Anathem

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:

x y

2 3

“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.”

x y

2 3 20

“Mark. Mark. Mark…”

x y

2 3 20

3 3.5 70

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 spinning across the floor. Would you agree?”

x y

2 3. 20

3 3.5 70

4 4. 120

5 4.5 170

6 5 220

7 5.5 270

8 6. 320

“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-spinning rotation that you plotted on your slate. So we need an extra dimension for it.” Grabbing the bottle, and keeping its heel pressed against the floor, I now tilted it up so that its neck was pointed up from the floor at an angle, like an artillery piece. “And what I’m doing here is yet another completely independent rotation.”

“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.”

“When you say ‘trajectory’ I think of something flying through the air,” Barb said, “but I don’t follow what you mean when you use that word in this twelve-dimensional space that isn’t like a space at all.”

“Well, let’s make it ultrasimple and restrict the bottle and the potato to the x-axis,” I said, “and ignore their rotations.” I moved them around thus:

“Can you use your slate to record their x positions?” I asked.

“Sure,” he said, and after a few moments, showed me this:

Bottle’s x Potato’s x

7 1

“I’m going to smash them into each other,” I said, “in slow motion, of course. Try to make a record of their positions, if you would.” And, much as before, I began to move the potato and the bottle in small increments, calling out “Mark” when I wanted him to add a new line to his chart.

“The bottle’s moving faster,” he observed, as we worked.

“Yeah. Twice as fast.” I ended up holding the potato on top of the bottle at 3.

Bottle’s x Potato’s x

7 1

6 1.5

5 2.

4 2.5

3 3.

“They just hit each other,” I said, “and so now they are going to bounce apart. But they are going to move slower, because the potato got mashed in the collision and some energy was lost.” With a little over-the-shoulder coaching from me, Barb added several postsmashup points to the table:

Bottle’s x Potato’s x

7 1

6 1.5

5 2.

4 2.5

3 3.

3.2 2.5

3.4 2.

3.6 1.5

3.8 1.

“There,” I said, letting go of the projectiles, and clambering back up to my feet. “Now, all of this action happened along a straight line. So, this is a one-dimensional situation, if you keep thinking in Saunt Lesper’s coordinates. Saunt Hemn, though, would do something here that might strike you as strange. Hemn would think of each row of the table as specifying a point in a two-dimensional configuration space.”

“Treat each pair as a point,” Barb translated, “so, the beginning point is (7, 1) and so on.”

“That’s right. Can you make a plot of that for me?”

“Sure. It’s trivial.”

“That’s weird!” Barb exclaimed. “It’s like Saunt Hemn has turned the whole situation inside out.”

“Well, give me the chalk for a minute and I’ll annotate it in ways that will help you make sense of it,” I said. A few minutes later, we had something that looked like this:

“The collision line,” I said, “is nothing other than the set of all points where the bottle and the potato happen to be at the same place-where their coordinates are equal to each other. And any theor, looking at this plot, even without knowledge of the physical situation-the bottle, the potato, the floor-can see right away that there is something special about that line. The state of the system progresses in an orderly and predictable fashion until it touches that line. Then something exceptional happens. The trajectory makes a hairpin turn. The points become more closely spaced-this means that the objects are moving more slowly, which means that the system has lost energy somehow. I don’t expect you to be bowled over by this, but maybe this can give you an inkling of why theors like to use configuration space as a way to think about physical systems.”

“There’s got to be more to it than that,” Barb said. “We could have just plotted this in a simpler way.”

“This is simpler,” I insisted. “It is closer to the truth.”

“Are you talking about the Hylaean Theoric World now?” Barb asked, half whispering and half gloating, as if this were just about the naughtiest thing that a fraa could do.

“I’m an Edharian,” I answered. “No matter what some people around here might think…that’s what I am. And naturally we seek to express what we are thinking in the simplest, most elegant way possible. In many-no, most-cases that are interesting to theors, Saunt Hemn’s configuration space does that better than Saunt Lesper’s space of x, y, and z coordinates, which you’ve been forced to work in until now.”

Something occurred to Barb: “The bottle and the potato each had six numbers-six coordinates in Hemn space.”

“Yes, in general it takes six numbers to represent the position of something.”

“A satellite in orbit needs six numbers too!”

“Yes-the orbital elements. A satellite in orbit always needs a six-dimensional Hemn space, no matter which coordinate system you use. If you’re using Saunt Lesper’s Coordinates, it leads to the problem you were complaining of earlier-”

“The xs and ys and zs don’t really tell you anything!”

“Yes. But if you transform it into a different six-dimensional space, using six different numbers, it becomes very clear, the same way that the bottle-potato scenario became clear when we chose an appropriate space in which to plot it. For a satellite, those six numbers are the eccentricity, the inclination, the argument of perihelion, and three others with complicated names that I’m not going to rattle off now. But just to name a couple of them: the eccentricity tells you, at a glance, whether or not the orbit is stable. The inclination tells you whether it’s polar or equatorial. And so on.”