«Is that your own plan?» asked Parantham.
«No.» Sida inclined her head toward her companions. «We might have chosen to ignore the bulge staring down at us, but we're still chasing mysteries of our own.»
«I see.» Parantham left no doubt that she wanted to hear more.
Fith said, «There are plenty of Interesting Truths to be found, even now.»
Though the quad words were slightly ambiguous, Rakesh understood immediately: «Interesting Truths» referred to a kind of theorem which captured subtle unifying insights between broad classes of mathematical structures. In between strict isomorphisms — where the same structure recurred exactly in different guises — and the loosest of poetic analogies, Interesting Truths gathered together a panoply of apparently disparate systems by showing them all to be reflections of each other, albeit in a suitably warped mirror. Knowing, for example, that multiplying two positive numbers was really the same thing as adding their logarithms revealed an exact correspondence between two algebraic systems that was useful, but not very deep. Seeing how a more sophisticated version of the same linkage could be established for a vast array of more complex systems — from rotations in space to the symmetries of subatomic particles — unified great tracts of physics and mathematics, without collapsing them all into mere copies of a single example.
Paba offered them a description of the work that the three friends were pursuing. Rakesh absorbed only the first-level summary, but even that was enough to make him giddy. Starting with foundations in the solid ground of number theory and topology, a glorious edifice of generalisations and ever-broader theorems ascended, swirling into the stratosphere. High up, far beyond Rakesh's own habitual understanding, no less than five compelling new structures that the trio had identified had started to reveal intriguing echoes of each other, as if they were, secretly, variations on a single theme. The elusive common thread had yet to be delineated, but it seemed plausible to Rakesh (albeit with all the fine details glossed over) that sufficient effort would eventually reveal one dazzlingly beautiful and powerful insight that accounted for the subtle fivefold symmetries they had charted.
Parantham said, «So much for the cliché that embodiment is the antithesis of abstraction.» She sounded impressed, and Rakesh suspected that she'd looked more closely at the work-in-progress than he had.
«I've always believed the opposite,» Fith replied firmly. «You don't need to turn every mathematical space into a kind of scape, and literally inhabit it, in order to understand it. Anchored in three dimensions, obeying mundane physics, we can still reason about any system you care to describe with sufficient clarity. That's what
general intelligence
means, after all.»
«How long have you been searching for something like this?» Rakesh asked.
«Thirteen hundred years,» Paba replied. Rakesh glanced at her précis; that was most of her life. «Not full-time,» she added. «Over the years, for one or two days in every ten or twenty as the mood has struck us.»
Sida said, «I've known people who've given their whole lives over to the same kind of search, but if they find nothing in a century or two they usually become discouraged. The only way we could do this was by refusing to make it the be-all and end-all. The only way we could afford to try was by ensuring that we could also afford to fail.»
«That sounds like a good strategy,» Rakesh said. He had never been drawn to the same ethereal heights himself, but he wondered if travelers could benefit from a similar approach. His youthful vow to leave his home world after exactly one thousand years, as if he'd expected fate to hand him the ideal destination at that very moment, seemed increasingly foolish. He might have passed another two or three centuries happily on Shab-e-Noor, if he'd found some way to make himself receptive to the kind of serendipity that had ultimately rescued him from the limbo of the node, without subjecting himself to the same miserable feeling that every day without success was wasted.
The five of them sat talking until noon, then the quads took them to the guest shelter to eat. Rakesh's body was flexible enough to make use of almost anything — or at the very least to survive its ingestion without harm — but the quads had a garden that was equally flexible. Instructed in his tastes, within half an hour the plants were able to form fruits and leaves that even his wild ancestors would have found nourishing and delicious. Fith insisted on cooking them into a spicy stew, using tools rather than his mouth to manipulate the ingredients, no doubt having been briefed by Massa's library on certain peoples' preference for food wholly unmasticated by others.
This, Rakesh thought, was the Amalgam at its best. Even these citizens who shared no molecular ancestry with him had made him welcome on their planet, in their town, at their meal. They had shared their ideas and discoveries, and listened attentively to his own stories and opinions.
His next hosts would be very different. For one and a half million years, the Aloof had made it clear that they needed no one's company, no one's stories, and no one's opinions but their own.
Nevertheless, it seemed that they wanted something now: some contact, some flow of information. It had started with Lahl, but Rakesh had no idea where it would end, or what the transaction would finally amount to. A disinterested exchange of scientific data? An act of trade, of mutual benefit? Munificence? Misunderstanding? Deception? Enslavement?
He and Parantham stayed with their friends until the stars of the bulge filled the sky, then they prepared themselves to walk among them.
6
«Three,» Zak said, «is a beautiful number. Three is what the map shows, which means somebody else who cared about these things believed that three was correct. And three makes perfect sense, if the weights come from something simple.»
He fell silent, brooding. He pushed against the wire he'd been holding and began to drift slowly away from Roi, back into the depths of the Null Chamber, but before he'd gone far he reached out and stopped himself.
«And yet?» Roi prompted him.
«And yet three is not what we found. What we found was
two and a quarter
.» Zak seemed torn between melancholy and excitement, as if he couldn't decide whether this strange result was simply a failure of his methods and reasoning, or a hint of some kind of deeper revelation, if only he knew how to decipher it.
