The Manifold Destiny - An Automorphic Tale
Copyright © 2005, David A. Epstein.
All Rights Reserved.
"It's a very simple problem, pal. Your exhaust system is all screwed up," said the auto mechanic. He was a master at his trade, able to quickly identify what was malfunctioning in a car and then fix it with equal alacrity. There was no question in my mind that he was right on target with his diagnosis, though he didn't check underneath the car, run tests on the catalytic converter or perform a smog check. I was convinced he was correct, not only because of his reputation, but from the confidence that profusely flowed from his persona. I didn't need to ask him about his methodology or how he reached his conclusion. I was a true believer.
"OK, then let's fix it," I said. I had this bad habit of invoking a "we" term to give me the illusion of being involved. I wanted to get in on the action, of course, even though I've never had any interest in car mechanics. I enjoyed the vicarious experiences of daily living that were somewhere just out of the realm of outright day-dreaming or fantasizing.
"Don't you want to first hear about what caused this problem?" he asked.
"You mean like the root cause?" I replied.
"Yeah, exactly. Not just the symptoms, mind you, but the underlying problem. That's what I'm going to attack."
"Nah, I'm not into root-cause thinking these days. I tend to believe the symptoms have lives of their own and have severed all communication with their alleged root cause."
The mechanic started snickering. "Ha. Just what the world needs; another abstract philosopher."
"Well, I've been getting out of that field, actually. I'm detoxing, you might say."
"OK pal, while you're sobering up, let me just say that the whole system stinks. That's where the problem starts."
"So you tend to believe that the car is messed up to begin with." I was meditating on that thought for a while. I wanted to explore its implications.
"Doc, stop thinking of the car as a physical object. You'll begin to think you own it and can do whatever you want with it."
"But you know very well that I DO own it."
"I'll just pretend I didn't hear that. You just remember that it's a very intricate being, like one of your own kids. Keep that in mind while I go write up your estimate." The mechanic walked past the empty car port into his office, shaking his head, interjecting with a few bursts of laughter.
The car was still running, which led me to think that the mechanic was somewhat absent-minded or forgetful. All the extra exhaust emissions couldn't be good for anyone nearby, not to mention it was wasting my gas. Then I thought about how gifted he is. He doesn't let anything get by him. He's undoubtedly letting the car freely run so he could monitor it's performance in an idle state. I convinced myself that this will "tell him everything he needs to know".
I sat down in a nearby chair. I stared at my car, thinking about how it takes me everywhere. The mechanic's words were seeping into me. I became absorbed with what he said. The auto -- the car isn't an object; it's like a kid, he said. It's as if I started giving it the benefit of the doubt. "Yep,I really love that guy. He really is my geographical connector; my loyal companion, indeed", I thought to myself. I need to treat it, errr, I mean, Him better. Perhaps, for a change, I should ask him where HE would like to go.
Constructive as I thought it may have been, I never was one to just sit in my chair and entertain idle thoughts. I loved exploring ideas, but I just couldn't do that sitting still. That's probably the main reason I seldom developed any thoughts I deemed to be worthy while I was in school. I just couldn't sit in my chair and listen to my teacher. I needed to walk around to get the creative juices flowing. The Chi, that Chi energy would circulate more freely once I was in fluid motion. Yeah, that's what it was.
So I arose from my chair. I saw the mechanic, in the office, writing up an estimate for the job. I considered going in and asking him how long it would take to complete it. Instead, I walked towards the car. I was thinking about how the engine was idly running, a steady motion of sound with some periodic fluctuations in emission patterns. Nah, I wasn't going to indulge in comparing the car to your average person. Metaphors, analogies, similies; these are all the devices of self-promoting attention seekers. I like to describe things on their own terms. A thing is a thing in its own self. It's unjust to liken it to something else unless its directly related to it. I was just interested in the sound patterns the engine was making. In particular, I thought about chaos theory. I surmised that the phase shifts must have islands of stability they latch onto.
Well, the hood was open, you see. How inviting was that? The loud sounds and emisions were certainly inhibitors, but not deterrents. I cautiously walked over to the car. Then, with my hands held in front of me, and head slightly turned away, I peaked in, slowly and carefully. I had this image lodged in my mind that I was perilously perched on the edge of a cliff, afraid I was going to fall into the rough waters below. That gave me a cold chill, in spite of the warmth of the car.
