Paper

A

 

In response, Michael S., who was forty-four years old, said to his wife (and former student): "It's something. It's not nothing. It has to do with dimensions."

His wife, Linda S. (whose last name, though it started with an S, was not the same as the Michael's last name), said: "'Furrowed eyebrows.'"

Michael and Linda were in Michael's atelier at their top of the house. Books were everywhere: children's books in thick dust, textbooks of '80s physics, Loebs of Plotinus and other Neoplatonists, cheap Signet reference works about oddballs through the ages, plus a good selection of novels, mostly sleazy. They were arrayed horizontally along the sloped walls and stacked deep in the corners, they were piled on the desk and under the desk, and some were opened and turned face-down for what had been months. There were so many books, Michael had to take his notes in a tiny space on his desk.

Michael leaned back and asked: "What's the name of those wooden things, the toys with the dowels and the connectors?"

"Tonka?"

"Aren't those cars? No, not them. I wonder how I can find some of those wood things. Maybe they're plastic these days. You know, you could build models of molecules using them. What are they called?"

Michael made a sketch of a fictional molecule. "These bits were the dowels, and then there's the nodes."

"Maybe I can go out to find some. Goodwill has lots of junk like that. What are you after it all for?"

"I'm trying to visualize more than the usual number of dimensions. I'm guessing I could come up with something I could understand better."

Linda said: "I'm sure your imagination works just fine." She kissed him and went into the other room of the attic.

In case you don't know, Linda was thirty-eight at the time, and Michael had in truth been merely a teaching assistant rather than a real professor when they had met. Linda was from a moderately wealthy family while Michael was a real country boy. The amusements created by this contrast so far showed no signs of exhausting themselves. Linda had white hair, and it had always been white. Michael meanwhile had black hair, and they made quite the couple.

In any case, Michael muttered as he tried to understand what he was trying to get at. "A three-dimensional object ... passing through a two-dimensional plane ... appears to be a ... plane too, but one that appears to originate from nowhere ... and return to nowhere. Therefore ... four-dimensional objects passing through three-dimensional space also appear, and then disappear. Is that the only solution? There's something I'm missing. It can't mean just that."

Linda called from the other room. "Tinkertoy."

Michael called out: "Yes! You remembered!"

"Look." Linda came back into the room with a large cylinder: Tinkertoy itself. "I found it in the corner. I just happened to look in the right direction."

"Geez, I guess that's mine! From childhood!"

"Seems likely, since you never throw anything out."

"Gimme gimme." She gave him the cylinder and he popped off the lid. "Wow, what a wood smell! Who knows how long it's been since I opened it last." He reached in and pulled out a handful of dowels. "I think this'll help me out. Maybe I never actually played with this stuff, I don't recall."

Linda had her hands to her forehead. Michael asked: "Oh, hon, are you feeling okay?"

Her eyes were closed. "I have to go lie down. I can probably get it to pass."

Linda was getting a migraine headache. She got them once a season or so. Michael couldn't imagine what they were like. He said: "Go down, lie down. Turn out the lights and try to think about a green dot." This was the only bit of folk wisdom he possessed; he was helpless otherwise.

She left the room in the atelier, and Michael was quiet as he messed with the Tinkertoys. First, he made a cube with twelve short dowels and eight connectors, with each connector plugged with three dowels. Then he took eight more dowels and stuck them onto the cube, one dowel at each connector, at about 135 degrees off each other dowel, so that the thing looked like an eight-spiked sea urchin. The last step was to connect the loose ends with twelve longer dowels such as to form a larger cube outside the smaller cube. Each connector connected four dowels, and everything fit perfectly, as if by a mysterious design. He turned the form over in his hands again and again. Finally he said: "It's a tesseract." He put it down on his desk and pondered it, taking notes about how it would behave if the dowels were made of a flexible material. He could pull it about, and turn all the isosceles trapezoids into squares; could he pull two at a time? What would happen then?

He made a number of sketches, and then Linda returned. She was looking somewhat better, so Michael felt okay asking: "So, are you feeling somewhat better?"

She nodded. "Much better. That crazy dream again."

"The room with all the machines."

