Half Circles, Quarter Circles, and So On


I’m making a game. It’s an interactive novel mixed with an RPG. It’s called Firmament. This is a piece of concept art I wrote while designing the world.

In this story, a boy learns an important lesson about binary search.


I was looking for Alexander. A few months ago, my assignment had been switched from changing out capacitors on a little patch in 440A to running integrity checks on some of the Unchanging Sums. I liked this. Working on capacitors meant that your hands would always smart after your Maintenance shift. The sums were easy, though. The worst that would happen with those is that your eyes might cross after a few hours of comparing the readouts to what was in the Manual.

I wasn’t looking for Alexander to tell him about my new assignment, though. Well, maybe in a way. He had asked me to tell him if I saw anything out of the ordinary, especially when I was on maintenance. This itself was out of the ordinary. My older brother had always seemed to have a richer inner world than anyone I knew, or some air of mystery, but in the past months I had noticed it take a different quality. I wasn’t sure if it was my maturing or a real change on his part, and in any case I had my own life going on around me, so it fell to the back of my mind. But I finally decided that what I’d been seeing on my new maintenance was strange. Strange in the way that Alexander can be strange. And so I was looking for Alexander.

The best place to find Alexander was at dinner. Part of his air of mystery was his hours of coming and going, but you could usually count on him to show up when the steel slots opened and slid out the evening’s portion. Plus, skipping dinner was one thing that didn’t fly with Ma. Our pod was big, one of the biggest in the Dome at thirty-three, but Ma made sure everyone knew there were consequences for missing dinner more than once or twice a week.

I spotted him at the head of one of the long, rectangular tables off to the side of the dining area of the pod. He was sitting with five or six of our creche-mates. I got my tray and sat down next to them.

Pop, Foley, Stew, Wino and Manny were at the table with us. Pretty much the usual suspects when it came to Alexander and his hangers-on. Myself included. They were talking about Wino’s new maintenance. She’d just gotten switched over to a notoriously fickle part of the Manual, but I couldn’t really make myself focus on the conversation. The more I saw the abstracted look on Alexander’s face, the more I psyched myself out thinking what he’d say when I told him about the sums.

“It just doesn’t make sense to me – oh, hey Chare,” she said as I walked up.

“Look, here, Chare knows. He worked capacitors for the last six months. Chare, did you ever find a little way to do the capacitors a bit faster that wasn’t in the Manual?” Wino asked.

“Oh, uhm, hey Wino, yeah, I did notice once that if you don’t use rosin core on a lot of those weird patches in the 430s and 440s that it just takes a lot better for some reason,” I said. “I figured that out on accident, though,” I added, glancing at Alexander unconsciously.

“So? Did you start doing it that way?” she asked.

“Wino, don’t make him self incriminate,” Alexander said.

“I don’t like it when you guys talk about this stuff. You make it sound like people change the Manual for any reason they want,” Pop said.

I’m not saying I did anything. Take the stick out of your ass, Pop,” Wino said.

“Sorry, Manual says it’s got to be up there until next sleep cycle. Book seven, chapter forty-three, right between the section on hot swapping chips and what to when the fucking Dome collapses,” Pop replied.

“The Dome isn’t going to fucking collapse because I didn’t walk until my feet bled crossing the Dome cleaning dust traps in the right order,” Wino said. I could tell this was a good argument. It was also an argument we had nearly beaten to death, and probably the reason we clung to Alexander’s domer coveralls together. Ma Bell, Uncle Jerry, Mike, a lot of the folks who really ran the pod were okay with us stretching our minds viz. the Manual, but Alexander would push us to have these arguments. Or to experiment with how loosely the Manual could be interpreted, and how looseness correlated with change. I think Ma Bell knew more about this than she let on, but I know Alexander did some things that would have taken a few kilocycles off of her life if she knew.

“Hey,” Alexander said, “keep it down. Don’t be stupid.” His voice was firm, deep, not at all matching his wiry frame. But it had a certain intensity to it that completely matched Alexander, with his glower, sharp face, dark sunken eyes. Pop and Wino kept going at a slightly reduced volume.

“How’s sums, Chare?” Stew asked me dully after halfheartedly watching them for a few minutes. He startled me out of thought.

