear from him.

"But...," he again stopped himself.

"Go on, Anders. Please." She leaned back against the wall, almost ready to fall over. Cousin!

"Well..." Again, complete silence. "Time for a short walk?" he asked.

They started to walk away from the library. It was chilly outside, but Farrah didn't mind the cold.

"What do you say to the idea that you and I...."

Don't say have dinner, or I will scream bloody murder and run.

"Well, first, let me explain," he said. He brushed his hair behind his ears like a young man might, and smiled a nervous smile. "There were several reasons I avoided you that weren't entirely noble."

"Go on." She kicked a small stone into the grass beside the sidewalk.

"First, I was your professor. Second, we got close because of my wife's death. Third, I'm so much older than you are. Fourth, I wasn't ready."

"OK. You're not my professor anymore, and what are the chances you'll be assigned to my committee if you're only here for a year? Not going to happen. I get the second one, and agree one hundred percent. You're not old! Ready for what?"

"Ready to let it go any further than it could be allowed to."

She wanted to put her arms around him, squeeze the breath out of him, and never let go, but kept walking, always looking ahead. "How far are we allowed to be with one another now?" she asked. After the words came out, she could not believe she had said them.

"You know what my trouble is?" he asked. "My trouble is that I want to defy Gödel's proof."

Gödel! He had slipped Gödel nonchalantly into a conversation, as she had guessed he could! A word from the wise was sufficient.

"You want to be both consistent...," she started.

"... and complete," he finished for her.

Farrah's heart was beating near her throat. If Anders kept this up, she would fall over onto the grass.

"Since you lost Jim, and I lost Joan, I felt I had to be very careful. What if--"

"--we found the wrong bond?" she now finished his thought.

"Exactly."

She hurried her pace until they came to a bench, and then sat down. When Anders sat beside her, she spoke again. "We could try it and see what happens," she offered. "Just try. I'm certainly fond of you." How childish a word fond sounded when spoken into English air.

Anders reached his hand toward hers, but only halfway. She had to reach out, too, and hold his hand. It was strong, but not too strong. Come to my arms, my beamish boy! Drink me!

"Can we agree on one thing?" he asked.

"What's that?"

"If Gödel was right, and let's assume he was," he began, "and we can't have it both ways, can we agree to settle for being complete, and leave the consistency to figure itself out?"

If seven bridges cross a river, just so, it is impossible to walk in a path such that each bridge is crossed but once. This was proven by Leonard Euler. Eulerian Paths are not all that different from Hamiltonian Circuits. Sir Hamilton was famous for his work with quaternions and imaginary numbers, and died the same year that a mediocre logician, but adventurous and well-loved children's writer, published "Alice in Wonderland." Carroll was famous for his imaginary world. As she sat with Anders Byrne on the bench, Farrah Donahue knew that it all made perfect sense, or nonsense, and smiled the Cheshire smile.

Copyright © 2003 Quinn Tyler Jackson
All rights reserved

About the Author

        Quinn Tyler Jackson has been writing since he was twelve. At various stages of his career, he has been an artist's apprentice, antiquarian bookseller's assistant, gas jockey, freelance editor, literary agent, stay-at-home father, and computer software and hardware consultant. Through it all, he has always written poetry and fiction and has usually, when presented with two paths, taken the one that holds the promise of enlightenment, however worn. To learn more about Mr. Jackson's writing, please visit his Homepage.

Images:
Top:"Blue and White: The Princess from the Land of Porcelain," James Abbott McNeill Whistler, 1864, Freer Gallery of Art, Washington