The Infinite Tides

by Christian Kiefer

  Of course he knew that numbers—all numbers—were meant to represent the physical world. That was their point and their purpose. But these numbers, these rows upon rows of numbers, described nothing he could point to and yet, eventually, when they reached the tens of thousands, they would apparently build into images of intricate and indeed infinite beauty: shapes nesting in secret within other shapes, order from what, to his eye, continued to look like chaos, the numbers drained even of the colors he had always seen them in, their forms now gray and lifeless.

  He was so absorbed that when his mother knocked on his door he did not at first understand what the sound could be. Then she said, the door still closed, “Honey, it’s about time to come down for breakfast.”

  “Breakfast?” he said. “OK.”

  He looked up from the books and papers, looked to the window where the first gray light of dawn was filtering through the ash trees, and then to the pages around him, sheet upon sheet filled, each and every one, with row upon row of numbers. He had stopped trying to graph the information at some point during the night, letting the computer’s concurrent tabulations do that work for him, and now he went to the screen. Upon its lit rectangular surface was displayed a filigreed shape not unlike a beetle, a shape generated by the same numbers he had been working with on the bed, numbers he returned to and looked at in confusion for they held within them no sense of order. No sense of order at all.

  He went to school that day and when he returned home he immediately went back to work on the same problems, once again neglecting to sleep, eating only because his mother called him down to the kitchen and then bolting his food, claiming he had homework to complete, that he was working on a project. He could see the numbers upon the page, hundreds of them, and they all seemed to have deserted him, as if they had gone inexplicably silent all at once. Mute. At yet he could see them, could move them and could reorder them and so he tried to make sense of whatever system they represented, for there was always the fact of the image itself, the graphed results of the algorithm, but the individual numbers did not seem to hold any relation. What he saw on the page was stochastic, chaotic. What he saw in the image, in the graphed results, was intricately organized and staggeringly beautiful.

  By morning he was so utterly exhausted that his mother took his temperature and told him that he needed to remain home from school, a decision he protested feebly and then returned to his room only to awake in the late afternoon with papers and pencils and calculator strewn around him and the colorless home computer still chattering upon his desk: the beetled screen more detailed now, lightning extending from the edges of the shell as if reaching out into the white space beyond the limits of the graph, perhaps reaching into the room itself.

  He returned the books to the library a few days later but the aftereffects remained with him for months and even years to come. In some ways he could trace his desire to become an astronaut to those two days where he seemed to fall down some mathematical wormhole, or rather not the desire to become an astronaut—he had been calling that his goal since he was in grade school—but the development of that goal as his singular focus. He could still feel the taste of those days upon his lips, the flavor of metal, the hint of rust on his tongue. He might have turned directly toward the abstract, might have pushed himself to better understand what was, in essence, the first and only mathematical encounter he had failed to understand. It was certainly his immediate thought to do so, because every problem had a solution. That was the fundamental rule of the numbers, the only rule that actually mattered. Everything could be solved. Everything. Why develop an unsolvable mathematics to describe something as intangible as beauty itself? What purpose could it serve?

  The truth was that even though he was inexplicably drawn to that beauty, the very idea of it terrified him because it confused what he had always understood as the purpose of the numbers themselves: fixedness, location, direction, force, mass. His ideas were of sharp angles, clear divisions, straight lines, elegant clean pistons, gears that flowed into each other without effort, jet propulsion, heat diffusion, power, energy. Perfect machines. Perfect, perfect machines. That was math. That was what the numbers had always been for and any other purpose or function they might have served was irrelevant. The numbers resolved into an answer, even if that answer was complex, and therefore a problem that could not be solved was not a problem at all. It was something else. A distraction. A mistake. An error.

  He would tell himself over and over that he wanted to become an astronaut because of the challenge and the pinnacle that this achievement would represent but he knew that there was a component of his desire that this explanation did not reach, for of course he could just as actively work on mathematics without becoming an astronaut. But after the fractals there remained within him a source of gravity he had never conceived of or comprehended, a force that seemed to pull at him from space, from the chaos itself, from the fractal and crystalline darkness that existed everywhere beyond him, as inexplicable and indomitable and infinitely beautiful as the pull of life and death itself. And yet he remained as practical as he ever had been, perhaps even more so, and he would take that practicality with him to spin at the fringe of whatever mankind did not know, would probably never know, about the whole of the universe outside Earth’s orbit, as if the abyss itself could be placed in a frame and as if such a framing would make it easier to understand. These thoughts not even conscious and yet present nonetheless.

  He was only a few steps away from his goal. This was what he had been focused on when Quinn’s school year started and it was what he was still focused on when that school year ended and another began. He was so close to becoming an astronaut that it was all he could think of. Perhaps he had assumed that the vector upon which he had imagined his daughter to move continued to guide her progress. Or perhaps he had simply lost track of her. By then he might have lost track of both of them. He had worked so hard, so very hard, every day, for so many hours, and then the phone call from the Astronaut Office came, telling him that he had been accepted into the training program at long last. He was almost there. He went home and told Barb the news and she squealed in delight and leapt into his arms.

