Ruz had been a metalworker for most of his life. It had taken Roi more than a dozen shifts to recruit him, but he had later admitted that the instant he'd seen Roi's «Rotator» — her contraption for demonstrating the Splinter's spin — he had been hooked in equal measure by a fascination with the idea that the world could be secretly turning, and a conviction that he could do a far better job at making the kind of gadgets needed to quantify that motion. Happily, his conviction had turned out to be entirely justified.
Roi drifted past the clockmakers and landed against the wall, close to the point where Tan was talking with a small group of students. «What is natural motion?» he asked. «Looked at closely, and in the absence of spin, it seems as if a weightless stone is trying to follow a straight path. Yet over large enough distances, that path can curve around into a circle, or other kinds of curves. What's happening?» He lifted up a complicated patchwork he'd made by gluing together dozens of fragments of skin. «See this line, marked across this surface?» He indicated a path he'd dyed in ink. «On every small piece of the surface, it's a straight line. But the line as a whole isn't straight; it can't be, because the surface itself isn't flat. So how can we determine exactly which paths can be made by small, straight lines joined together in this way? That will depend on the way the parts of the surface are connected to each other. We need a precise, mathematical expression of the nature of that connection, in order to understand which paths are as straight as they can be, given the geometry of the surface.»
Roi listened closely. She, Zak and Ruz had lured Tan away from the sign-age team where he'd honed his geometrical skills. Calculating distances through the tunnels of the Splinter had given him both an extraordinary facility with numbers and a powerful sense of how they could be used to analyze paths, shapes and motion.
«Keep in mind,» Tan continued, «that there is one ingredient in the idea of natural motion that doesn't show up when we study a surface like this. Zak has argued that the natural path for any stone you throw from a given point depends, not just on the direction you throw it, but also upon its speed. The natural path of the Splinter appears to be a circle, but an object that starts out on that circle and travels in the same direction as the Splinter will still follow a path with a different shape if it's moving faster or slower than the Splinter. So we need to find a way to incorporate that into our geometrical scheme. We need to merge the idea of
speed
with the idea of
direction
.»
Roi had to make an effort to tear herself away. She had heard Tan explain these ideas many times, but on each occasion the concepts became a little clearer, a little bit more precise. If he ever reached the point where they were defined with sufficient mathematical rigor to allow her to start making calculations, she hoped she could find a way to merge them with Zak's other principle — that the true weights everywhere summed to zero — and then she might finally be able to start mapping the possibilities for the Splinter's past and future.
She clambered across the wall to the crevice where Zak was resting. She tapped the adjacent rock gently, and after a moment a single claw emerged from the crack.
«It's Roi,» she said, «I've brought you some food.»
«Thank you.» Zak slid out on to the wall, awkwardly. Roi opened her carapace and took out a bundle of food. She'd spent half the shift collecting it, but she did not begrudge the effort. Zak was old, his body was failing, but she had no intention of letting him starve to death.
Zak ate slowly, in silence. Roi no longer asked him what hurt and what didn't; she gathered that almost everything did.
When he'd finished, he surveyed the activity in the chamber with a satisfied air. Roi could see the meal dissolving smoothly inside him, unhindered by the obstructions she'd noticed the last few times. Clearly the rest had done him some good.
«How are you finding things on your travels?» he asked.
«What do you mean?» She'd returned from the last recruiting expedition with two young students, but that had been several shifts ago, and she'd reported the result to him then.
«How do people think of us? Word must have spread out from the Calm by now, that there's a new team here, doing a new kind of work.»
«Ah.» It was a good question, but a difficult one to answer. «I wouldn't say that there's any particular resentment directed against us. Nobody likes having their team-mates taken, but recruitment is recruitment, it's a fact of life.»
«And work is work?» Zak pressed her. «The mere existence of a team is its own justification?»
Roi replied cautiously, «It seems that way. Most people don't consider themselves experts in the history of the Splinter, to the point of declaring 'There has never been a team like that before'. Work is whatever a group of people do, and most of us take it for granted that what other teams do is useful in some way. There might be only five or six jobs that literally everyone knows about and understands, but that doesn't mean people are hostile or suspicious toward all the rest.»
Zak pondered this. «I've been wondering at what point we'll need to let some of our own members get poached.»
Roi was startled. «Can we afford that? Our numbers are still very low.»
«Can we afford not to?» Zak replied. «It's not just a matter of being sure we play the game, being sure our existence is accepted. It would also be of value if some of our ideas could spread outside the team itself. Almost every child learns writing and simple arithmetic; they're parts of the culture that have managed to move beyond the specializations where they originated. Imagine if the facts about weight and motion could acquire the same status.»
Roi could see where this was heading. «So by the time the next division of the Splinter is imminent, everyone will have at least a basic understanding of what's going on. It won't be necessary to try to educate them from scratch.»
«Is that too ambitious?» Zak wondered.
«I don't know. Tell me when the next division is coming.»
Zak emitted a sarcastic rasp. «I have a feeling you'll know that before I do.»
«Don't count on it.» In truth, the idea of being able to predict the event still seemed almost as strange and metaphysical a prospect to Roi as the thing itself.
«When is the next overview meeting?» Zak asked.
«Two shifts from now.»
«I think I'll attend.»
Roi was pleased. «It will be good to have you there. You've been away for too long.» She hadn't been close to anyone near the end of their life before, and she was never sure what to expect. Zak's strength came and went, and every time it declined she was afraid he was dying, but a few shifts' rest, some good news about the team, and a problem worth thinking about were often enough to revive him. He'd never travel all the way to the garm-sharq edge again, but he might survive at the Null Line for dozens more shifts.
