Schild's Ladder

by Greg Egan

  The Left Hand had detected a microscopic lowering of the horizon against the backdrop of stars, which did prove that the Planck worms had corroded the far side into vacuum hundreds of thousands of kilometers away. But the line of sight from the Left Hand to the new horizon still only penetrated twenty or so meters below the surface where the border would normally have been; the growing crater could be as shallow as that limit, or it could be a million times deeper.

  Tchicaya waited. The fireflies could still wink out at any moment. The Left Hand's engines weren't powerful, and it carried only a small reserve of fuel, but it could adapt to a shift in the border's velocity of a few meters per second.

  After ten minutes, nothing had changed. The fireflies were still visible. The border was outracing them.

  That did not mean that there was no hope left. But to move the Left Hand faster than the fireflies, to have any chance at all of catching the border, he would need the shuttle.

  He was useless on his own, now. It all came down to three Preservationists, and whether or not the hint of life in the far side had been enough to change their minds.

  Tchicaya woke his father with a tug of the hand.

  “What is it?” His father squinted at him blearily, but then he smiled and put a finger to his lips. He climbed out of bed and scooped Tchicaya into his arms, then carried him back to his own room.

  He put Tchicaya down on the bed and sat beside him.

  “You can't sleep?”

  Tchicaya shook his head.

  “Why? What's wrong?”

  Tchicaya didn't need to have the truth coaxed out of him. “I don't want to get older,” he said. “I don't want to change.”

  His father laughed. “Nine isn't old. And nothing's going to change tomorrow.” It was his birthday in a few hour's time.

  “I know.”

  “Nothing's going to change for you, for years.”

  Tchicaya felt a flicker of impatience. “I don't mean my body. I'm not worried about that.”

  “What, then?”

  “I'm going to live for a long time, aren't I? Thousands of years?”

  “Yes.” His father reached down and stroked Tchicaya's forehead. “You're not worried about death? You know what it would take to kill a person. You'll outlive the stars, if you want to.”

  Tchicaya said, “I know. But if I do...how will I know that I'm still me?”

  He struggled to explain. He still felt he was the same person as he'd been when he was seven or eight, but he knew that the creature of his earliest momeries, of three or four, had been transformed inside his skin. That was all right, because an infant was a kind of half-made person who needed to be absorbed into something larger. He could even accept that in ten year's time, some of his own feelings and attitudes would be different. “But it won't stop, will it? It won't ever stop.”

  “No,” his father agreed.

  “Then how will I know I'm changing in the right way? How will I know I haven't turned into someone else?” Tchicaya shuddered. He felt less dread now that he wasn't alone, but his father's mere presence couldn't banish this fear entirely, the way it had banished the terrors of his childhood. If a stranger could displace him, step by step over ten thousand years, the same thing would be happening to everyone. No one around him would be able to help, because they'd all be usurped in exactly the same way.

  His father conjured up a globe of the planet and held it toward him, a luminous apparition painted over the gray shadows of the room. “Where are you, right now?”

  Tchicaya turned the globe slightly with a gesture, then pointed to their town, Baake.

  “Here's puzzle for you,” his father said. “Suppose I draw an arrow here, on the ground in front of you, and tell you it's the most important thing there is.” He marked the globe as he spoke. “Wherever you go, wherever you travel, you'll need to find a way to take this arrow with you.”

  This was too easy. “I'd use a compass,” Tchicaya said. “And if I didn't have a compass, I'd use the stars. Wherever I went, I could always find the same bearing.”

  “You think that's the best way to carry a direction with you? Reproducing its compass bearing?”

  “Yes.”

  His father drew a small arrow on the globe, close to the north pole, pointing due north. Then he drew another on the opposite side of the pole, also pointing due north. The two arrows shared the same compass bearing, but anyone could see that they were pointing in opposite directions.

  Tchicaya scowled. He wanted to claim that this was just a perverse exception to an otherwise reasonable rule, but he wasn't sure that was the case.

  “Forget about north and south,” his father said. “Forget about the stars. This arrow is your only compass; there is nothing else to steer by. You must take it with you. Now tell me how.”

  Tchicaya stared at the globe. He drew a path leading away from Baake. How could he duplicate the arrow as he moved? “I'd draw another arrow, each time I took a step. The same as the one before.”

  His father smiled. “Good. But how would you make each new one the same?”

  “I'd make it the same length. And I'd make it parallel.”

  “How would you do that?” his father persisted. “How would you know that the new arrow was parallel to the old one?”

  Tchicaya was unsure. The globe was curved, its geometry was complicated. Maybe it would be simpler to start with a flat surface, and then work his way up to the harder case. He summoned a translucent plane and drew an arrow in black. On command, his Mediator could duplicate the object faithfully, anywhere else on the plane, but it was up to him to understand the rules.

  He drew a second arrow and contemplated its relationship with the first. “They're parallel. So if you join the two bases and the two tips, they make a parallelogram.”

  “Yes. But how do you know that they make a parallelogram?” His father reached over and skewed the second arrow. “You can tell that I've ruined it, just by looking, but what is it that you're looking for when you see that?”

