4
The sun was rising ahead of them as they reached the top of the hill. Sando turned to Joan, and his face became green with pleasure. “Look behind you,” he said.
Joan did as he asked. The valley below was hidden in fog, and it had settled so evenly that she could see their shadows in the dawn light, stretched out across the top of the fog layer. Around the shadow of her head was a circular halo like a small rainbow.
“We call it the Niah’s light,” Sando said. “In the old days, people used to say that the halo proved that the Niah blood was strong in you.”
Joan said, “The only trouble with that hypothesis being that you see it around your head . . . and I see it around mine.” On Earth, the phenomenon was known as a “glory.” The particles of fog were scattering the sunlight back toward them, turning it one hundred and eighty degrees. To look at the shadow of your own head was to face directly away from the sun, so the halo always appeared around the observer’s shadow.
“I suppose you’re the final proof that Niah blood has nothing to do with it,” Sando mused.
“That’s assuming I’m telling you the truth, and I really can see it around my own head.”
“And assuming,” Sando added, “that the Niah really did stay at home, and didn’t wander around the galaxy spreading their progeny.”
They came over the top of the hill and looked down into the adjoining riverine valley. The sparse brown grass of the hillside gave way to a lush violet growth closer to the water. Joan’s arrival had delayed the flooding of the valley, but even alien interest in the Niah had only bought the archaeologists an extra year. The dam was part of a long-planned agricultural development, and however tantalizing the possibility that Joan might reveal some priceless insight hidden among the Niah’s “useless abstractions,” that vague promise could only compete with more tangible considerations for a limited time.
Part of the hill had fallen away in a landslide a few centuries before, revealing more than a dozen beautifully preserved strata. When Joan and Sando reached the excavation site, Rali and Surat were already at work, clearing away soft sedimentary rock from a layer that Sando had dated as belonging to the Niah’s “twilight” period.
Pirit had insisted that only Sando, the senior archaeologist, be told about Joan’s true nature; Joan refused to lie to anyone, but had agreed to tell her colleagues only that she was a mathematician and that she was not permitted to discuss her past. At first this had made them guarded and resentful, no doubt because they assumed that she was some kind of spy sent by the authorities to watch over them. Later it had dawned on them that she was genuinely interested in their work, and that the absurd restrictions on her topics of conversation were not of her own choosing. Nothing about the Noudah’s language or appearance correlated strongly with their recent division into nations—with no oceans to cross, and a long history of migration they were more or less geographically homogeneous—but Joan’s odd name and occasional faux pas could still be ascribed to some mysterious exoticism. Rali and Surat seemed content to assume that she was a defector from one of the smaller nations, and that her history could not be made explicit for obscure political reasons.
“There are more tablets here, very close to the surface,” Rali announced excitedly. “The acoustics are unmistakable.” Ideally they would have excavated the entire hillside, but they did not have the time or the labor, so they were using acoustic tomography to identify likely deposits of accessible Niah writing, and then concentrating their efforts on those spots.
The Niah had probably had several ephemeral forms of written communication, but when they found something worth publishing, it stayed published: they carved their symbols into a ceramic that made diamond seem like tissue paper. It was almost unheard of for the tablets to be broken, but they were small, and multitablet works were sometimes widely dispersed. Niah technology could probably have carved three million years’ worth of knowledge onto the head of a pin—they seemed not to have invented nanomachines, but they were into high-quality bulk materials and precision engineering—but for whatever reason they had chosen legibility to the naked eye above other considerations.
Joan made herself useful, taking acoustic readings farther along the slope, while Sando watched over his students as they came closer to the buried Niah artifacts. She had learned not to hover around expectantly when a discovery was imminent; she was treated far more warmly if she waited to be summoned. The tomography unit was almost foolproof, using satellite navigation to track its position and software to analyze the signals it gathered; all it really needed was someone to drag it along the rock face at a suitable pace.
From the corner of her eye, Joan noticed her shadow on the rocks flicker and grow complicated. She looked up to see three dazzling beads of light flying west out of the sun. She might have assumed that the fusion ships were doing something useful, but the media was full of talk of “military exercises,” which meant the Tirans and the Ghahari were engaging in expensive, belligerent gestures in orbit, trying to convince each other of their superior skills, technology, or sheer strength of numbers. For people with no real differences apart from a few centuries of recent history, they could puff up their minor political disputes into matters of the utmost solemnity. It might almost have been funny, if the idiots hadn’t incinerated hundreds of thousands of each other’s citizens every few decades, not to mention playing callous and often deadly games with the lives of the inhabitants of smaller nations.
“Jown! Jown! Come and look at this!” Surat called to her. Joan switched off the tomography unit and jogged toward the archaeologists, suddenly conscious of her body’s strangeness. Her legs were stumpy but strong, and her balance as she ran came not from arms and shoulders but from the swish of her muscular tail.
“It’s a significant mathematical result,” Rali informed her proudly when she reached them. He’d pressure-washed the sandstone away from the near-indestructible ceramic of the tablet, and it was only a matter of holding the surface at the right angle to the light to see the etched writing stand out as crisply and starkly as it would have a million years before.
