Convergent Series

by Larry Niven

  Yesterday, a conqueror surveying his realm, Guppry had stood at the edge of a balcony and become less than a gnat in his own eyes. A stairway with a railed landing ran a little way down; then miles and miles of naked cliff pocked with caves; then no more cliff. Just blue sky swirled with clouds, and flocks of legless birds, and a sinuous cloud-streamer, gilded by sunset, that might almost have been a golden dragon except for its size. Eternity was down there. You could fall until you died of thirst, and who would bury you then?

  Sarol was saying, "King Charl had other reasons for building there. The army has a standard maneuver for invaders. Get them between us and the edge and push. Even seasoned troops tend to panic if they're new to the edge. You probably felt it yourself. You haven't lived that long in Halceen."

  Guppry shuddered, then smiled, because Sarol hadn't had the chance to try that on him.

  Sarol said, "I never heard of a strategy that would save anyone from an internal enemy."

  Guppry lost his smile. "We're nobody's enemy. We only want justice for all, an end to the rift between nobles and—"

  "Spare me. Are you taking us through the forest?"

  The path split here. They had been riding through farmland; now one branch veered right to circle between the forest and more farmland. The other branch went straight in between the tremendous trunks.

  "Unless you'd rather go around," Guppry said. "We'd be another four hours—"

  "No, that's fine."

  ***

  If the near-bloodlessness of Guppry's coup was remarkable in that place and age, sti1l it could not have happened elsewhere.

  What had happened was simple enough. King Sarol had been careless. In the middle of a drought, he had gone wooing in Zarop-Opar. One of the ladies who rode with him now was going home, and her lovely face was not sad but enraged: first a princess, then a queen, now she was one of three wives of an unemployed politician. Sarol should have known better than to stay so long. In times of drought people become annoyed; they tend to depose the reigning king, or witch doctor, or president.

  Sarol came home to find that Guppry had charge of Halceen, including most of the army, and the armory. Some army men had died in abortive resistance. Many more served Guppry, and the rest had bowed to necessity. Those nobles still loyal to Sarol were prisoners, comfortably housed, but hostages.

  Sarol's retinue might have made a glorious last stand; but there were women among them.

  Guppry had allowed the king a conditional surrender.

  Complicated oaths were sworn between them. The King swore never to strive to rule any land again. He swore not to seek revenge. Guppry in his turn swore that Sarol and his followers would suffer exile only, that no citizen would harm them; and he bound his army to the same oath.

  What made it work was this. Oaths were binding in that place, as binding as natural law; which is to say, any magician who could break his oath could also make fire burn backward, and he was a rare man indeed.

  And so they rode to exile: the King, and twelve soldiers, and five of their wives, and three of the King's.

  The tremendous trees cut the noon sun to a twilight speckled with brilliant dots. Between the huge trunks there was plenty of room for mounted men. The path grew less and less visible. Many came to the forest, to hunt or to pick wild berries and mushrooms, but few rode through it.

  A dragon sat on his haunches and, grinning, with flame licking his lips like a tongue, watched them pass.

  Now you saw another division in the group. The Halceen soldiers and nobles paid the dragon almost no attention. But eight of Guppry's men were Castolan and two were from further away, and their eyes were wide and white. They couldn't meet the dragon's eyes. The dragon liked that. He was not there to defend his territory, or to snatch a meal; he had tested the power of the spell that guarded the path. He enjoyed frightening travelers.

  Guppry's voice did him credit; it was almost normal. "I'd think you'd get a lot of suicides, there at the edge."

  Sarol looked around. "Of course we do. From neighboring countries too. There's a temple, the Order of the Black Mercy, half a mile from the palace. We, try to steer the suicides to them, and they try to make their last hours more peaceful."

  "I was thinking. We should fence off the parts of the palace that lead to the edge."

  "Guppry, you're a Castolan, you don't think at all like a Halceen. Why did my good Halceen people follow you?"

  A little smugly Guppry replied, "Sometimes it takes a stranger to see the truth and point it out. After that, anyone could see that they were being robbed by the nobles. They only had to look up toward the palace to see where their taxes went."

  "You've been through the palace. Everything in it is five generations old. I spent almost nothing on the palace. The new cistern was about it."

  "The dance floor in the pavilion—"

  "That's new, yes, but a government has other expenses, Guppry. Salaries for officials, the army, cleaning crews and gardeners, prophets... That one I could have skipped."

  "I bought your prophets, Sarol."

  "They must be long gone by now, hey? They saw you'd win, and they saw what would follow. What I started to say was, a Halceen wouldn't be racking his wits for ways to stop a suicide. Suicide is a kind of last refuge."

  "One you didn't take."

  "But I chose not to. It was there for me. Guppry, I believe that's Zarop-Opar ahead."

  The trees didn't thin out. They ended abruptly, where kings' oaths prevented mere woodcutters from crossing. Beyond, the path became a paved road that curved to the left around Mount Demonhead and ran to Zarop the capital.