“Oh, uh, good, Stew. My hands are already looking better. It’s pretty much as easy as everyone says,” I said. “Except for a couple of sums where you have to switch the summing functor,,” I quickly added. I looked down at my food for a second and then gave Alexander another glance. I was not having trouble with any summing functors. I wanted to show him the strange sums, but I had been under the impression that whatever Alexander was interested in was between the two of us only. Our pod was close, and the crechemates even closer, but that was Alexander. He had never said so outright to me, but I knew there was plenty of things he kept from us. Whatever he was looking for, I couldn’t be sure who was on the inside and who was on the outside.

Stew didn’t seem to notice. “I’ll take a couple of dense functors over cap testing. It’s taken me two hours longer than the Manual says every night this week,” he said with a distant sigh. I knew this was important to Stew. He was a thin, quiet boy, also with his own inner world, but a very different one than the intense and charismatic Alexander.

“Chare, you’re having trouble with the summing functor?” Alexander asked as he popped out of his fugue. “I worked sums a few hectocycles ago. I know what sums you’re talking about. Do you want me to show you?” he asked.

“Sure. I’m done now,” I said with relief.

We left our podspace and walked together quietly until about 400.

“What did you find?” Alexander said.

“Was I, uhm, too obvious?” I asked.

“No,” he replied.

“The sums have been wrong. I have a list of the pages. They’ve been wrong for a few cycles now. The sector starting at 0xFF5A43BF. But they’re not just wrong, they’re…” I paused, trying to think of the right word to use, “…coherently wrong,” I finally decided.

“It started five cycles ago. The sum on the middle page of my sector was off. Eventually I found some entropic bits. The Manual tells you how to fix entropied sectors, and the sums came out right after I fixed htem. The next day, I found a bad sum on the page right between the first page, and the middle page I had fixed yesterday. But what I couldn’t explain was that the offsets into both pages were the same. Location 1,347,” I said.

I expected Alexander to show some surprise here, but he just kept walking. His face was turned toward a point in space, like a man thinking.

I waited a moment to see if he was going to say anything. “If it was just that, I might not have said anything. Entropied sectors happen all the time. You fix them, sometimes they stick around for a few cycles, but they always go away,” I said. “But every cycle, a new page would be corrupted in the same way. The same bits at the same location, just a different page,” I continued.

“Yesterday, it was the same thing. Sector 0xFF5A43BF, page 0x8. I fixed it. But today…” I trailed off. “Uhm, Alex?” I asked. I was the only one who would ever call him Alex.

“Hm?” he replied. His thought was disrupted for a moment.

“This is, uhm, this is going to make me sound, well, kind of crazy. Not just weird.”

“That’s fine. I wouldn’t have asked if I didn’t want to know.” he said.

“Okay,” I said. “Today, the Manual was wrong.” That caught his attention. He wasn’t abstracted in thought anymore. His face darkened, and his eyes burned. I waited again to see if he would say anything. But nothing, so I kept going.

“Uhm, yeah. The sum changed. Not the one the Dome spit out. The sum in the manual. I distinctly remember checking it yesterday, I remember exactly what the Manual said. I kept track of the sums that the Dome was giving me every day, because it was so strange. Yesterday, I fixed an entropic bit at that sector so the sum would match the Manual. Today, the Manual had changed to say the sum was the one I fixed yesterday. The Manual changed, Alexander.”

We didn’t slow as we walked. And Alexander didn’t say anything. We approached the sector where I’d been doing sums. I slowed to a stop to key in, but Alexander kept walking. “Come on,” he said. “Not here.”

We kept walking, circling back around to the 390s, taking a rarely-used tunnel through some of the sectors for the Dome’s self-patching functionality. I was pretty familiar with this part of the Dome, but I wasn’t sure where we were until we popped out near a wide walkway near our pod on surface level. We pulled into our pod. Alexander made a clear effort to tone back his intensity through the common room, where the group that tended a few years older than me and my crechemates were engrossed in a loud card game. No one noticed him, and we slipped into his room.

“What do you think of what you just told me? Does anything stick out to you?” he asked me as soon as we were inside.

Besides the Manual changing, I thought? But being sarcastic to Alexander wasn’t a good idea. He deserved more respect than that, anyway. “Well, the offsets. If, you know, if it was random, there’s no way that it would be the same offset four days in a row.”