The first thing I noticed was the vibration of the motor. It was idling at a rapid pace. Each motion was slight, but noticeable. The aggregation of these perturbations was more intense than the individual motions; each one appeared to build upon its predecessor. Then, I could see the fan belt. It was amusing to watch it perpetually whirrrrrrrrr. This captivated my attention for a couple of minutes. I thought about all the "little people" that worked on that conveyer belt. That's what kept that engine going, I playfully thought. I was wondering why I couldn't see the pistons moving up and down. I guess it was occurring beneath the engine, not visible to curious onlookers like me. Nevertheless, I peered a little closer. I felt a gust of heat blow right into my face. A couple drops of some oily substance flew right by. That was followed by a suction. It was intensifying. I anchored my hands on the front of the car, but to no avail. I was cycloning in to ...
Where was I?
I was sucked into something, spiraling in. But wait, I'm quite aware I'm standing right at the edge of the car. I'm right where I was a second ago. Yet, somehow, some way, I'm on the other side. That was a startling contradiction. I wanted to explore its implications, but ... WHOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOSH once again.
The colors changed. So did the terrain, for I was now in a dark chamber. I was breathing heavily, and my heartbeat was pulsating more rapidly than before. My sense of touch was enhanced. I more perceptibly felt the air surrounding me. It was penetrating the pores of my body. I thought the mechanic came over and did something to the car. Perhaps while I was mentally drifting off, he slipped underneath and fixed the emissions problem. I glanced over at the office; but where did it go?
Then I heard this loud male voice. It sounded like a drill sergeant. "LISTEN UP NOW." I looked around, I even checked behind. I looked up, I looked down; but I didn't see anyone.
"I SAID LISTEN UP. THAT MEANS YOOOU!". Where was that voice coming from? Who was speaking to me in that manner? "I WANT ALL OF YOU PATHETIC RODENTS TO ASSEMBLE INTO TWO GROUPS." Assemble into two groups? To whom and what was he refering to?
Before I could think about it, I saw some people emerge into view.
"Now soldiers, all of you to with last names beginning with A to M to go to the left. The rest of you weasils, go to the right." Whoever was speaking toned down a bit. The people looked at each other, confused. Some uttered "what's going on", others put their hands up in an "I don't know" formation, still others shook their heads in disbelief.
"DO YOU HAVE TROUBLE UNDERSTANDING ME? MOVE IT, NOW, ON THE DOUBLE." In cross migrational motions, some strolled to the left, and others to the right. Several people bumped into each other; there was an occasional shove here and there. After observing this movement for a while, I headed to the right. We completed this task in a few seconds, though I can't imagine anyone doing it without some trepidation or bewilderment. If the rest were like me, we were mesmorized and fearful of the voice from the unknown.
One of the people did speak up. "Who are you? Why don't you show yourself?"
"NEVER MIND THAT. THE ONLY THING YOU NEED TO CONCERN YOURSELF WITH IS FOLLOWING ORDERS." Out of nowhere, the enquiring gentleman was zapped with a large electrical spark.
"OK, let's try again," instructed the unseen person. "Those in the left group, you are the sweepers. You will be known as Team S. And those in the right group, you are the blowers. You will be known as Team B. GOT THAT?"
"What are we supposed to do. Why are we here anyway," asked a nearby hapless soul.
"QUIET. IF ANYONE ASKS ANY QUESTIONS AROUND HERE, IT'S GONNA BE ME." Silence rapidly permeated the dark cavern.
"As I was explaining, the sweepers are Set S, err, I mean Team S, the blowers Team B. You'll both be in open groups, meaning that some of you can stroll into the other group, while new people will join either of the two. You'll see."
While I was wondering what was going to happen next, a gust of smoke and soot blew into the cavern. We all started coughing, and several of us were wiping our eyes. It's as if we were attending an antiwar rally and the police sprayed us with tear gas and dropped a ton of crap on us. It was unsightly, smelly, and certainly unjust, to say the least.
"Great, now you know what you're up against," spoke the unseen drill sergeant. "The sweepers are going to fan out and sweep that dirt away. And the blowers, well, ...". He abruptly stopped speaking. I saw a few sparks fly past us. This triggered some vaporizing action that caused the blowers to float above the sweepers. We were floating in air. Some of us looked at each other with fear and disbelief. A few of us gasped, while I myself felt this shocking sensation spread throughout my body. The sergeant spoke again: "Floaters, blowers, whatever your are, start blowing that carbon monoxide crap out of the area."
"Should we work together or separately," asked one of my fellow blowers.
"I SAID, NO QUESTIONS. Anyway, start sweeping, start blowing, all of you. Sometimes you'll work together, other times separately, like when you're in different pipes. You'll figure it out."