She sat down in the only other chair available. "The machines with all the circles in them. humming, rotating back and forth. It's all very disquieting. What's this?" She picked up the tesseract.

"It's called a tesseract. It's a three-dimensional representation--a shadow, in fact--of a four-dimensional, ah, hypercube I suppose. If I had a good geometry book maybe I could figure it out. Anyway, this is what we here in 3D space see of a 4D hypercube if it was stable in our space. Four dowels at each point, all of them ninety degrees off the others. Which, here, is impossible. But not there, in 4D. It'd be the most ordinary object in the world." He got up and went to his textbook collections. "I thought I had a basic geometry here...."

"Why don't you look it up on the computer?"

Michael stopped. Computer? What could that.... "Yes, of course. The computer." He turned to his desk and looked to where his monitor and keyboard sat. He sat down again and said: "I'll check Wikipedia. Or some geometry website."

 

 

B

 

As if things weren't settled down enough, things settled down some more as Michael and Linda settled into the twilight, and ate dinner, and relaxed themselves on the sofa to watch Scenes from a Marriage. Michael was still thinking about dimensions, but Linda wasn't. The television was big, and a light blackness when turned off, and Michael wished for that light blackness during some of the proceedings of the motion picture. His mind went forward to what could possibly happen in the morning, when a certain female would arrive at his office near the university. He shook his head and thought about dimensions some more.

"What the hell is this?" exclaimed Linda. "Marianne and Johan have kids? They suddenly have kids? They weren't even mentioned! Or did I miss it?"

Michael thought for a moment. "I don't know. You never see them. It's like they don't exist."

"I think Bergman's making it up as he goes along. Or maybe he just doesn't care about kids."

"It could be any of the above, I suppose."

The film dragged drearily onward. When it finally came to an end, Michael went upstairs to do a bit more work. Linda came up after a while. "So, how's it going?"

He tapped his forehead proudly with a pencil. "I'm trying to understand where ideas go when we're not using them."

Linda sighed because he was at it again. "Can't you leave that for the office?"

Michael tapped his temple with an unmanicured index finger. "This is my office."

"It needs some re-decorating. Maybe a pillow and some blankets."

He plunged on. "The ideas don't just get stored. They get moved into active memory by some force. They don't come all crashing out at once like Fibber McGee's closet. They are pulled in, and I think they're pulled in by lines like these." He'd picked up his tesseract again. "The ideas are all connected by lines of attraction, through as many dimensions as it takes. So I think that our thoughts, the ones we experience, through only one dimension, the dimension of, uh, stream of thought, is connected to a near-infinite number of other dimensions, like, one for each idea. And there we are again, inhabiting one dimension but with access to an incredible number of other dimensions."

"Hah! You have to prove the existence of the first dimension first. Time."

He rubbed his eyes. "Back to the drawing board."

They went downstairs and got into bed. While Linda fell asleep immediately, Michael wondered and wondered and saw all the dimensions hooked up in an incredible fashion, and he fell asleep and found himself in a department store at the bottom of three sets of escalators. Signs with long numbers hung over them and he couldn't tell which was greater than which, so many were the digits employed. He shrugged and got onto the middle one, as string-saturated arrangements of some Beatles song played somewhere distantly. The escalator took him up to a vast plane with innumerable escalators going up or down, while written on the floor in parquet tiles was the word UNCONSCIOUS. Someone said he was on the right track. He asked: "The escalators? This one?" He went up another one, and found nothing on that level, which was disappointing. One escalator was green which he figured meant he was going in the right direction. In the distance he saw his childhood home--a cabin near a copse--but he didn't want to go there. Instead he found there was a ladder stretching up high overhead. The ladder became successively other ladders as he climbed up and up and he didn't get fatigued. At the top he met a man in dingy clothes who was standing at a maître d's lectern, who told him: "You can get anywhere from here. No reservation necessary. Would you like to take a look at the menu?" Michael said: "Sure." The man looked down at the lectern, then at Michael again. "The menu. I asked you about the menu," said both simultaneously. "I believe we are in the menu. You can find what you want yourself. It may take some time, though."

Michael woke up because Linda was moving around, and he said: "What's up? Where are you going?"