She bid him farewell, and launched herself across the chamber to the point on the web where her own equipment was set up. Every shift, she counted a few cycles of the three periodic motions to see if anything had changed in their relationship. Once Ruz's clock was declared trustworthy as a standard in its own right she'd start using it to measure the absolute durations of the cycles, but until then she was content to record the ratios between them.
She set everything in motion and then watched patiently, counting the passage of the cycles using a trick she'd picked up from Gul, a recruit who'd worked in a storage depot: sliding a series of stones threaded on wires, rather than trusting everything to memory or wasting precious skin by making a scratch for each event. Though all three motions slowly diminished over time — however thin the air the stones were moving through, however well-greased the pivots on the Rotator's spinning bar — the periods she was measuring were unaffected, and as long as each cycle could be clearly tracked this gradual decay caused no problem.
As Roi watched the stones, in her mind's eye she pictured the way their paths might have looked to some impossible cosmic observer, floating in the Incandescence high above the Splinter's orbit. The problem of how these paths wrapped around the Hub entranced and infuriated her. If the Map of Weights could be believed, then long ago — and, presumably, further from the Hub — anything falling freely would have traveled endlessly along the same closed curve. Whether it was simply going around in a circle, or whether it was also detouring up and down or in and out made no difference, because the periods for all three motions were the same. Now, it was as if something had taken that simple pattern and squeezed and twisted it, forcing the different cycles to break ranks, and yet miraculously preserving Zak's balance of weights.
She finished her count. In eighty-five cycles of the shomal-junub stones, the plane of the rotating bar turned sixty-eight times, and the looping stone completed forty-five loops. These numbers hadn't changed since she'd begun measuring them.
Roi recorded the results with the usual mixed feelings. Any change would be the cause of great excitement, the start of a new opportunity to prise apart the mysteries of weight and motion. The numbers had spoken eloquently when she and Zak had first identified the three cycles, but their silence since then had been disappointing.
At the same time, she knew that any change would mean far more than an intellectual impetus for the team. If the weights increased, the strength of the rock beneath her would be tested, and everyone in the Splinter would be at risk. However great her hunger for revelation, she could not deny a powerful sense of relief that the numbers continued to seem immutable, and that she might yet live out a quiet life merely contemplating their mysteries without ever feeling their sting.
The overview meeting was held in a chamber a few dozen spans from the Null Line. This place was large enough for the whole team to fit, clinging to the walls, but not so large that people could split up into individual project groups with the members audible only to each other.
Tan spoke about his group's continued efforts to explain natural motion geometrically. «First, we need to extend the idea of direction to include speed. We can understand the direction 'three spans garm for every one span rarb', so why not also include the idea of speed, and talk about 'three spans garm for every heartbeat that passes'? But then, if we talk about the garm direction, the rarb, and the shomal, there is a fourth simple direction we must add to the list: time. In fact, every path that's traveled includes some component in that direction; we can't travel garmwards a single span without some time also passing.
„Once we can describe both
speed
and
direction
in the same framework, it makes sense to understand
natural motion
and
spin
as two aspects of the same thing. When an object is weightless, that means its velocity is simply following the geometry it encounters: there is no rock, no claw, pushing against it, so the only thing that can influence it is the way empty space itself is shaped. Similarly, when an object isn't spinning, the directions it carries with it must be following that same general rule. We know that the directions tied to the rock of the Splinter aren't following that rule, because of the swerve weight“ — the sideways weight of motion connected to the Splinter's spin—„that we see if we treat those directions as fixed. But I believe the directions tied to the frame of the Rotator obey the same laws as natural motion, and
that
is why we can declare that it's the Splinter that is spinning, not the frame, however compelling the opposite scenario must seem to a casual observer.“
In Tan's view, at every point in space and every moment in time it ought to be possible to summarise the effects of the local geometry with a simple mathematical rule for the way directions and velocities were „naturally carried“ along any given path. Zak had proposed that circular orbits around the Hub, with a certain period that depended on their size, comprised one form of natural motion. Tan wanted to find a single rule that could account for that, and also the behavior of the Rotator: a single template into which he could insert a direction or a velocity in order to calculate how much (if at all) it was changing, compared to the dictates of geometry. Feed in the Splinter's velocity, and the answer would be: this is natural motion, there is no change. Feed in the direction
garm
and the answer would be: this direction is constantly turning, at a certain rate, around the shomal-junub axis. Feed in any direction tied to the Rotator's frame, and the answer would be: there is no change.
If Tan's ideas were dizzyingly abstract, the next speaker proved to be an antidote. Bard had been a miner, searching out and extracting metal, and he had a bluntly practical approach to his new team's work that side-stepped speculation in favor of tangible results.
„We have no way of knowing exactly why the weights changed in the past,“ Bard declared. „The Splinter seems to have shifted closer to the Hub, but it isn't clear what made that happen. Was it a gradual effect, spread over many generations, or was there a sudden, violent change in the wind that forced us off our earlier path and into our present orbit?
«The wind on the garmside pushes us faster along our orbit, which tends to move us away from the Hub, while the wind on the sardside acts to slow us down and bring us closer to the Hub. If everything about the Splinter was perfectly symmetrical, the two influences would balance exactly. I doubt that the symmetry is perfect, but even if it's not, we've been unable to measure the consequences in the short time that this team has existed.»
«However,» he continued, «whether these dangerous shifts come slowly or quickly, it seems likely to me that the Splinter would be safer if we could move it further from the Hub. If we could reduce the weights, taking them back to the values they had before the last division, there would be a far greater margin for surviving any subsequent change.»