Rali was not a mathematician, and he was not offering his own opinion on the theorem the tablet stated; the Niah themselves had had a clear set of typographical conventions which they used to distinguish between everything from minor lemmas to the most celebrated theorems. The size and decorations of the symbols labeling the theorem attested to its value in the Niah’s eyes.
Joan read the theorem carefully. The proof was not included on the same tablet, but the Niah had a way of expressing their results that made you believe them as soon as you read them; in this case the definitions of the terms needed to state the theorem were so beautifully chosen that the result seemed almost inevitable.
The theorem itself was expressed as a commuting hypercube, one of the Niah’s favorite forms. You could think of a square with four different sets of mathematical objects associated with each of its corners, and a way of mapping one set into another associated with each edge of the square. If the maps commuted, then going across the top of the square, then down, had exactly the same effect as going down the left edge of the square, then across: either way, you mapped each element from the top-left set into the same element of the bottom-right set. A similar kind of result might hold for sets and maps that could naturally be placed at the corners and edges of a cube, or a hypercube of any dimension. It was also possible for the square faces in these structures to stand for relationships that held between the maps between sets, and for cubes to describe relationships between those relationships, and so on.
That a theorem took this form didn’t guarantee its importance; it was easy to cook up trivial examples of sets and maps that commuted. The Niah didn’t carve trivia into their timeless ceramic, though, and this theorem was no exception. The seven-dimensional commuting hypercube established a dazzlingly elegant correspondence between seven distinct, major branches of Niah mathematics, intertwining their
most important concepts into a unified whole. It was a result Joan had never seen before: no mathematician anywhere in the Amalgam, or in any ancestral culture she had studied, had reached the same insight.
She explained as much of this as she could to the three archaeologists; they couldn’t take in all the details, but their faces became orange with fascination when she sketched what she thought the result would have meant to the Niah themselves.
“This isn’t quite the Big Crunch,” she joked, “but it must have made them think they were getting closer.” “The Big Crunch” was her nickname for the mythical result that the Niah had aspired to reach: a unification of every field of mathematics that they considered significant. To find such a thing would not have meant the end of mathematics—it would not have subsumed every last conceivable, interesting mathematical truth—but it would certainly have marked a point of closure for the Niah’s own style of investigation.
“I’m sure they found it,” Surat insisted. “They reached the Big Crunch, then they had nothing more to live for.”
Rali was scathing. “So the whole culture committed collective suicide?”
“Not actively, no,” Surat replied. “But it was the search that had kept them going.”
“Entire cultures don’t lose the will to live,” Rali said. “They get wiped out by external forces: disease, invasion, changes in climate.”
“The Niah survived for three million years,” Surat countered. “They had the means to weather all of those forces. Unless they were wiped out by alien invaders with vastly superior technology.” She turned to Joan. “What do you think?”
“About aliens destroying the Niah?”
“I was joking about the aliens. But what about the mathematics? What if they found the Big Crunch?”
“There’s more to life than mathematics,” Joan said. “But not much more.”
Sando said, “And there’s more to this find than one tablet. If we get back to work, we might have the proof in our hands before sunset.”
5
Joan briefed Halzoun by video link while Sando prepared the evening meal. Halzoun was the mathematician Pirit had appointed to supervise her, but apparently his day job was far too important to allow him to travel. Joan was grateful; Halzoun was the most tedious Noudah she had encountered. He could understand the Niah’s work when she explained it to him, but he seemed to have no interest in it for its own sake. He spent most of their conversations trying to catch her out in some deception or contradiction, and the rest pressing her to imagine military or commercial applications of the Niah’s gloriously useless insights. Sometimes she played along with this infantile fantasy, hinting at potential superweapons based on exotic physics that might come tumbling out of the vacuum, if only one possessed the right Niah theorems to coax them into existence.
Sando was her minder too, but at least he was more subtle about it. Pirit had insisted that she stay in his shelter, rather than sharing Rali and Surat’s; Joan didn’t mind, because with Sando she didn’t have the stress of having to keep quiet about everything. Privacy and modesty were nonissues for the Noudah, and Joan had become Noudah enough not to care herself. Nor was there any danger of their proximity leading to a sexual bond; the Noudah had a complex system of biochemical cues that meant desire only arose in couples with a suitable mixture of genetic differences and similarities. She would have had to search a crowded Noudah city for a week to find someone to lust after, though at least it would have been guaranteed to be mutual.
After they’d eaten, Sando said, “You should be happy. That was our best find yet.”
“I am happy.” Joan made a conscious effort to exhibit a viridian tinge. “It was the first new result I’ve seen on this planet. It was the reason I came here, the reason I traveled so far.”
“Something’s wrong, though, I think.”
“I wish I could have shared the news with my friend,” Joan admitted. Pirit claimed to be negotiating with the Tirans to allow Anne to communicate with her, but Joan was not convinced that she was genuinely trying. She was sure that Pirit would have relished the thought of listening in on a conversation between the two of them—while forcing them to speak Noudah, of course—in the hope that they’d slip up and reveal something useful, but at the same time she would have had to face the fact that the Tirans would be listening too. What an excruciating dilemma.