  She seemed excited but later that same day, before Quinn came home from school, she told him that she did not think it was necessary that they all move to Houston with him, that he should go ahead and start the training and they would talk about moving her and Quinn later, after he had finished the training and was working regular hours at Johnson Space Center. “You’re going to be busy all the time, Keith,” she had said. “And we’ll just be a distraction anyway. You can come home when you can. On weekends or whenever you have a break. And Quinn’s just settled into her new school. Remember what happened when we moved last time? That’s hard on a kid like her.”

  He tried to raise counterarguments but she shot them down one after another and there was that central important fact: that he was going to be busy, very busy, all the time. Still, he could not understand why she would want him to start his training without them. But maybe that was not it at all. Maybe she was only stepping aside so he could embrace his training more fully, so that he could charge, unencumbered, toward his destiny, toward their destiny, for they had chosen it together and now it was almost upon them. Maybe she was right and it was premature to move the entire family to accommodate the training period. They could be a family again in Houston when he was done with training. They had their entire lives, after all, and the best part had nearly arrived at last.

  “OK,” he said, simply and definitively. “OK, we won’t move.”

  Of course he had thought then that Quinn would be relieved for the same reasons that Barb had raised, for indeed he did remember her dramatic reaction to their last move: the weeks of sulking, the angry slamming of doors. Quinn was thirteen years old now and she was already doing the kind of work he had not even known was possible until he was eighteen or nineteen. They had a gifted program at the junior high, one s
ignificantly more advanced than the program at her previous school, and the level of mathematics to which she was being introduced was staggering and exciting. He still brought up the academy sometimes, continuing to hope that she might enroll there for high school, but she did not seem any more enthusiastic about the idea. And yet he could not help but think how much greater her experience at the academy would be nor could he understand why anyone would choose something lesser whan a greater solution presented itself, especially because her enthusiasm for mathematics had only grown in intensity. Sometimes she would call him to the kitchen table—still where she did most of her homework—and would ask him to doublecheck her numbers or would tell him about some project she was doing about black holes or perpetual motion machines or something else and he could feel her excitement, her discovery, the path of her forward motion.

  The evening he was to tell her his good news, she was working on a math paradox called Hilbert’s Hotel, a puzzle he had forgotten about entirely concerning a hotel with an infinite number of rooms, all of which were full, and the various guests that arrive looking for a vacancy. She told him this with a smile on her face, as if imparting some impossible wisdom to her father, as if finally she had something to tell him that he did not know and he played along with that notion.

  “So what happens then when an infinity of guests arrive and all of them are looking for a room?”

  “I don’t know,” he said. “They tell them the hotel is full?”

  “No, Dad, it’s an infinite hotel. Remember?”

  “Right,” he said.

  “Pay attention.”

  “I am.”

  “Infinite hotel. Infinitely full. Infinite people arrive and they all want rooms. What do they do?”

  “I don’t know.”

  She paused as if for dramatic effect. Then she said, “They ask every other guest to move down one room.” She was really smiling now. Beaming. In his memory it was like there was light shining from her. From her face. From all parts of her at once.

  “How so?” he said.

  “Look,” she said. She took a scrap of paper, already mostly covered with numbers, and wrote n on it and then said, “If n is a room with a guest then n moves to n plus one and then—” her pencil moving as she spoke, finishing the line and then sliding the paper around so it faced him. “There,” she said.

  He looked at it, at the scribbles of numbers and lines and equations rambling across the page. “Nice,” he said. “That’s interesting.”

  “That’s awesome.”

  “That’s what I meant. Awesome. What’s that called again?”

  “Hilbert’s Hotel. I can’t believe you haven’t heard of that. It’s kinda famous.”

  “Yeah, maybe I have. Seems familiar. I probably forgot.”

  “Oh yeah, you forgot. I doubt it.”

  He smiled at her. “So do I,” he said. He leaned over the paper and took the pencil from her and wrote two symbols:

  ℵa

  “What’s that?” she asked.

  “Aleph,” he said. “It’s the symbol for the cardinality of infinite sets.”

  “The whatsit?”

  “The cardinality of infinite sets.”

  She stared at him.

  “The aleph symbol is the cardinality. It just means infinity here.”

  “Infinity like the hotel?”

  “Exactly like the hotel.”

  “So then the little a is the set?”

  “Correct.”

  “Cool.”

  “So how would you apply that to the hotel?”

  She sat for a moment. Then she said, “Well, the hotel at start would be aleph-a.”

  “Maybe,” he said. “Most mathematicians would probably call it aleph-null.”

  “Null why?”

  “Because that’s the smallest possible set.”

  “OK, so the empty hotel would be aleph-null.”

  “Yeah, OK,” he said.

