The ancient Orithenans suspected, but didn’t know how to prove, that the tiles of the Teglon were aperiodic: that no pattern would ever repeat. Again, solving the Teglon would have been easy—it would have been automatic—with square or triangular tiles, or any tile system that was periodic. With aperiodic tiles, it was impossible, or at least very unlikely, unless you had some Godlike ability to see the whole pattern in your head at once. Metekoranes had believed that the final pattern existed in the Hylaean Theoric World, and that the Teglon could only be solved by one who had developed the power of seeing into it.
Suur Spry was clearing her throat. I looked up. I was squatting at the edge of a system of tiles fifty feet wide. It was getting hot.
“Sorry,” I said.
“Some people use sticks to push them around. Saves wear and tear on the back.”
“We should probably get out of here, huh?”
“Soon,” she allowed.
First, though, I followed her about as she showed me the remnants of the ancient buildings. All the roofs were gone, of course. Some pillars still stood, and a few courses of stone that had once been walls, now half-buried in blocks that had tumbled down from above. But mostly we were looking at foundations, floors, stairs, and plazas. Active parts of the dig were gridded with string, a geometric touch Adrakhones would have appreciated. The rocks were annotated with neatly brushed letters and numbers put down by diggers of centuries past. Up above, I knew, was a sort of museum where they’d placed many of the artifacts they had found, including presumably the cast of Metekoranes. I imagined that museum should be dark. Nicely ventilated. And cool. “Okay, let’s get out of this barbecue pit,” I proposed, and heard no argument from Suur Spry.
We had stayed later than expected. Partly because it had been fascinating. But—and this probably didn’t say much for my character—mainly because this was the one thing I could do on this journey that would seem almost as cool as Jesry’s space adventure.
My body had healed to the point where it was willing to cut me a little bit of slack, and so during the early part of the climb I was babbling about the Teglon just like all of those geometers of yore who’d gone crazy over it. Soon enough, though, my injuries began talking to me, and excitement was snuffed out by pain. The remainder of the hike was a long silent trudge. Another sluice-bath was called for. I fell asleep. When I woke up it was late afternoon. Orolo was on kitchen duty. I helped him. But we didn’t really get to talk about anything. So more than one of my three days had been gobbled up just like that. Before we retired that evening I warned Orolo we must speak of important things the next day. So after breakfast the next morning we hiked back up to the meadow.
* * *
Sconic: One of a group of Praxic Age theors who gathered at the house of Lady Baritoe. They addressed the ramifications of the apparent fact that we do not perceive the physical universe directly, but only through the intermediation of our sensory organs.
—THE DICTIONARY, 4th edition, A.R. 3000
“After I landed at Bly’s Butte,” Orolo said, “I was like one of those poor cosmographers, just after the Reconstitution, who couldn’t use his atom smasher any more.”
“Yes, I saw that telescope,” I told him, “the pictures you tried to take of the icosahedron…”
He was shaking his head. “I could not see a thing with that. So, my work concerning the aliens had to be based on what I could observe.”
This didn’t make sense to me. “All right,” I said, “what was that?”
He looked at me, mildly startled, as if it ought to have been obvious. “Myself.”
I was nonplussed. Which only showed that I was dealing with the same old Orolo. “How would self-observation help you understand the Geometers?” I asked. For I had already mentioned to him that this was the term people were now using to denote the aliens.
“Well…the Sconics are not a bad place to start. Remember fly-bat-worm?”
I laughed. “Just got a refresher on that a couple of weeks ago. Arsibalt was explaining it to an extra who wanted to know why we didn’t believe in God.”
“Ah, but that’s not what fly-bat-worm says,” said Orolo. “It says only that pure thought alone doesn’t enable us to draw any conclusions one way or another about things that are non-spatiotemporal—such as God.”
“True.”
“The same observations that the Sconics made about themselves must also be true of aliens’ brains. No matter how different they might be from us in other respects, they must integrate sensory givens into a coherent model of what is around them—a model that must be hung on a spatiotemporal frame. And that, in a nutshell, is how they come to share our ideas about geometry.”
“But they share more than that,” I pointed out, “they appear to share the idea of Truth and of Proof.”
“It is a reasonable enough supposition,” Orolo said with a cautious shrug.
“More than that!” I protested. “They emblazoned the Adrakhonic Theorem on their ship!”
This was news to him. “Oh, really? How cheeky!”
“Didn’t you see it?”
“I remind you that I was Thrown Back before I saw the last picture that I took of the alien ship.”
“Of course. But I assumed you had taken other pictures before then—had been taking them for a long time.”
“Streaks and blobs!” Orolo scoffed. “I was only learning how to capture a decent image of the thing.”
“So you never saw the geometric proof—or the letters—or the four planets.”
“That’s correct,” Orolo said.
“Well, there’s much more that you have to know, if you want to think about the Geometers! All kinds of new givens!”
“I can see how excited you are about those new givens, Erasmas, and I wish you all the best in your study of them, but I’m afraid that for me it would all prove a distraction from the main line of the inquiry.”
“The main line—I don’t know what you mean.”