The two groups were accommodating and performed their respective functions. At first, we were all doing our own thing. We were blowing in various directions, and quite naturally, at different rates; it was very counterproductive. Then, we formed a V-shape formation and synchronized our efforts to blow the smoke in a uniform direction. It appeared the sweepers were doing the same; it just took them a while longer to get their act together and achieve similar results. Suffice it to say that we were blowing and sweeping our stuff straight ahead towards the pipage.
"OK, that's a good start. Keep it up," said the sergeant. "Now we're seeing the beginning of a productive enterprise."
I was noticing that some of the dirt was rising from the floor of the cavern. The blowers were not only blowing smoke, as they were instructed to do, but some dirt as well. That certainly made things much messier. And more people started to cough more regularly. One of the blowers slightly descended to blow some dirt down to the floor. This allowed one of the sweepers to collect more dirt to push forward towards one of the pipes.
I decided to ask a question. "It's obvious that smoke and dirt sometimes mix together. Heck, it's doing it right now. How should the two groups work together to sort this out?"
"I SAID NO ... Well, that turns out to be a good one, and I was just going to explain it anyway. Yes, there will be times when they'll stick together. You'll try to separate them so that you can each concentrate on your own substance; but sometimes it won't happen. In those cases, the blowers will be blowing some dirt, and the sweepers will be sweeping some smoke."
"So we'll have to coordinate our efforts. We can't just worry about blowing smoke, but also dirt that the sweepers will eventually have to sweep."
"Ah, we have here a regular know-it-all," replied the sergeant. "Maybe he would like to explain the relationship between the two groups of people".
"Well, yes, there must be a functional relationship between ...". I was rudely interrupted, unsurprisingly.
"SHUTUP. I'LL DO THE EXPLAINING AROUND HERE ... There is a mapping, let's just call it M. Now, abracadabra, and poof." Magically, a huge 'M' materialized out of the thin air, with a bright red light shining on it. "You see the M now, don't you kids. This M is gonna take the work of you sweepers." He stopped for a second and pointed to the other group of people; the red light was now shining on them. Then he resumed speaking. "And it will continuously apply it to the work of you blowers." Naturally, he was now pointing at us, and concurrently, so did the red spotlight. "Got that? At every point of this mapping, we can determine how much change in productive sweeping is enhancing the productive output of you blowers." Indeed, I could see this cross-pollenation before my eyes. There were these animated, neon-like directional arrows flowing through the M, originating from the sweepers and pointing to us. It was a virtual reality demonstration, of sorts.
"Conversely, there is an inverse mapping that takes the work of you blowers and applies it to the work of you sweepers. At every point, we can determine how much change in productive blowing is enhancing the productive output of the sweepers." Not surprisingly, the directional arrows of the M were now originating from us and pointing to the sweepers.
One of the men spoke to someone nearby, and uttered, "what is he saying".
"IT'S CALLED A DIFFEOMORPHISM, YOU IMBECILE. AND IT'S THE FOUNDATION OF ...". He paused for a couple seconds. "Well, you'll see."
As we resumed our tasks, working together and separately, some of us were switching roles, just as the sergeant prophesized. The red light flickered between these people. A couple of the blowers who were out of breath agreed to become sweepers. And a couple minutes after that, two of the sweepers who's hands were getting tired floated upwards to become blowers. There were also a few new people who mysteriously appeared in the premises. We started explaining to the new blowers what to do, and we placed them in the V-shape formation.
"OK, let me continue," said the sergeant. "I'm gonna explain what really makes this work. This mapping M that I was just talking about." We saw the bright red M appear again.
"The diff something or another," someone replied.
"That's right. The diffeomorphism. Well, you take a subset, I mean, a section of all possible work outcomes, like this." A luminscent, blue sphere appeared and was vibrating with a low-pitched humming sound. Slowly but surely, it surrounded all of the sweepers. "This sphere is an example of what's called an n-dimensional manifold. To make this operational, the mapping of the work of you sweepers within that sphere ..." Again, he stopped for a few seconds to point to that group of people. The directional arrow of the bright red M now passed through the blue sphere, pointing right at us. Then he resumed again. "Must equal the work of you blowers within this ..." Once again, he paused, and I was surprised to see another sphere emerge from the blue one. This new sphere was light green, and it expanded rapidly to encompass the entire volume of the cavern. It engulfed everyone. "within this n-dimensional space, uhh, I mean this green sphere."
"What the ...", replied one of the sweepers.