“You should have brought a communications link with you,” Sando suggested. “A home-style one, I mean. Nothing we could eavesdrop on.”
“We couldn’t do that,” Joan said.
He pondered this. “You really are afraid of us, aren’t you? You think the smallest technological trinket will be enough to send us straight to the stars, and then you’ll have a horde of rampaging barbarians to deal with.”
“We know how to deal with barbarians,” Joan said coolly.
Sando’s face grew dark with mirth. “Now I’m afraid.”
“I just wish I knew what was happening to her,” Joan said. “What she was doing, how they were treating her.”
“Probably much the same as we’re treating you,” Sando suggested. “We’re really not that different.” He thought for a moment. “There was something I wanted to show you.” He brought over his portable console, and summoned up an article from a Tiran journal. “See what a borderless world we live in,” he joked.
The article was entitled “Seekers and Spreaders: What We Must Learn from the Niah.” Sando said, “This might give you some idea of how they’re thinking over there. Jaqad is an academic archaeologist, but she’s also very close to the people in power.”
Joan read from the console while Sando made repairs to their shelter, secreting a molasseslike substance from a gland at the tip of his tail and spreading it over the cracks in the walls.
There were two main routes a culture could take, Jaqad argued, once it satisfied its basic material needs. One was to think and study: to stand back and observe, to seek knowledge and insight from the world around it. The other was to invest its energy in entrenching its good fortune.
The Niah had learned a great deal in three million years, but in the end it had not been enough to save them. Exactly what had killed them was still a matter of speculation, but it was hard to believe that if they had colonized other worlds they would have vanished on all of them. “Had the Niah been Spreaders,” Jaqad wrote, “we might expect a visit from them, or them from us, sometime in the coming centuries.”
The Noudah, in contrast, were determined Spreaders. Once they had the means, they would plant colonies across the galaxy. They would, Jaqad was sure, create new biospheres, reengineer stars, and even alter space and time to guarantee their survival. The growth of their empire would come first; any knowledge that failed to serve that purpose would be a mere distraction. “In any competition between Seekers and Spreaders, it is a Law of History that the Spreaders must win out in the end. Seekers, such as the Niah, might hog resources and block the way, but in the long run their own nature will be their downfall.”
Joan stopped reading. “When you look out into the galaxy with your telescopes,” she asked Sando, “how many reengineered stars do you see?”
“Would we recognize them?”
“Yes. Natural stellar processes aren’t that complicated; your scientists already know everything there is to know about the subject.”
“I’ll take your word for that. So . . . you’re saying Jaqad is wrong? The Niah themselves never left this world, but the galaxy already belongs to creatures more like them than like us?”
“It’s not Noudah versus Niah,” Joan said. “It’s a matter of how a culture’s perspective changes with time. Once a species conquers disease, modifies their biology, and spreads even a short distance beyond their homeworld, they usually start to relax a bit. The territorial imperative isn’t some timeless Law of History; it belongs to a certain phase.”
“What if it persists, though? Into a later phase?”
“That can cause frictio
n,” Joan admitted.
“Nevertheless, no Spreaders have conquered the galaxy?”
“Not yet.”
Sando went back to his repairs; Joan read the rest of the article. She’d thought she’d already grasped the lesson demanded by the subtitle, but it turned out that Jaqad had something more specific in mind.
“Having argued this way, how can I defend my own field of study from the very same charges as I have brought against the Niah? Having grasped the essential character of this doomed race, why should we waste our time and resources studying them further?
“The answer is simple. We still do not know exactly how and why the Niah died, but when we do, that could turn out to be the most important discovery in history. When we finally leave our world behind, we should not expect to find only other Spreaders to compete with us, as honorable opponents in battle. There will be Seekers as well, blocking the way: tired, old races squatting uselessly on their hoards of knowledge and wealth.
“Time will defeat them in the end, but we already waited three million years to be born; we should have no patience to wait again. If we can learn how the Niah died, that will be our key, that will be our weapon. If we know the Seekers’ weakness, we can find a way to hasten their demise.”
6
The proof of the Niah’s theorem turned out to be buried deep in the hillside, but over the following days they extracted it all.
It was as beautiful and satisfying as Joan could have wished, merging six earlier, simpler theorems while extending the techniques used in their proofs. She could even see hints of how the same methods might be stretched further to yield still stronger results. “The Big Crunch” had always been a slightly mocking, irreverent term, but now she was struck anew by how little justice it did to the real trend that had fascinated the Niah. It was not a matter of everything in mathematics collapsing in on itself, with one branch turning out to have been merely a recapitulation of another under a different guise. Rather, the principle was that every sufficiently beautiful mathematical system was rich enough to mirror in part—and sometimes in a complex and distorted fashion—every other sufficiently beautiful system. Nothing became sterile and redundant, nothing proved to have been a waste of time, but everything was shown to be magnificently intertwined.
The New Space Opera Page 14