“Evenedrician datonomy,” Orolo said, as if this ought to have been quite obvious.
“Datonomy,” I translated, “that would be study, or identification, of what is given?”
“Yes—givens in the sense of the basic thoughts and impressions that our minds have to work with. Saunt Evenedric pursued it late in his life, after he got locked out of his atom smasher. His immediate forerunner, of course, was Saunt Halikaarn. Halikaarn thought that Sconic thought was badly in need of an overhaul to bring it in line with all that had been discovered, since the time of Baritoe, about theorics and its marvelous applicability to the physical world.”
“Well—how’d he make out?”
Orolo grimaced. “Many of the records were vaporized, but we think he was too busy demolishing Proc and kicking away all the ankle-biters Proc sent after him. The work fell to Evenedric.”
“Has it been an important thing to the Lineage?”
Orolo gave me a queer look. “Not really. Oh, it’s important in principle. But notoriously unsatisfying to work on. Except when great alien ships appear in orbit around one’s planet.”
“So, then…are you finding it satisfying now?”
“Let’s be quite direct and say what we mean,” Orolo said. “You fear that I’m navel-gazing. That on Bly’s Butte I pursued this line of inquiry, not because it was really worthwhile, but simply because I didn’t have hard givens about the Geometers. And that now that we have evidence that they are, or were, physically and mentally similar to us, this line of inquiry should be dropped.”
“Yes,” I said, “that’s what I think.”
“I happen to disagree,” Orolo said. “But things have changed between us. We are no longer Pa and Fid but Fraa and Fraa, and fraas disagree, cordially, all the time.”
“Thank you but it has certainly felt like a Pa/Fid conversation to this point.”
“Largely because I have a bit of a head start on you.”
I let this polite nothing pass without comment. “Lis
ten, if I can tear you away from Evenedrician datonomy, we have to talk about Saecular stuff for a minute.”
“By all means,” Orolo said.
“Several of us were Evoked to a Convox at Tredegarh,” I said, for, unbelievably, Orolo had not yet expressed any curiosity as to why or how I’d turned up at Orithena. “One of the others was Fraa Jad, a Thousander. He accompanied me and Arsibalt and Lio to Bly’s Butte—”
“And saw the leaves on the wall of my cell there.”
“He—Jad—figured out quickly—weirdly quickly—that you had gone to Ecba and, I guess, that you had ideas about the Geometers that he wanted to know more of.”
“It was neither quick nor weird,” Orolo said. “All of these matters are connected. It would have been obvious to Fraa Jad as soon as he walked in.”
“How? Do you guys communicate? Violate the Discipline?”
“What do you mean, ‘you guys’? You are carrying around some melodramatic idea of the Lineage, aren’t you?” Orolo said.
“Well, just look at this place!” I protested. “What is going on?”
“If I got interested in meteorology,” Orolo said, “I’d spend a lot of time observing the weather. I would come to have much in common with other weather-watchers whom I’d never met. We would think similar thoughts as a natural result of observing the same phenomena. Nine-tenths of what you think of as mysterious Lineage machinations is explained by this.”
“Except that instead of watching the weather you’re thinking about Evendrician datonomy?”
“Close enough.”
“But there was nothing about Evenedric or datonomy on the wall of your cell for Fraa Jad to see. Just material pertaining to Orithena, and a chart of the Lineage.”
“What you identified as a chart of the Lineage was really a sort of family tree of those who have tried to make sense of the Hylaean Theoric World. And it turns out that if you trace the branches of that tree and, so to speak, prune off all the branches populated by fanatics, Enthusiasts, Deolaters, and dead-enders, you end up with something that doesn’t look so much like a tree any more. It looks like a dowel. It starts with Cnoüs and runs through Metekoranes and Protas and some others, and about halfway along you encounter Evenedric.”
“So Fraa Jad, looking at your tree-pruned-down-to-a-dowel, would guess immediately that you must be working on Evenedrician datonomy.”
“And would assume I was doing so in hopes of gaining upsight as to how the Geometers’ minds must be organized.”
“What about Ecba? How’d he guess you went to Ecba?”
“This math was founded by people who lived in the same cells where Fraa Jad has spent his whole life. He would know or surmise that if I could get to this place they would let me in the gates and provide me with food and shelter—quite obviously a better existence than what I could manage at Bly’s Butte.”
“Okay.” I was feeling relieved of a burden I’d been carrying since that day above Samble. “So there’s not a conspiracy. The Lineage doesn’t communicate through coded messages.”
“We communicate all the time,” Orolo said, “in the way I mentioned.”
“Meteorologists watching the same cloud.”
“That’s good enough for this stage of our conversation,” Orolo said. “But you haven’t yet unburdened yourself of whatever terribly important-seeming message or mission you brought in the gates with you. What errand has Fraa Jad sent you on?”
“He said ‘go north until you understand.’ And I guess that part of the mission is accomplished now.”
“Oh really? I’m pleased that you understand. I’m afraid I am still full of questions about these matters.”