"Go back to sleep. The girls are calling. Maybe they've scared one another about something."

Linda left the bedroom. Michael closed his eyes. Having daughters was tough, but wouldn't having sons be worse?

 

 

C

 

He slept for some time, then Linda returned to the bedroom. She was sobbing quietly. Michael turned over and looked at her; she was facing away, with her face in her hands. "What is it? What's wrong?"

Linda quietly gasped. "My mother!"

"What about her?"

Linda lay down and grasped him. "The phone rang, at ten-thirty."

"I didn't hear it."

"We didn't answer it. It was Peter, asking me to call him back. I just got the message now. The light was blinking. I called him, and he gave me the news."

"What news?"

"My mother died last night."

Michael sat up. "What? How?"

"Her liver gave out, and she died of poisoning or something."

"How can your liver 'give out?'"

"She'd had the condition for some time."

"I didn't know anything about this."

"It was pretty obvious."

"Still."

"Well, anyway. I'm going tomorrow morning. Will you come with me?"

"I would, but ... I've got a big staff meeting in the afternoon. I can come in the evening. How would that be?"

Linda sighed. "I guess that would work. Maybe it's even preferable."

They continued talking for about a half hour, going over the details and memories of a woman's life. There was a lot that Michael didn't know at all, or had forgotten. Finally they settled down and got some sleep.

In the morning, quietly, they explained what was going on to the girls, who couldn't figure out what was going on. Linda got them into the car and drove off to go to her parents' place which had once been on the very edge of a suburb but which had since become the centre of a whole new town and she wept along the way at time the destroyer of all.

All the way along the streetcar journey to his university Michael came back again and again to the thought that he would never again see his mother-in-law and this upset him for he liked--he'd liked--her. The cross-streets moved from the future into the past as he saw them all as if for the first time. He didn't know if the places he was looking at were months or years or decades old; all he knew was that they were new to him. He got off the transit at the usual spot and went up seven floors to his office. He sat down at his desk and mused about time for about five minutes before Trixie came happily in and closed the door behind her. He had her sit down and he told her that Linda's mother had died and that everything was depressing all around. Trixie cried a little in sympathy before getting a typically bright idea. "Stand up." She undid his pants and said, "You're not entirely unhappy," and proceeded to blow him. Though this was not an unusual greeting for a Monday morning it was, under the circumstances, odd enough to be thrilling. He kissed her and they both sat down again properly, just as if the physical sensations of dampness, taste, and warmth didn't ever exist.

"She's gone to other dimensions," said Michael.

"She's out of time, in two senses," said Trixie.

"I'll be out of town for the rest of the week."

"I'll miss you."

"Yes. Have you ever thought about where things go when you're not looking at them?"

"I figure they stay where they are."

"When you're looking at that plant there"--pointing to a particular sad specimen on the windowsill, formerly a geranium‑‑"and look away, and look back, is it the same plant?"

"Generally speaking, I do assume it's the same plant. Philosophically speaking, I have no proof it's the same plant."

"So it's not the same plant."

"No. Even if it is the same plant, it's aged."

"I was messing around with a tesseract yesterday. It's the 3d shadow of a 4d cube."

"I know what a tesseract is."

"All the dimensions have to be essentially of the same nature."

"So time isn't a dimension?"

"Time is indeed a dimension."

"Does entropy affect length or width?"

"Maybe not. But, if you stretch something, and it breaks, can't that be considered a type of entropy?"

"That's very doubtful. Anyway, I think time is special in some way. Maybe dimensions are illusory. Maybe it's all in our heads."

"All I want to figure out is where words go when we're not using them."

"That's an interesting question." She stood up and lifted her skirt. She was not a fan of cotton. "Remember using this?"

They cleared off the usual space on the desk, and did it.

In the afternoon was the staff meeting during which the members of the department jostled over minutiae in attempts to promote their courses and the courses of their good friends for the following semester. The meeting came to no conclusion (for it was never meant to), and by about three Michael was back in his office with Trixie. He told her about his travel plans out to the suburbs and she consoled him for he cried a little. He said, "So, I've got to go home and pack up some stuff and leave."

She blinked and laughed and pointed. "You've already done that."