  “Then the infinitely full hotel would be aleph-a.”

  “Aleph-one. Remember, the sets would be numbered. The a is just the variable.”

  “Right right,” she said. “So aleph-one.”

  “OK,” he said. “Then one more guest arrives and that would be what?”

  “Aleph-two, I guess,” she said.

  “I guess so,” he said. “So what does that mean?”

  “I don’t know.”

  “Think about it.”

  She did and then said, tentatively, quietly: “Some infinities are smaller than other infinities?”

  “Or larger,” he said.

  “Or larger,” she said. And then, after a moment, she said, “Holy crap!” He smiled at her. “Yeah, holy crap,” he said.

  “That’s awesome.”

  “Well, yeah, kind of.” He wondered if he could get the same response if he taught her the mathematics of building something, how the numbers could predict the size or density or dimensions a piece of metal would need to be, how the numbers could be made into something solid and useful and tangible. Maybe he would try that next time. “You can’t do anything with it but it’s fun to think about,” he said.

  “Yeah, awesome.” She sat looking at the paper, at the pencil marks they had both made upon it.

  “Glad you’re having fun.”

  “Yeah,” she said. “This is heavy stuff.”

  He smiled again. “Wait until something you do is in space,” he said. “That’s the heavy stuff.”

  “Yeah.” She looked up at him now. Perhaps she saw him then differently than she had before or perhaps she was just looking at her father. He did not know and would never know. But she looked at him for a long time and said nothing and then her eyes returned to the paper again.

  “So about that,” he said. “I have some news.”

  “You heard?”

  “I heard.”

  “Tell me tell me tell me.”

  He paused and then he said, “I’m in.”

  “Oh!” she screamed and she threw her arms around him. “I knew it. I knew you’d get in. I knew it.”

  “Well, so did I,” he said, smiling. “I’m glad you’re excited.”

  “Yeah, I’m excited. Of course I’m excited. It’s awesome news.” She released him and sat staring up at his face.

  “So here’s the thing. Your mom and I have talked about it. You know the training is all in Houston?”

  “Yeah, we’re moving. I get it. Bummer but OK.”

  “No, wait, the idea your mom and I talked about is that I would fly back and forth. At least for a while.”

  “What do you mean?”

  “It would just be for a while. You know. Until we figured out how well it was working.”

  “How often?”

  “I’ll be home most weekends, I think.”

  She looked away and then turned toward him again, her eyes luminous. She looked like a woman then, or like the woman she might become. “But you’ll hardly ever be here,” she said to him.

  “I’ll be here every weekend.”

  “Most weekends, you said.”

  “Yeah, most weekends.”

  “Then what about the rest of the week?”

  “During the week I’ll be in Houston,” he said, “and some weekends I probably won’t be able to come home either.”

  “Shit,” she said.

  “Quinn.”

  “Shit, shit, shit,” she said. Her eyes were already bright with tears.

  “We think it’s the best idea.”

  “How is that the best idea?”

  “I’d be away those weekends even if you and your mother moved to Houston with me,” he said. The first tear slid down her cheek. “Look,” he said, “Your mom and I talked about it and we thought it would be better if you didn’t have to move again. I mean the school here seems really good and you’re learning all kinds of stuff I didn’t even know about until I was in college.”

  “I don’t care about that,” she said.

  “I kn
ow,” he said, “but it’s important.”

  “No, it’s not.”

  “Yes, it is. You’re gifted, Quinn.”

  “It’s not my fault,” she said, and now the tears really started to come, pouring down her face, and he knelt next to her on the kitchen floor and pulled her toward him, her head tilting to his shoulder, her arms wrapping around him as she wept.

  “Don’t say that,” he said. “It’s not anyone’s fault. It’s great. I’m proud of you.”

  “I don’t want you to go.”

  “I have to.”

  “Why?”

  “Because I’m going to be an astronaut.”

  “Then let me come with you,” she said.

  He tilted her away from him, his heart a hard burning knot in his chest. “You and I are different from normal people,” he said. “We have a gift. That’s why I have to be an astronaut and that’s why you have to stay here and learn, because that’s your gift.” The words came slow and halting, perhaps because he had never spoken of her ability in these terms. Or perhaps because he knew there were schools in Houston that could teach her, schools probably even better than the academy. But Barb had made her case and that discussion was over. And it would be easier for everyone this way: less turbulent, less confusing. This was what he told himself.

  “I don’t care,” she said. Her tears had wet his shirt through at the shoulder but she had stopped crying.

  “Yes you do,” he said. “It’ll be OK. You’ll see.”

  And he believed it too, believed, in that moment, that the arrow he had made for himself would continue its long upward motion. As if there was no possibility of return, no aphelion or perihelion to chart the long spiral of his orbit around whatever false burning center he had made.

  Know this. That the things that go into the fire are forever changed. That all you have ever done can be measured not by distance but by circumference. That these twin spirals of smoke: they are your life, rising in curls.