Anathem

“Right.”

“And how many servings in a small square?”

“Four.”

“So each triangle has enough cake for how many servings?”

“Two.”

“And the square that’s made up from four such triangles has enough cake for—”

“Eight servings,” he said, and then realized: “which is the problem we were trying to solve before!”

“We’ve been trying to solve it the whole time,” I corrected him, “it just takes a minute or two. So, can you cut us eight servings then, please?”

“That’s it,” I said.

“We can eat now?”

“Yes. Do you see what just happened?”

“Uh…I cut eight equal servings of cake?”

“You make it sound easy…but it was hard, in a way,” I said. “Remember, a few minutes ago, you knew how to cut four servings. That was easy. You knew how to cut sixteen. That was easy too. Nine, no problem. But you didn’t know how to cut eight. It seemed impossible. But by thinking it through, we were able to come up with an answer. And not just an approximate answer, but one that is perfectly correct.”

CALCA 2: Hemn (Configuration) Space

A supplement to Anathem by Neal Stephenson

IT JUST SO HAPPENED that in our comings and goings we had kicked over an empty wine bottle, which was resting on the kitchen’s floor like this:

The floor had been built up out of strips of wood, set on edge in a gridlike pattern, which put me in mind of a coordinate plane.

“Get a slate and a piece of chalk,” I said to Barb.

I felt a little guilty bossing him around like this, but I was cross at him for not helping me with the drain. He didn’t seem to mind, and it didn’t take him long to fulfill the request, since slates and chalks were all over the kitchen. We used them to write out recipes and lists of ingredients.

“Now indulge me for a second and write down the coordinates of that bottle on the floor.”

“Coordinates?”

“Yes. Think of this pattern as a Lesper’s coordinate grid. Let’s say each square in the floor pattern is one unit. I’ll put a potato down here, to mark out the origin.”

“Well, in that case the bottle is at about (2, 3),” Barb said, and worked with the chalk for a moment. Then he tipped the slate my way:

“Now, this is already a configuration space—just about the simplest one you could possibly imagine,” I told him. “And the bottle’s location, (2, 3), is a point in that space.”

“It’s the same as regular two-dimensional space then,” he complained. “Why didn’t you say so?”

“Can you add another column?”

“Sure.”

“Notice that the bottle isn’t straight. It’s rotated by something like a tenth of p—or in the units you used to use extramuros, about twenty degrees. That rotation is going to become a third coordinate in the configuration space—a third column on your slate.”

Barb went to work with the chalk and produced this:

“Okay, now it’s starting to look like something different from plain old two-dimensional space,” he said. “Now it’s got three dimensions, and the third one isn’t normal. It’s like something I had to learn once in my suvin—”

“Polar coordinates?” I asked, impressed that he knew this. Quin must have spent a lot of money to send him to a good suvin.

“Yeah! An angle, instead of a distance.”

“Okay, let’s learn something about how this space behaves,” I proposed. “I’ll move the bottle, and whenever I say ‘mark,’ you punch in its current coordinates.”

I dragged the bottle a short distance while giving it a bit of a twist. “Mark.”

“Mark. Mark. Mark…”

I said, “So, this set of points in configuration space is like what we’d get if I accidentally kicked the bottle and sent it skidding and spi

“Sure. That’s kind of what I was thinking!”

“But I moved it in slow motion to make it easier for you to take down the data.”

Barb didn’t know what to make of this very weak attempt at humor. After an awkward pause, I plowed ahead: “Can you make a plot now? A three-dimensional plot of those numbers?”

“Sure,” Barb said uncertainly, “but it’s going to be weird.”

“The dotted line track on the bottom shows just the x and the y,” Barb explained. “The track that it made across the floor.”

“That’s okay—it’d be confusing otherwise, if you’re not used to configuration space,” I said. “Because part of it—the xy track that you plotted with a dotted line—looks just like something that we all recognize from Adrakhonic space; it just shows where the bottle went on the floor. But the third dimension, showing the angle, is a completely different story. It doesn’t show a literal distance in space. It shows an angular displacement—a rotation—of the bottle. Once you understand that, you can read it directly off the graph and say ‘yeah, I see, it started out at twenty degrees and spun around to three hundred and some degrees while it was skidding across the floor.’ But if you don’t know the secret code, it doesn’t make any sense.”

“So what’s it good for?”

“Well, imagine you had a more complicated state of affairs than one bottle on the floor. Suppose you had a bottle, and a potato. Then you’d need a ten-dimensional configuration space to represent the state of the bottle-potato system.”

“Ten!?”

“Five for the bottle and five for the potato.”

“How do you get five!? We’re only using three dimensions for the bottle!”

“Yeah, but we are cheating by leaving out two of its rotational degrees of freedom,” I said.

“Meaning—?”

I squatted down and put my hand on the bottle. The label happened to be pointed toward the floor. I rolled it over. “See, I’m rotating it around its long axis so that I can read the label,” I pointed out. “That rotation is a completely separate, independent number from the kick-spi

“So we’re up to five,” Barb said, “for the bottle alone.”

“Yeah. To be fully general, we’d want to add a sixth dimension, to keep track of vertical movement,” I said, and raised the bottle up off the floor. “So that would make six dimensions in our configuration space just to represent the position and orientation of the bottle.” I set the bottle down again. “But as long as we keep it on the floor we can get along with five.”

“Okay,” Barb said. He only said this when he totally got something.

“I’m glad you think so. Thinking in six dimensions is difficult.”

“I just think of it as six columns on my slate, instead of three,” he said. “But I don’t understand why we need six completely new dimensions for the potato. Why don’t we just re-use the six that we’ve already got for the bottle?”

“We sort of do,” I said, “but we keep the numbers in separate columns. That way, each row of the chart specifies everything there is to know about the bottle/potato system at a given moment. Each row—that series of twelve numbers giving the x, the y, and the z position of the bottle, its kick-spin angle, its label-reading angle, and its tilt-up angle, and the same six numbers for the potato—is a point in the twelve-dimensional configuration space. And one of the ways it starts to get convenient for theors is when we link points together to make trajectories in configuration space.”