She was pointing at a piece of luggage beside his desk. He looked at it with curiosity. "Well, what do you know. I guess I've already taken care of that."

"Maybe it's come to you ... courtesy of ... the twilight zone." She doo-bee-doo-bee-doo-bee-doo-beed the theme.

He remembered packing it and carrying it along on the streetcar. He remembered how he'd hoped no-one would ask about it, and how no-one did.

"Sorry," he said: "I'm a bit distracted."

"No worries."

"I didn't expect any."

They kissed one last time. Would their paths ever cross again? This was a question for which they had no answer, and they didn't even think to ask it. Michael took his bag to the elevator. Trixie was lost from his sight. Would they ever cross paths again?

 

 

D

 

He left the building and walked down the three steps to the boulevard and along the nine feet to the sidewalk. He was thinking about Ulysses as he went to the streetcar stop and looked to see if one of them was approaching. Nothing was within sight, and he wondered if Leopold Bloom as he travelled the streets of Dublin on 16 June 1904 suspected he was living for one day a pattern established by Homer. There was no evidence that Bloom knew; if memory served Michael right, Homer was conspicuously absent from the allusions in the novel, as if Homer had never existed. A streetcar came into sight. However, there is no doubt that Homer existed in the space of the novel, since Shakespeare existed in the novel, and Shakespeare wrote Troilus and Cressida, which was based on Homer (albeit probably in Latin versions); thus Bloom could have known he was following Homer's pattern, yet failed to recognize it. Michael got onto the streetcar and sat down to think some more. Bloom knew Homer, and the lives of Bloom and Ulysses (Latin again!) were united as if Bloom was Ulysses; as if Ulysses accompanied Bloom as if in four dimensions running precisely along-top (or along-below) Bloom's experiences, with the two sets of quadra-dimensionality connected by some other dimension or dimensions with neither consciousness (Bloom's nor Ulysses') able to articulate the connection. On top of all that was the writer, with his own dimensions: eight years, compared to ten years, compared to 16 June 1904.

All the lights were off when he got home: an ordinary enough occurrence in the summer, but less ordinary in winter; furthermore, it was portentously quiet, which led him, a moment after his arrival, to call out: "Hello?"

A noise came from up above, then Linda came down the stairs. She was casually carrying the tesseract. "Hello," she said. "I was just looking at this thing. Each vertex is the terminus of four edges."

Michael replied: "Now imagine each vertex being the terminus of five edges, or six, or seven."

She looked at the tinkertoy. "I can't."

"Neither can I.

"We appear to be stuck in our feeble dimensions."

"We can't let everything happen at once, can we?"

"Well. So: anything cooking for dinner?"

They went into the kitchen and sat down at the table. Chicken was served, alongside some salad.

Linda said: "My mother called today."

Michael leaned back. "I was just thinking about her today, wondering if she was all right. So what's up?"

"She'd been sick with a cold, but she's okay now. Why were you thinking about her?"

"I dunno. I had some kind of an idea there was something wrong. Kind of funny that there was something wrong: though by the time I thought it, the idea was out-of-date."

"You were very much behind the times there. From now on, please think your ideas at least twenty-four hours earlier."

"That may solve my problem. Thanks for the tip!"

They watched a documentary on Getflicks that evening all about the World War made by a renowned documentarian who liked slide shows all doctored up with the clicking of the frames changing because that was the way people watched everything during the World War and that was an established historical fact. Michael was idly thinking about a certain person, in a certain place, thousands of trillions of miles away, who was at the centre of a great battle against a hostile alien invasion. This person was at that moment estimating how many projectiles they had left in the garrison. The person was jotting down sums and multiplying by an estimate he called 'bullet-power' and it was very loud due to the shouts and the screams outside on the terraform. He opened up another crate and yanked the excelsior out only to find out it contained more guns instead of more bullets. "Guns?!" Thus they had a lot less weaponry than his calculations had predicted. In his mind's eye he surveyed the whole colony, looking for a building that could be sacrificed as some sort of a giant alien-trap. Lure the aliens into the trap, and then blow it up. Michael was engrossed in the man's dilemma because Michael knew that it was a life-or-death situation. The man, who was thousands of trillions of miles away, checked his smart-watch for a map of the colony. Sure enough, there was a rec facility with a swimming pool and a track outside. To get there, he would have to cross the plaza.