«I can't be sure that my measurements were correct,» Roi confessed. «I was as careful as I could be, but—
Zak cut her off. „This is not down to you. I took many measurements myself. A few other people helped me, as well. Throughout the Splinter, whoever was doing the measuring, the result was the same: moving garm or sard increases your weight by two and a quarter times as much as moving shomal or junub. Not three times. Nowhere, never, is it three.“
„Perhaps there's some error in the navigation signs,“ Roi suggested. „Perhaps the weight itself distorts the way the signage teams mark out their distances.“
„No. I checked that. I found small, random discrepancies, not some systematic distortion. We all make errors: me, you, the signage teams. Enough to mistake two for two and a quarter, perhaps. But not three.“
„Apart from the map, then,“ Roi said, „why can't the true value be two? If you had never found the map, would you be satisfied with two?“
Zak made a chirp of wry admiration. „That's a good question. Maybe I've been fooling myself. Maybe I've let a mapmaker who I've never even met corrupt my idea of simplicity.“
„Tell me,“ Roi begged him, „why you think it should be three. How can one answer be favored over another? How can anything in the world be more than chance?“ This was what she'd come here to learn: the answer to the impossible question that had dragged her away from her work team, away from everything she knew and trusted.
„If I'm right,“ Zak said, „then weight is all about motion, and motion is all about geometry. That's where the simplicity comes from.“
With these cryptic words hanging in the air, he led Roi along the wire, deep into the Null Chamber. She struggled to remain calm; it was hard enough being weightless anywhere, but at least in the tunnels there'd been rock all around her to assuage the feeling that she was constantly falling. Here, out in the middle of the chamber with only Zak's flimsy web to hold on to, the confusion and contradictions were starker. The fact was, Roi had no trouble supporting herself with the weakest of single-clawed grips, and even if she released that claw-hold accidentally she'd easily have time to regain it. She had probably never been in less danger of
falling
in her life. So why did the lack of weight, the very thing that guaranteed her safety, also make her feel that she was forever on the verge of being dashed against the walls of the chamber?
„Here at the Null Line, with no wind and no weight to confuse us, how do things move?“ Zak took a stone from his carapace and tossed it gently away from them. „What do you see?“
Roi said cautiously, „As far as I can tell, that stone moved smoothly in a straight line until it hit the wall.“
„Good. I don't expect you to be certain about anything from one crude experiment, but for the sake of argument let's suppose that's true:
weightless things move smoothly in a straight line
. And I'll tell you something more from my own experience, which you can confirm for yourself when you feel more confident: once I've given myself a push and started moving across the chamber, it doesn't matter how fast I make myself travel; except for the slight touch of the air passing over me I really can't feel any difference. Weightlessness is weightlessness, as long as you're moving smoothly, and the only thing that stops you moving smoothly is contact with a wall, or a wire.“
Zak led her to a small apparatus attached to the wire that marked the Null Line. It was a tube of susk cuticle, containing a spring with a stone at one end, much like the one she used to measure weights. Here, of course, the spring was unstretched, and the stone lay beside a mark on the tube that indicated no weight at all.
The end of the tube opposite the stone was attached to the wire by a small loop that allowed it to pivot. Zak flicked the tube and set it spinning, the free end sweeping out a circle while the other remained fixed. „What do you see?“
„The spring is stretched now,“ Roi observed. „As if the stone had weight.“
„Yes.“ Zak reached over and gave the tube another sharp tap, setting it moving faster. „And now?“
„It's stretched even more. As if the weight had increased.“
„Good. Now let's put some numbers to this.“
Zak took a sheet of cured skin from his carapace, and had Roi count while the tube spun around, to judge how quickly it made each revolution. Six times, they spun the tube and recorded both the time it took to complete a circle and the weight indicated by the stretching of the spring. A special kind of pointer that could only move one way under pressure from the stone made it possible to read the weight off the scale after the tube was brought to a halt; squeezing the pointer made it narrower and allowed it to be slid back, resetting the weight.
Zak said, „Multiply the weight by the time, and then by the time again.“
Roi stared at the skin, as if the answers might simply leap into her mind, but nothing happened. „I can't do that,“ she admitted. She understood the concept, but when it came to manipulating actual figures she had only been taught how to add and subtract. „None of my teams ever needed multiplication.“
„All right, don't worry, I'll teach you later.“ Zak moved down the list of figures, rapidly scratching in the results. Although the individual times and weights varied greatly, the numbers produced by his calculation — weight by time, by time again — were all similar, all close to two hundred and seventy.
Roi was mystified. „Two hundred and seventy? What does that mean?“
„Nothing. Ignore the particular value, it's just a measure of such things as how fast you count, and how we assign numbers to weights. The important thing is, we always get the same value, however fast or slow the stone is moving. There's a rule here, there's a pattern.“
„Not a very simple one,“ Roi protested.
„Be patient.“
Zak modified the experiment, shifting the spring and the stone further along the tube, doubling the stone's distance from the pivot. Six more times they spun the tube. When Zak calculated the same quantity again, it was no longer fixed around two hundred and seventy, but had doubled to five hundred and forty.
He repeated the experiment again, then again, each time with the spring shifted further.
„Now we divide by the distance. Weight multiplied by time, by time again, divided by distance.“ All the numbers this new calculation produced were more or less the same, regardless of the distance from the pivot. By combining all the variables in this way, a constant value again emerged.
Roi had no idea why. She said, „Spinning the tube around gives the stone weight, I can understand that much. But these numbers.“
Zak replied, „
Why
does the stone acquire weight?“
She stared at the apparatus, and struggled to articulate the reason why this phenomenon hadn't greatly surprised her. „A stone without weight moves in a straight line. This stone moved in a circle, so it couldn't still be a stone without weight.“
„All right, that's logical. But what made it move in a circle, when I struck it? As opposed to the one that flew straight across the chamber?“
„This one's tied to a spring. The spring holds it back.“
„Exactly,“ Zak said. „The spring forces it to follow a circle, frustrating its preference to move in a straight line. And the effort, the toll this takes on the spring, shows in the spring's extension. Just as the effort it takes for the spring to keep the stone from falling, when they're far from the Null Line, shows in the same way.“