by Greg Egan
«Now we reduce the speed.»
Roi said, «Now I'm confused. I thought this weight was supposed to increase with speed.»
«It does. But greater speed also gives it less time to act before the stone hits the target. By making the stone move more slowly, the weight that bends its path becomes weaker, but the extra time the stone spends in flight more than compensates for that.»
Zak was right. With the spring half-compressed, the stone emerged less rapidly, but the mark it made was shifted to the sard side of the Null Line by about twice the width of the stone itself. In a third experiment, with the compression reduced further, the sardwards shift was even more pronounced.
Roi could picture what was happening very clearly, now. While the stone was in flight, the Splinter was turning, carrying the wire and the target board a small way garmwards, leaving the un-rotated path of the stone to strike the target askew.
She said, «Why can't we use this to measure the speed of the rotation?»
«It's a crude experiment,» Zak insisted. «If I release the stone several times in succession with the spring compressed by exactly the same amount, the point where it strikes the target still varies. And how can I know how fast the stone is moving? It's traveling too quickly for me to time its motion accurately.»
«Let it move more slowly, then. You'll get a larger effect as well.»
«There's a catch,» Zak said ruefully. «The more the stone gets pushed away from the Null Line, the more it comes under the sway of the ordinary garm-sard weight as well. What we're measuring will no longer be one simple thing. When you combine that with the uncertainties in the aim and the velocity, I don't think there's much hope of getting a meaningful number out of the results.»
Roi could see how daunting the complications were, but she wasn't ready to give up. «Can I try it? Slowly? Just to see what happens?»
«Of course.»
She squeezed the plunger down to the first notch, the smallest compression possible, then released it. The stone emerged at an absurdly leisurely pace, and even as she watched, it veered visibly sardwards. By the time it had traveled less than half a span toward the target its path had turned sideways, and not long after that it had swung around so far that it was level with the spring-shot again, albeit some distance sard of it. While continuing to move sardwards, its velocity along the Null Line had been completely reversed.
Roi said, «This is not what I'd expected.»
«It's just following the rules,» Zak said.
Roi moved aside to let the stone pass her. Eventually, its sardwards drift seemed to level out, and it was simply moving backward, parallel to the Null Line, much faster than it had been moving when it had left the tube. Its direction continued to change, though; the relentless sideways tug of the weight of motion was stronger for this stone than it was for the wind, and it began to veer back toward the Null Line.
As the stone approached the Null Line, the sharqwards part of its motion slowed, leveled out, and reversed, so it was now heading back toward the spring-shot. This didn't last long, though. When it reached the Null Line itself, almost grazing it, the stone executed a small loop that took it first sardwards, and then — in a replay of its original manoeuvre upon being launched — swung it around sharqwards once more. It was far behind the spring-shot, let alone the target, and showed no sign that it would ever come close to either again. Rather, it seemed to be cycling back and forth between the Null Line and a certain distance sardwards, while it drifted — mostly, though not constantly — ever more sharqwards.
Roi approached Zak. «What's the simple explanation? I suppose I can accept that the sardwards weight combined with the weight of motion did all of that, but there must be an easier way to understand it.»
Zak said, «Think about the orbit of that stone. The stone was always on the sardside of the Null Line, so its whole orbit was larger than the orbit of the center of the Splinter. Larger orbits have longer periods, so the stone took longer than we did to go around the Hub. That's why it drifted backward. It wasn't as fast as us.»
«But it started out faster,» Roi protested.
«That's true. At the same distance from the Hub, where its orbit touched ours, it was faster than us. That's why we weren't constantly outpacing it, and it didn't move backward all the time. But over a complete orbit, we were faster.»
This made sense, but Roi still wasn't satisfied. «Why didn't the stones you launched go backward? Was that because they were moving faster than mine?»
«Definitely not!» Zak was emphatic. «The only difference was, they hit the target before they could swing around and go into reverse. If we had taken the target away — and the walls of the chamber too, if necessary — then those stones would have followed exactly the same kind of path as your one. The fact that they were moving faster made their paths larger, and we only saw a small part of each path, but other than that everything about them was the same.»
«I see.» The whole point of Zak's version had been to concentrate on the very start of the motion, before the garm-sard weight could complicate things. «Can I try something else?»
«Anything,» Zak said.
She detached the spring-shot, then reattached it pointing in the opposite direction: sharq along the Null Line. Now the stone would be moving backward from the start, making it slower than the Splinter at the point where their orbits touched.
Its path followed exactly the same pattern as before, except that rarb became sharq, and sard became garm. After leaving the spring-shot the stone veered garmwards; halted its leisurely sharqwards progress and went rapidly into reverse; reached a maximum distance garmwards from the Null Line and started back toward it; then, close to the Null Line, performed a small loop that took it back into the same cycle, albeit many spans rarb of where it had begun.
Roi said, «Its orbit was smaller than ours, so it was racing ahead of us?»
«Yes.»
«And the way it moved away from the Null Line and then back again, that's because the orbit wasn't a perfect circle?»
