Terrible Events: A poorly documented worldwide catastrophe thought to have begun in the year-5. Whatever it was, it terminated the Praxic Age and led immediately to the Reconstitution.
Thelenes: A great theor of the Golden Age of Ethras, protagonist of many Dialogs, mentor to Protas. Executed by the Ethran authorities for irreligious, or at least disrespectful, teachings.
Theor: Any practitioner of theorics, which see.
Theorician: Nearly equivalent to theor, but with slightly different connotations. “theorician” tends to be used of one who is devoted to highly specific, detailed, technical work, e.g., carrying out elaborate computations.
Theorics: Roughly equivalent to mathematics, logic, science, and philosophy on Earth. The term can fairly be applied to any intellectual work that is pursued in a rigorous and disciplined manner; it was coined by Diax to distinguish those who observed the Rake from those who engaged in wishful or magical thinking.
Thousander: Informal term for a Millenarian (see).
Throw Back: An informal term meaning to subject an avout to the aut of Anathem.
Throwback: An ex-avout who was Anathematized.
Tredegarh: One of the Big Three concents, named after Lord Tredegarh, a mid-to-late Praxic Age theor responsible for fundamental advances in thermodynamics.
Triangle Ark: Alternate term for the Kelx faith or one of its arks.
Unarian: An avout sworn not to emerge from the math or to have contact with the outside world until the next Annual Apert. Informally, “One-off.”
Upsight: A sudden, usually unlooked-for moment of clear understanding.
Uraloabus: Prominent Sphenic theor of the Golden Age of Ethras who, if the account of Protas is to be credited, committed suicide after being planed by Thelenes.
Uthentine: A suur at Saunt Baritoe’s in the Fourteenth Century A.R. who, along with Erasmas, founded the branch of metatheorics called Complex Protism.
Vale-lore: Martial arts. Associated with the Ringing Vale (see).
Valer: An avout of the Ringing Vale; one who has, therefore, devoted his or her entire life to the martial arts.
Vlor: An informal contraction of Vale-lore (see).
Voco: A rarely celebrated aut by which the Saecular Power Evokes (calls forth from the math) an avout whose talents are needed in the Saecular world. Except in very unusual cases, the one Evoked never returns to the mathic world.
Vout: An avout. Derogatory term used extramuros. Associated with Saeculars who subscribe to iconographies that paint the avout in an extremely negative way.
Warden Fendant: A hierarch charged with defending the math or concent from Saecular interlopers, by all means up to and including physical violence, and typically overseeing a staff of more junior hierarchs trained to carry out such functions.
Warden of Heaven: During the years leading up to the time in which Anathem is set, a popular religious leader who obtained Saecular power by claiming to embody the wisdom of the mathic world.
Warden Regulant: A hierarch charged with maintaining the Discipline intramuros, empowered to conduct investigations and to mete out penance. Technically subordinate to the Primate but ultimately answerable to the Inquisition, and empowered to depose the Primate in certain exceptional circumstances.
Wick: In Complex Protism, a fully generalized Directed Acyclic Graph in which a large (possibly infinite) number of cosmi are linked by a more or less complicated web of cause-and-effect relationships. Information flows from cosmi that are more “up-Wick” to those that are more “down-Wick” but not vice versa.
CALCA 1: Cutting the Cake
A supplement to Anathem by Neal Stephenson
“LET’S SAY THAT EACH serving will be a square, the same width as the spatula. Go ahead and cut in one corner of the pan.”
Dath cut the cake thus:
and then made more cuts thus, to produce the four servings I’d asked for:
“I can’t believe you’re doing this!” Arsibalt muttered.
“If it worked for Thelenes…” I muttered back. “Now shut up,” and I turned my attention back to Dath who was awaiting further instructions. “How many servings do we have there?” I asked him.
“Four,” he said, slightly unnerved by my ridiculously easy question.
“Now, what if you cut a similar figure but with sides twice as long? So instead of each side being two units—two spatula-widths—it would be—?”
“Four units?”
“Yes. We have four servings here already—if you doubled the size of the figure, how many people could we serve then?”
“Well, two times four would be eight.”
“I agree that two times four is eight. Go ahead and try it,” I said. Dath made more cuts thus:
Halfway through, he saw his error and made a wry face, but I encouraged him to keep going until he was finished. “Sixteen,” he said. “We actually have sixteen servings. Not eight.”
“So, just to review: when we cut a square grid that is two units on a side, we get how many servings?”
“Four.”
“And you just told me that a four-unit grid gives us sixteen. But what if we only wanted eight servings? How many units would our grid have to be?”
“Three?” Dath said, cautiously. Then his eyes dropped to the cake and he counted it out. “No, that gives nine servings.”
“But we’re getting warmer. And now an important thing has changed, which is that you know you don’t know.”
Dath’s eyebrows went up. “That’s important?”
“It’s important to us in here,” I said.
I couldn’t remember what Thelenes had done next when he had done this with a slave-boy on the Plane six millennia ago, and had to ask Orolo.
I spun the cake around, presenting Dath with an unmarked corner. “Go ahead and cut one square big enough for four servings. You don’t have to cut the individual servings out of it.”
“Can I make lines on the frosting?” he asked.
“If it helps.”
With some hints and nudges from Cord, Dath produced a square like this:
“Good,” I said, “now add three more squares just like it.”
Extending lines he’d already made and adding some new ones, Dath enlarged it to this:
“Now, remind me, how many servings can we get out of that whole area?”
“Sixteen.”
“All right. Now look only at the square in the lower right-hand corner.”
“Is there a way you can divide it exactly in half with only one cut?”
He got ready to slice along one of the dotted lines, but I shook my head. “Arsibalt here is very particular about his cake and he wants to be sure no one gets a larger slice than him.”
“Thank you very much, wise Thelenes,” Arsibalt put in.
I ignored him. “Can you make one cut that’s guaranteed to satisfy him? The pieces don’t have to be square. Other shapes are okay—like triangles.”
With that hint, Dath made a cut like this:
“Now, do the others the same,” I said. He made it like this:
“When you made the first diagonal cut, you cut a square exactly in half, right?”
“Right.”
“And is the same true of the other three diagonal cuts and the other three squares?”
“Of course.”
“So, let’s say I rotate the pan and you look at it this way”:
“What shape do you see in the middle there?”
“A square.”
“And how many servings worth of cake are contained in that square?”
“I don’t know.”
“Well, it’s made up of four triangles, right?”
“Yeah.”
“Each of those triangles is half of a small square, right?”
“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:
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 posi
tion 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.