  “The distances aren't the same anymore.” Tchicaya traced them with his finger. “From base to base and tip to tip, it's different now. So to make the second arrow a copy of the first, I have to make sure that it's the same length, and that its tip is as far away from the first one's tip as the bases are from each other.”

  “All right, that's true,” his father agreed. “Now suppose I make things more difficult. Suppose I say you have no ruler, no tape measure. You can't measure a distance along one line and duplicate it on another one.”

  Tchicaya laughed. “That's too hard! It's impossible, then!”

  “Wait. You can do this: you can compare distances along the same line. If you go straight from A to B to C, you can know if B is exactly half the journey.”

  Tchicaya gazed at the arrows. There was no half journey here, there was no bisected line in a parallelogram.

  “Keep looking,” his father urged him. “Look at the things you haven't even drawn yet.”

  That clue gave it away. “The diagonals?”

  “Yes.”

  The diagonals of the parallelogram ran from the base of the first arrow to the tip of the second, and vice versa. And the diagonals divided each other in two.

  They worked through the construction together, pinning down the details, making them precise. You could duplicate an arrow by drawing a line from its tip to the base you'd chosen for the second arrow, bisecting that line, then drawing a line from the base of the first arrow, passing through the midpoint and continuing on as far again. The far end of that second diagonal told you where the tip of the duplicate arrow would be.

  Tchicaya regarded their handiwork with pleasure.

  His father said, “Now, how do you do the same thing on a sphere?” He passed the globe over to Tchicaya.

  “You just do the same thing. You draw the same lines.”

  “Straight lines? Curved lines?”

  “Straight.” Tchicaya caught himself. Straight lines, on
a globe? “Great circles. Arcs of great circles.” Given any two points on a sphere, you could find a plane that passed through both of them, and also through the center of the sphere. The arc of the equatorsized circle formed where the plane cut through the surface of the sphere gave the shortest distance between the two points.

  “Yes.” His father gestured at the path Tchicaya had drawn, snaking away from their town. “Go ahead and try it. See how it looks.”

  Tchicaya copied the arrow once, a small distance along the path, using the parallelogram construction with arcs of great circles for the diagonals. Then he had his Mediator repeat the process automatically, all the way to the end of the path.

  “That's it,” Tchicaya marveled. “We've done it.” A lattice of diagonals ran along the path, marking the way, carrying the arrow forward. No compass, no stars to steer by, but they'd found a way to copy the arrow faithfully from start to finish.

  “It's beautiful, isn't it?” his father said. “This is called Schild's ladder. All throughout geometry, all throughout physics, the same idea shows up in a thousand different guises. How do you carry something from here to there, and keep it the same? You move it step by step, keeping it parallel in the only way that makes sense. You climb Schild's ladder.”

  Tchicaya didn't ask if the prescription could be extended beyond physics; as an answer to his fears, it was only a metaphor. But it was a metaphor filled with hope. Even as he changed, he could watch himself closely, and judge whether he was skewing the arrow of his self.

  “There's one more thing you should see,” his father said. He drew a second path on the globe, joining the same two points but following a different route. “Try it again.”

  “It will be the same,” Tchicaya predicted confidently. “If you climb Schild's ladder twice, it will copy the arrow the best way, both times.” It was like being asked to add up a dozen numbers twice, grouping them in different ways. The answer had to be the same in the end.

  “So try it again,” his father insisted.

  Tchicaya complied.

  “I've made a mistake,” he said. He erased the second ladder, and repeated the construction. Again, the second copy of the arrow at the end of the path failed to match the first.

  “I don't understand,” Tchicaya complained. “What am I doing wrong?”

  “Nothing,” his father assured him. “This is what you should expect. There's always a way to carry the arrow forward, but it depends on the path you take.”

  Tchicaya didn't reply. He'd thought he'd been shown the way to safety, to persistence. Now it was dissolving into contradictions before his eyes.

  His father said, “You'll never stop changing, but that doesn't mean you have to drift in the wind. Every day, you can take the person you've been, and the new things you've witnessed, and make your own, honest choice as to who you should become.

  “Whatever happens, you can always be true to yourself. But don't expect to end up with the same inner compass as anyone else. Not unless they started beside you, and climbed beside you every step of the way.”

  Tchicaya made the globe vanish. He said, “It's late. I'd better go to sleep now.”

  “All right.” His father stood as if to leave, but then he reached down and squeezed Tchicaya's shoulder. “There's nothing to be afraid of. You'll never be a stranger, if you stay here with your family and friends. As long as we climb side by side, we'll all change together.”

  “Tchicaya? Can you hear me?”

  It was Mariama.

  “Loud and clear,” he said. “Are you all right?”

  “That depends what you mean by me. My Qusp is fine. Parts of my Mediator got fried; I only have a short-range IR link left. My body's not a pretty sight, but it's recovering.”

  The signal was coming to him via the Left Hand; she'd freed the shuttle and gone there in person. The long-range transceivers in both the module and the shuttle must have suffered irreparable radiation damage, which said something about the likely state of her body.

  “What about the others?”