«Right,» Zak said. «We remain a constant distance from the Hub, but there are orbits like this that draw closer to the Hub and then move away again.»
Roi contemplated this. «What if we could put a stone into an orbit that wasn't a perfect circle, but was still the same size as ours, overall? With the same period?»
Zak didn't reply immediately, but his posture made it clear that he was intrigued by her suggestion. «That could be very useful,» he said eventually. «We ought to see it execute a fixed, cyclic motion instead of running away across the chamber.»
Roi detached the spring-shot from the Null Line again, and attached it pointing sardwards: perpendicular to the Null Line, «halfway between» the two directions she'd already tried. She was acting purely on instinct, and even as she tightened the clips she wondered if launching the stone away from the Hub meant she'd be putting it into an orbit that would keep it perpetually on one side of the Null Line. But then, shooting the stone along the Null Line itself, which seemed more symmetrical in that respect, certainly didn't work, so what she was trying made as much sense as anything.
She squeezed the plunger one notch, then released it.
The stone veered sideways as it emerged, but not as sharply as it had in the previous two experiments. As it moved, it picked up pace, but nowhere near as rapidly as before. Roi was surprised; she'd half expected the sardwards weight to take over and drag the stone into a frenzied spiral as the weight of motion twisted its ever-quickening flight. Instead, the stone continued to turn in a smooth, shallow arc, still progressing sardwards while swinging around ever more to the sharq.
Eventually, its sardwards motion leveled off, about two spans from the Null Line. It was moving perhaps three times faster now than it had when she'd launched it. It continued to swing around gently, coming back toward the Null Line, while its sharqwards speed lessened.
As it approached the Null Line, Roi tensed. It was no longer travelin
g sharqwards, but it would probably perform the same annoying little loop as the others, and then it would be lost to them, drifting away across the chamber.
It didn't. It crossed straight over the Null Line, at about the same speed as it had left the spring-shot, and veered to the rarb. The symmetry was unmistakable: it was performing exactly the same kind of motion as it had when she'd fired it, only with garm in place of sard and rarb in place of sharq. If that symmetry held true, there was only one place where it could cross the Null Line again.
When the stone finally approached the spring-shot, Roi thought it might collide with it, but her aim hadn't been that perfect. It was close, though. The stone passed less than half its width from the tube before continuing on around the same closed loop as before.
«I can't believe I missed this,» Zak said. «A new periodic motion! Congratulations!»
Roi said, «What are we seeing, exactly? Is this showing us the Splinter rotating?»
«What we're seeing is a stone in orbit, moving back and forth between its nearest and furthest points from the Hub,» Zak replied. «If we try to explain that from a Splinter's eye-view, the motion will depend on the strength of the garm-sard weight, as well as the speed of the Splinter's rotation. Once I would have said that those two things should combine in a very simple way, but now I'm not so sure. The two and a quarter has taught me to be more cautious.»
Roi launched another stone directly sardwards, this one faster than the first. The loop it followed was larger, but the shape was the same — about three times as long as it was wide — and the faster stone completed each circuit in the same time as its slower companion. These stones were spending half their time sard of the Null Line, moving more slowly than the Splinter, and the other half garm of the Null Line, moving faster, so over time they were keeping pace both with the Splinter and with each other. Surely that meant that each cycle they completed marked the time for all three to complete an orbit around the Hub? And surely the Splinter's rotation around its axis shared the same period, too, as a matter of simple geometry?
Zak said, «I know what we should do.» He found an empty tube, attached it to the Null Line aligned shomal-junub, then placed a stone in its mouth and let it begin its slow fall. Now they could compare the two kinds of motion directly, without having to worry about the accuracy of their counts to time the periods.
It soon became obvious that the periods were not the same: the looping stones were taking far longer to complete each cycle than the stone falling shomal and then junub. For a while, Roi wondered if the slower cycle might be exactly twice as long as the faster one — and if some simple aspect of the geometry that she'd neglected could make sense of this — but that hope proved misplaced. The shomal-junub stone completed seventeen cycles while the looping stones completed nine. There was nothing simple about that.
Zak seemed forlorn at first, but then he proclaimed, «There's something encouraging about the way these numbers demolish half of my assumptions, yet the whole notion of orbits seems to survive. Watching these stones, can you honestly tell me that you don't believe they're going around the Hub?»
Roi said, «The idea still makes sense, but we're missing something.»
Zak regarded the shomal-junub stone. «If orbits still make sense at all, then this stone tells us how much time passes for something orbiting at a small angle to the Splinter's orbit to come back to the same height above us each time. The stone doesn't go wandering off along the Null Line, so the periods of the two orbits must be the same. But what if the place where the orbits are farthest apart isn't fixed? What if that point moves around? Then this need not be telling us how long an orbit actually takes.»
He moved to the looping stones. «And when you deform an orbit so it's no longer circular, what if the point of closest approach to the Hub isn't fixed either? That point, too, might wander around.»