"SILENCE. YOU DON'T HAVE TO UNDERSTAND EVERYTHING. JUST LISTEN UP. All you need to know is the work of the sweepers and blowers influence each other through the presence of this spatial, mathematical manifold. And that is why you need to keep working and get all of THIS CRAP OUT OF THIS EXHAUST MANIFOLD. Through the pipes thy destiny lies. Onwards to the catalytic converter, the great omni purifier."
Most of the blowers entered the pipe where more smoke was present, while the sweepers entered the one with more dirt and soot. Some of each group joined with the majority of the other group. There were a few sweepers working with the blowers, and some blowers working with the sweepers. This made perfect sense since the soot mixed with the smoke. Blowing ground dirt would create more work for the blowers; it made greater sense for the sweepers to move it on the ground.
I kept blowing in the pipe and continued to rapidly move forward. Our group tried staying in V-formation, as this was an effective means of clearing out the smoke in the region. Unfortunately, the pipe was too narrow for us to retain our arrangement. We eventually disbanded into a few semi-autonomous lines, located at different vertical levels; some of us were at the bottom of the pipe while others floated to the top. The sweepers actually were working faster than the rest of us. They were better organized, but even if they were not, they had the distinct advantage of being relegated to the bottom of the pipe. We struggled to keep up with them.
It wasn't until about an hour later that we emerged from the pipe. We were greeted by the manifold docent who directed us to the catalytic converter. "You just go that way," she said.
"Where are the sweepers?" I asked.
"They're already there."
"And they're waiting for us?"
"You'll just have to see for yourself."
In a matter of a few minutes, we made our way to the catalytic converter. The sweepers were nowhere to be found. I figured they had already passed through the converter. I spoke with other blowers and we decided to enter on our own. As we entered, we were sucked into a much darker chamber. The interior was platinum plated. We saw carbon monoxide and hydrocarbons enter the compound. It was utterly smelly and unsightly, to say the least. We were all coughing again, hacking up a storm. My eyes were teary and blurry to the point that I could barely see in front of me. I inadvertantly bumped into several people. As this escalated, we were getting quite irritable.
I could feel the tension build up in "the room". I was ready to scream, when ...
"OK, LISTEN UP NOW. You've been blowing for hours, but you haven't completed the job. Not all the smoke has been blown into the catalytic converter."
"That's a bunch of nonsense," I replied. "We were extremely careful and systematic how we blew it."
"No, you be silent. You're just pulling our leg."
"I'LL BE PULLING A LOT MORE THAN THAT IF YOU DON'T..."
Suddenly, it became even darker. I could no longer hear the sergeant shouting at us. I turned to discuss this latest development with the others. As I spoke, I instinctively realized that something was amiss. Nobody responded to me. There was absolute silence. I called out again; but there was no response. I looked around for them, moving about rapidly, but in scatterbrain directions. I felt abandoned. It was getting darker and smellier. Then, I was splattered with a grimy substance. It felt absolutely disgusting. I started to scream, much more out of pervasive frustration than any residual fear. I screamed and screamed, and screamed some more; until, I felt like I was transmogrified into the sergeant. Without any warning, the light streamed into view. Within a few seconds, the interior was becoming cleaner. The carbon monoxide and other pollutants had dissipated. Although I still felt trapped, it felt refreshing to be rid of the schmutz. And then ...
"It's like one moment I'm drowning in a horrible oil spill in the ocean, then it's mysteriously cleaned up, and now it's like I'm soaking in a warm bathtub," I said out loud, without really thinking about it. Then, someone tapped me on the shoulder. I turned around, and he said to me: "You like using those cheap metaphors, don't you?" It was the mechanic. He was giving me the bill, and maybe the bull as well. My car was now situated right in front of me. This was perplexing, for I don't remember how I got there.
"When, when did you appear?" I asked.
"You were dilly-dallying around while I was fixing it."
"But I thought..."
"Of course you thought," he interrupted. "That's part of the problem. Well, in any case, your exhaust system is working fine now. You shouldn't have any problems."
"We'll thank you."
"Don't mention it. Now let's go into the office and you'll pay your bill."
I hesitated for a moment. I started drifting off in aimless thoughts. Then, he tapped me on the shoulder, and handed me this figure made of paper. I took it and looked at it. It was an orgami-like male doll. The mechanic said, "It's a man I fold. Get it?" I did not. I looked down towards the ground not knowing what to say. Then, he tapped me again, and said, "Come on, let's go in ... ON THE DOUBLE!"