  “Wael and Alejandro received similar exposure. They helped me get the shuttle unglued, but they weren't interested in sticking around, with no mod cons and such poor company. Birago's body seemed to be in better shape than mine, but the builders halted his Qusp, so he's as good as departed. When I left, the other rebels were all in a bad way; some of their bodies had reverted to undifferentiated goo, and even in the ones that were still intact and breathing, I'd be surprised if their minds have survived the repair process.”

  She was probably right; the bodies would make liberal use of apoptosis to kill off radiation-damaged cells, and there was no reason for them to treat neural tissue any differently.

  Mariama said, “I went to the Right Hand first, but it had already scribed the Planck worms. It wasn't pursuing the border down, but I gave it a nudge in the opposite direction, too fast for it to reverse. If we find some use for it, I could go and drag it back, but I'm hoping the Left Hand will be enough.”

  “It will have to be.” Nothing they did to the Right Hand would render it trustworthy.

  “Branco told me about the toolkit Yann gave you, while he was cutting us loose, but I didn't have time to get a copy myself. The simplest thing might be if you send it to me now, before I go chasing the border.”

  “What?” Tchicaya stared at the red-shifted stars above the horizon, checking the view for any sign that he'd departed from reality and was hallucinating this entire encounter. “Why would that be simplest? You're coming to get me, aren't you?”

  “That would be an awful waste of fuel. You don't need to be here, physically.”

  Tchicaya was silent for a moment. She was right about the fuel, but he couldn't accept what she was proposing.

  “That's not true,” he said. “If I stay out here, I'm going to lose radio contact, eventually. From sheer distance in the long run, but if the border has taken on a complicated shape, I might lose my line of sight much sooner.”

  “Then give me the key to the Left Hand. With that, and the toolkit, I can manage everything.” She sighed. “Don't be precious about this. I don't like the idea of leaving you to drift away, but there are more important things at stake here. The time and the fuel I used fetching you could make all the difference to the far side.”

  Tchicaya felt a flicker of temptation. He could wash his hands of everthing, and wake beside Rasmah on Pfaff. Mariama was being perfectly logical; time was against them, and apart from the secondhand skills that he could easily sign over to her, he was superfluous.

  He wanted to trust her. Hadn't she earned that? They'd had no end of differences, but she had always been honest with him. It seemed petty and mean-spirited to keep on doubting her.

  The trouble was, he didn't trust his own motives. Thinking the best of her would be the perfect excuse to absolve himself of all responsibility.

  He said, “I'm not handing you anything. If you care so much about the far side, you'd better come and get me.”

  Mariama remained seated at the front of the shuttle as Tchicaya clambered out of the airlock. He nodded a greeting, and tried to smile. Her Exoself would be discouraging her from doing anything to interfere with her body's healing, by means both gentler and more precise than a blanket of agony; extrapolating from the raw pain of the minor burns he'd willingly experienced as a child was absurd. Still, the sight of her weeping, blistered skin made his guts tighten.

  He said, “Hitchhiking in space isn't so bad. I've waited longer for a ride, on land.”

  Mariama replied through the IR link. “Try showing more flesh. That always works wonders.”

  On their way back to the Left Hand, Tchicaya received the first good news he'd heard since the moratorium vote. The horizon had stopped falling. The Left Hand was no longer seeing new stars creeping into view.

  That in itself didn't fix the depth of the lost region everywhere, but the particular geometry was suggestive. The new horizon was exactly where it would have been if the
Planck worms had failed to penetrate the signaling layer, where the vendek population changed abruptly, a hundred kilometers into the far side.

  As they approached the Left Hand, the news became even better. The fireflies had finally begun to vanish, and the timing of their deaths confirmed the best possible scenario: the border had retreated to the signaling layer, and no further.

  Tchicaya was elated, but Mariama said, “Don't assume this is the new status quo. Birago wasn't exactly confiding in me toward the end, but if what he's done here bears any resemblance to the work I was involved in with Tarek, the Planck worms won't have given up at the first obstacle.”

  “Meaning what?”

  “They'll mutate. They'll experiment. They'll keep on varying themselves, until they find a way to break through.”

  “You knew how to do that? You had it all worked out?”

  “No,” she admitted. “But as soon as you showed us the vendeks themselves, they provided an awful lot of inspiration. Tarek and I didn't pursue that, but don't expect Birago to have passed up the opportunity.”

  They docked with the Left Hand, and carried it down to the point where the fireflies were disappearing.

  Regaining alignment with the border took almost an hour, as a cycle of increasingly delicate adjustments brought the stylus into range. Once that was achieved, Tchicaya scribed a series of probes that would spread out laterally as well as moving straight in, improving their chances of gaining a comprehensive picture of the Planck worms. Unsurprisingly, now that the signaling layer was infected with Planck worms and exposed to vacuum, it was no longer vibrating, no longer tapping out primes. Tchicaya longed to discover the mechanism that had driven it, but he had to stay focused; trying to dissect the far-siders' ruined SETI equipment—if that was what it was—had to take second place to dealing with the plague the beacon had been unable to deter on its own.