“Right. That leaves us with two possibilities. The first: There is some hardware malfunction. One that might not be covered by the Manual, or might be so deep that it’s been lost for an era. The second: External intervention,” he said.

“Uhm, Alex, I don’t like either of those. I mean, we talk about how the Manual might be flawed and stuff, but, but that’s just talk, right? Just philosophy?”

“Everything is talk until it isn’t, Chare,” he said. I didn’t have anything to say. We talked more about what each possibility might mean. It was a strange conversation. I felt that Alexander wasn’t telling me everything that he knew. He kept asking me questions, as if he wanted me to piece this together myself. I felt us going in circles.

“What are we going to do?” I asked. “I don’t know how you could even figure out something like this.”

“Hm. I think it’s pretty simple, Chare. Each sector that we Maintain maps onto something that the Dome does. All we have to do is look. Find something different about the Dome. That’s our advantage. The Dome never changes. Even the smallest difference will stick out,” he said.

“It’s not easy. It could turn out to be nothing. A simple hardware malfunction, something we find in the errata of the Manual. But you have to look,” he said.

I couldn’t think of anything to say. Alexander looked like he was debating something again. I saw him decide, and he took me by the shoulders. “You’re not the only one looking, Chare. I just can’t tell you everything. It’s not because I don’t trust you. It’s just…” he trailed off. His confidence and dark energy parted for just a moment to a tenderness. “If I told you, and I was wrong, it would be very bad,” he said softly.

His softness terrified me. Alexander was not soft. “OK,” I managed.

“But I want you to think about what you saw. Think hard. And keep looking,” he said. I gave a meek nod, and left for my quarters. I wasn’t old enough to have my own quarters yet. When I came in, Stew was in a corner writing something slowly on a pad. Wino and Foley were sitting on the high bed, playing some two person card game. I didn’t want to talk to anybody, so I went to my bed, sealed off the entry into the small, boxed chamber, and put my pillow around my head.

Think. Think hard. I was bright. Sums was easy, but I wasn’t doing sums because I was stupid. The Dome materialized any necessity or luxury we needed – there was no need for anything resembling industry, nor innovation, nor anything in fact that the material world provided. Besides the fact that the Dome provided, there was no means for us to do our own material transformations. In other words: No raw materials. There was simply the Dome. Its smooth, cold metallic walls, some assorted flora and fauna, the slots which birthed our material needs. The easier your job, the more society recognized your worth by returning to you more of your time. Sums wasn’t the easiest, or least time consuming job, but for my age it was a sign of respect of my ability. An ability that could not begin to make headway on the problem at hand.

I thought again about the offsets. It wasn’t just the fact that the Manual had changed. Not that that wasnt a paradigm shifting event if it was true. It was the way it had changed. It was like a blind man distinguishing objects by touch until he found the one with the right shape and texture. The blind man stuck with me. If another citizen was trying to change the Dome, would they not be a blind man? What extra-Manual insight do we have into the Dome? The answer, of course, is none, which is why the Dome has persisted in stasis for the whole of our civilization. The Dome is far too large and too complex to simply induce bit entropy at random in hopes of a meaningful change. No being of appreciable complexity could come to be this way – things were designed, things followed order.

If someone knew of a bit pattern to change the Dome to their end, but didn’t know where to put it – what then? I didn’t know how this was possible, but this was how Alexander thought. Assume first, let the implications fall out, repeat. I kept thinking of the blind man. The blind man wants a specific fruit, of a specific shape. There are uncountably many fruit before him. He must have a way to discard vast swaths of fruit as too small or too large – checking them one at a time was simply unfeasible. I couldn’t fathom how one could do so without foreknowledge of the fruits, foreknowledge of the Dome.

The pattern of replacing pages, moving on to a smaller section of the sector, and trying again; I couldn’t piece together what the mechanism was, but there had to be something there. The blind man must be eliminating the swaths of fruit wholesale. But, on second thought, it almost didn’t matter what the mechanism was. The fact was, there was a pattern. There was intelligence. This was no simple hardware failure, no random permutations of pages in a sector with malformed bits. This was a search, a focused attempt to find just the right location to do…something. In fact, the speed with which the desired sector and page were found didn’t remind me of a search.

It reminded me of a hunt. And I had to find the hunter.