Roi struggled to picture what he was describing. «So these other orbits wouldn't close up? The Splinter would follow a perfect circle, but these stones would be weaving up and down, or back and forth around that circle, never quite repeating their paths?»
«Yes.»
Roi was dismayed. «If the things we thought were landmarks can't be trusted to stay still, how can we ever decide how long it takes for the Splinter to complete an orbit?»
Zak said, «Good question.»
Neither of them had the answer to it, so they set about calculating what Roi's looping stones actually did tell them. They worked side by side until the end of the shift, slept, then worked through two more shifts.
Finally, they had templates describing the relationship between three things: the strength of the garm-sard weight, the period of the Splinter's rotation, and the period of the looping stones. These calculations made no assumptions at all about the existence of «orbits around the Hub»; they just followed the effects of the weights directly — though they did rely on a correct understanding of how spin contributed to weight.
When Zak inserted the numbers, the template told them that the Splinter was rotating with a period about one and a quarter times the shomal-junub cycle.
If you believed in orbits, this meant that for a stone in a tilted orbit to return to its highest elevation was the fastest thing. For the Splinter to rotate around its axis took a little longer. And for a stone in an eccentric orbit to return to its greatest distance from the Hub took longer still.
Three phenomena, three different times.
«Where has all the simplicity gone?» Zak lamented.
Curiously, if he fed his original assumptions into the templates — if the shomal-junub weight was equal to the hidden rarb-sharq weight that was balanced by the spin, and if the garm-sard weight was, in total, three times as much — then all three periods would have been identical. The number three really would have made things very simple.
Roi took a break from the Null Chamber, and traveled a short way into the garmside to give herself some weight again, lest she lose too much strength. Even as she headed out of the Calm into the sights and sounds of ordinary life, she couldn't stop thinking about motion and orbits. At the end of each shift, when her mind had once filled with the images of weeds, now she saw stones, bouncing and looping and swerving in front of her. When she woke, her first thought was always of finding a new way to check Zak's conclusions. Their calculations deriving the Splinter's spin from the looping stones could be flawed. Or the weight measurements they had fed into the templates could be wrong.
Zak's simple experiment, when he'd launched a stone along the Null Line, had been compelling: it had made it obvious that the Splinter was turning while the stone was in flight. There had to be a way to measure the Splinter's rotation directly using that effect, without allowing the complications of the garm-sard weight to intrude. If you could keep the stone moving somehow — without ever letting it go too far from the Null Line — its path would act as a reference against which the turning of the Splinter could be judged.
How could you rein it in, though, without stopping it completely?
When Roi found the answer, she turned around and headed straight back to the Null Chamber. Zak wasn't there when she arrived, but she felt no hesitation about helping herself to his stores of material. They were a work team, now. These things were their common tools, not a lone eccentric's hoard.
Zak arrived just as she was putting the finishing touches to her apparatus, trial and error having led to some changes in the original design. Two equally heavy stones were glued firmly to the ends of a small bar. The bar was free to pivot around its center, where it was threaded by a stiff metal wire, which was bent around into a flat rectangular supporting frame large enough for the bar to turn continuously without obstruction. Another pivot, opposite the bar, attached the frame to the wire of the Null Line; this pivot left the frame free to rotate around the shomal-junub axis.
After they'd exchanged greetings, Zak watched in silence as Roi greased the pivots, marked the initial alignment of the frame on a card
fixed to the wire above it, then gave the bar a flick to set it spinning.
An earlier version — with the bar spinning around one of its ends, and a single stone at the other — had been unbalanced, shuddering mercilessly, causing the frame to slip back and forth. This design seemed to have fixed that problem. All Roi could do now was wait.
Slowly but unmistakably, the plane of the spinning bar turned. Or, stayed fixed while everything in the chamber, everything in the Splinter, wheeled around it.
Zak said simply, «Who can doubt it now?»
There was no need to measure the speed of the stones. There were no elaborate calculations to perform. One rotation of the frame corresponded to one rotation of the Splinter, if they understood anything at all.
They set up a shomal-junub stone a short distance away, to compare the motions. After a while, no doubt remained that the period of this new phenomenon agreed with their earlier calculations, based on the more complex motion of the looping stones. The plane of the spinning bar took one and a quarter times longer to complete one turn than it took for the shomal-junub stone to complete one cycle.
Roi didn't know what to feel. She was relieved to see two lines of evidence converging on the same answer for a change, but she'd actually been hoping that this experiment might yield a different result, one that removed some of the complexity that had begun to infest the theory of orbits.
«Where did all the simplicity go?» she joked, echoing Zak's refrain.
«I think I know where some of it went,» he replied. «I didn't dare mention this before, because I wasn't confident in our results. But now that you've confirmed the period of the spin, it doesn't seem so foolish any more.»
Roi said, «Go on.»
Zak had been forced to give up his beloved three, and the simple assumption that the shomal-junub weight would be equal to the hidden rarb-sharq weight, which the spin canceled out exactly. Since the shomal-junub cycle was faster than the spin, the shomal-junub weight was stronger than both the spin weight and its equal, the rarb-sharq weight.
