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 multi-tablet works were sometimes widely dispersed. Niah technology could probably have carved three million years' worth of knowledge on to 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 further 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.
Through 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 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 nearindestructible 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 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 labelling 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 seem 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."