“You know what I mean!” I snapped. “He also implied I was to come back to Tredegarh later. That he’d see to it I didn’t get in trouble. I guess he wanted me to fetch you. To bring you back to the Convox.”
“In case I’d developed any ideas, concerning the Geometers, that might be useful,” Orolo hazarded.
“Well, that’s the point of a Convox,” I reminded him, “to be useful.”
Orolo shrugged. “I’m afraid I don’t have enough givens to work with, concerning the Geometers.”
“I’m sure that all the givens that there are to be had, are available at Tredegarh.”
“They are probably collecting exactly the wrong sort of information,” he said.
“So go there and tell them what to collect! Fraa Jad could use your help.”
“For me and Fraa Jad and some others of like mind to try to change the behavior of this Saecular/Mathic monstrosity called a Convox sounds like politics, which I am infamously bad at.”
“Then let me try to help!” I said. “Tell me what you’ve been doing. I’ll go back to the Convox and look for ways to use it.”
The most charitable way to interpret the look Orolo now gave me was affectionate but concerned. He waited for my brain to catch up with my mouth.
“Okay,” I said, “with a little help from some of the others, maybe.” I was thinking of the conversation I’d had with Tulia before Eliger.
“I can’t advise you on what to do at the Convox,” he finally said, “however, I am happy to explain what I’ve been up to.”
“Okay—I’ll settle for that.”
“It won’t help you—in fact, it’ll probably hurt you—at the Convox. Because it will sound crazy.”
“Fine. I’m used to people thinking that we are crazy because of the whole HTW thing!”
Orolo raised an eyebrow. “You know, on balance I think that what I’m about to discuss with you is less crazy than that. But the HTW”—he nodded in the direction of the Orithena dig—“is a cozy and familiar form of craziness.” He paused for a few moments, returning his gaze to me.
“Who are you talking to?” Orolo asked.
I was wrong-footed by this bizarre question, and took a moment to be certain I’d heard the question right. “I’m talking to Orolo,” I said.
“What is this Orolo? If a Geometer landed here and engaged you in conversation, how would you characterize Orolo to it?”
“As the man—the very complicated, bipedal, slightly hot, animated entity—standing right over there.”
“But depending on how a Geometer sees things, it might respond, ‘I see nothing there but vacuum with a sparse dusting of probability waves.’”
“Well, ‘vacuum with a sparse dusting of probability waves’ is an accurate description of just about everything in the universe,” I pointed out, “so if the Geometer was not able to recognize objects any more effectively than that, it could scarcely be considered a conscious being. After all, if it’s having a conversation with me, it must recognize me as—”
“Not so fast,” Orolo said, “let’s say you are talking to the Geometer by typing into a jeejah, or something. It knows you only as a stream of digits. Now you have to use those digits to supply a description of Orolo—or of yourself—that it would recognize.”
“Okay, I’d agree with the Geometer on some way to describe space. Then I’d say, ‘Consider the volume of space five feet in front of my position, about six feet high, two wide, and two deep. The probability waves that we call matter are somewhat denser inside of that box than they are outside of it.’ And so on.”
“Denser, because there’s a lot of meat in that box,” Orolo said, slapping his abdomen, “but outside of it, only air.”
“Yes. I should think any conscious entity should be able to recognize the meat/air boundary. What’s on the inside of the boundary is Orolo.”
“Funny that you have such firm opinions on what conscious beings ought to be able to do,” Orolo warned me. “Let me see…what about this?” He held up a fold of his bolt.
“Just as I can describe the meat/air boundary, I can describe how bolt-stuff differs from both meat and air, and explain that Orolo is wrapped in bolt-stuff.”
“There you go making assumptions!” Orolo chided me.
“Such as?”
“Let?
??s say that the Geometer you’re talking to has been inculcated in his civilization’s equivalent of the Sconics. He’ll say, ‘Wait a minute, you can’t really know, you’re not allowed to make statements about, things in themselves—only about your perceptions.’”
“True.”
“So you need to rephrase your statement in terms of the givens that are actually available to you.”
“All right,” I said, “instead of saying, ‘Orolo is wrapped in bolt-stuff,’ I’ll say, ‘When I gaze at Orolo from where I’m standing, I see mostly bolt, with bits of Orolo—his head and his hands—peeking out.’ But I don’t see why it matters.”
“It matters because the Geometer can’t stand where you are standing. It has to stand somewhere else, and see me from a different angle.”
“Yes, but the bolt wraps all the way around your body!”
“How do you know I’m not naked in back?”
“Because I’ve seen a lot of bolts and I know how they work.”
“But if you were a Geometer, seeing one for the first time—”
“I’d still be able to surmise that you were not naked in back, because if you were, the bolt would hang differently.”
“What if I got rid of the bolt and stood here naked?”
“What if you did?”
“How would you describe me to the Geometer, then? What would meet your eye, and the Geometer’s?”
“I would say to the Geometer ‘From where I stand, all I see is Orolo-skin. From where you stand, O Geometer, the same is likely true.’”
“And why is it likely?”
“Because without skin your blood and guts would fall out. Since I can’t see a puddle of blood and guts behind you, I can infer that your skin must be in place.”