 

 

E

 

At thirteen minutes after two in the morning, Michael folded himself, was unfolded, unfolded himself, and was folded, though some number of dimensions. Time folded upon him, and he was seventeen and a day, at the birth of the sun, grave-rotting, and a zygote simultaneously; space unfolded upon him, and he was in bed, at the Jinsekikōgen town hall, awake or asleep on many planets and stars, on a tree, in a twig, staggering through Bramalea; and ﻼ▒Ѡ folded upon him, and he was 'above' the symbogistical tallahedium, and 'beneath' the tashuanistovicoll catyx, and 'beside' all N of the SaѮŎﱵӺﯘM ἮồﻻЇіᾧͽϠíᶎדּǼ. He had connexions everywhere at once and for all time past and present, connexions through all the dimensions but one, connexions to everyone everywhere, including you, who are still connected though you aren't ordinarily aware of it; he was stretched and he was shrunk yet ever being with that ineluctable I that was his point on his plane in his solid, folding while unfolding always 90° from all else, never 89° or 91°, always 90° precisely, was Michael.

 

 

F

 

Their answering machine had a message on it in the morning; someone had sent a 'silent' message in the middle of the night. Linda listened to it, then she said to Michael: "Bad news. My mother's fallen down some stairs. Sprained her ankle and she's in the hospital. I'm going to go see her. Think you can come?"

Michael said: "Sorry, I've got this big staff meeting to go to, and to prepare for. The meeting won't come to any conclusions; they're never meant to. Still, I have to make my appearance."

Linda got ready to leave, and Michael got ready to leave. They would be going in opposite directions, so at the door, Linda with her purse and Michael with his satchel, kissed farewell.

Michael's satchel rattled. Linda said: "You're taking your Tinkertoys?"

Michael nodded. "I think there's something I can work out. In any case, I have to show them what I'm working on; you can't draw it because you lose a whole dimension when it's only on paper."

"It has to be more dimensional than paper."

Michael blushed. "The end result will be paper, though. It'll have to be boiled down, like a French sauce."

"Flatten here, flatten there, soon it's flat as paper."

Michael took the streetcar to work. Trixie was idly sketching Fred Flintstones when he got there. "Ah. Monday!" she cried. "Yes, Monday," said Michael, who hoisted her onto the desk for a most satisfying round of cunnilingus. Following that, Trixie returned the favour, with more cunnilingus. They were both feeling good after that, though they couldn't keep from petting and teasing their damp pussies for a while, as they talked seriously about philosophy. He told her about what had happened to Linda's mother--she'd tripped over an ottoman leg--and Trixie told him about her plans to walk up the Eiffel Tower. At one point he moved his satchel, and the dowels and connectors inside the cardboard tube inside his satchel rattled.

"What's in there?" Trixie asked.

"Oh, something I'm working on. Like a model." He snapped his fingers. "Hey, I can show you what it's all about. Close your eyes."

"Better still, I'll go get some coffee."

Trixie skipped out of the office. Would she ever return? Meanwhile, Michael opened his satchel and pulled out his cylinder of tinker toys. He dumped everything out on his desk and separated the long dowels from the short dowels. With twelve short dowels and eight connectors he made himself a cube. He made a second cube off what he had built; that is to say, he took eight short dowels and four more connectors to build a cube against the cube, both sharing a square plane. The hard part came next. Using two connectors and four dowels, he made another cube that married two squares, one from each of the cubes he'd constructed, as two squares of a cube 90° off (though they had been 180° off originally). The thing he had made was already beyond three dimensions, but he had to continue. With one connector and three dowels he made a fourth square 90° off the first three. There he had it. He had made a form consisting of eighteen surfaces, twenty-eight edges, and fifteen vertices. He turned it over in his hands, marvelling at how simple and beautiful his 4D form was after all.

Trixie came back, with coffee for both. She saw the thing which she had thereunto figured could not be made. She thought she should drop the coffee in astonishment, but didn't.

Michael looked at her and said: "See? It's not so hard to do."