  Guppry asked a little maliciously, "Where do you go from here, Sarol?"

  Sarol laughed. "I'd thought of climbing Mount Demonhead. For the view, you know." And he rode out of the trees, followed by his knights and his ladies and theirs. King's oaths didn't stop them; they were escorting a Zarop-Opar princess. Guppry's men stopped, and so did Guppry.

  Sarol turned his horse. "There's a marvelous view of Halceen from Mount Demonhead. You can see the whole country. I'd like to see what happens there at sunset tomorrow."

  "What do you mean?"

  "When you looked over the edge, did you happen. to catch sight of something big and serpent-shaped, golden in color?"

  "I saw a cloud formation that—"

  "Those taxes that so irritated my people. Part of that money, a good part, went to hunters. We needed a lot of meat, and a variety too, to feed him. We had to buy salt, too. We didn't spend that much money on the army, Guppry, because we didn't need a big army. Came a big army against us, we sent the Worm after it."

  "Worm?"

  "I won't give you his real name. He lives in a cave below the edge, most of the time. At night he comes up to take food from our arras, if we speak the rites correctly. We never did get enough suicides. Before King Charl came, the Worm used to come over the edge to feed. He'd crack open bungalows to get at what was in them, the way we break a melon open... but King Charl stopped that. They made him king for that."

  "You're sworn not to seek vengeance, Sarol!"

  "I know it. I don't need to seek anything. I don't think we'll climb Mount Demonhead after all; we'll want to be a good way from the edge. Good-bye, Guppry. Good luck."

  In these lands of magic and fabulous beasts and men who keep their oaths, strangers become delightfully gullible. A native of the edge countries can make a fool of even the wisest outlander. Later he may listen to his common sense when a man of the world's edge is speaking. But common sense is not a good guide either.

  The thing to remember is this. Some men are liars.

  Of Guppry's forty men, not one had been of the palace staff; none had lived on the edge itself. They were frightened. They looked to Guppry. Mindful of his new dignity, Guppry put a sneer on his face as he turned to watch Sarol's retinue depart. "Our retired King is a great liar," he said cheerfully.

  Certainly a practical joke need not be considered as vengeance...
/>
  Certainly Sarol's oath could have forced him to give warning...

  Distant laughter drifted back from Sarol's lords and ladies. None were looking back. Considering their load, the horses were making all good speed.

  And Guppry turned back toward the edge, as Sarol must have known he would, because Guppry had no choice at all.

  Cautionary Tales

  Taller than a man, thinner than a man, with a long neck and eyes set wide apart in his head, the creature still resembled a man; and he had aged like a man. Cosmic rays had robbed his fur of color, leaving a gray-white ruff along the base of his skull and over both ears. His pastel-pink skin was deeply wrinkled and marked with darker blotches. He carried himself like something precious and fragile. He was coming across the balcony toward Gordon.

  Gordon had brought a packaged lunch from the Embassy. He ate alone. The bubble-world was a great cylinder whose landscape curled up and over his head: yellow-and-scarlet parkland, slate-colored buildings that bulged at the top. Below the balcony, patterned stars streamed beneath several square miles of window. There were a dozen breeds of alien on the public balcony, at least two of which had to be pets or symbiotes of other aliens; and no humans but for Gordon. Gordon wondered if the ancient humanoid resented his staring... then stared in earnest as the creature stopped before his table. The alien said, "May I break your privacy?"

  Gordon nodded; but that could be misinterpreted, so he said, "I'm glad of the company."

  The alien carefully lowered himself until he sat cross-legged across the table. He said, "I seek never to die."

  Gordon's heart jumped into his throat. "I'm not sure what you mean," he said cautiously. "The Fountain of Youth?"

  "I do not care what form it takes." The alien spoke the Trade Language well, but his strange throat added a castinetlike clicking. "Our own legend holds no fountain. When we learned to cross between stars we found the legend of immortality wherever there were thinking beings. Whatever their shape or size or intelligence, whether they make their own worlds or make only clay pots, they all tell the tales of people who live forever."

  "It's hard not to wonder if they have some basis," Gordon encouraged him.

  The alien's head snapped around, fast enough and far enough to break a man's neck. The prominent lumps bobbing in his throat were of alien shape: not Adam's apple, but someone else's. "It must be so. I have searched too long for it to be false. You, have you ever found clues to the secret of living forever?"

  Gordon searched when he could, when his Embassy job permitted it. There had been rumors about the Ftokteek. Gordon had followed the rumors out of human space, toward the galactic core and the Ftokteek Empire, to this Ftokteek-dominated meeting place of disparate life forms, this cloud of bubble-worlds of varying gravities and atmospheres. Gordon was middle-aged now, and Sol was invisible even to orbiting telescopes, and the Ftokteek died like anyone else.

  He said, "We've got the legends. Look them up in the Human Embassy library. Ponce de Leon, and Gilgamesh, and Orpheus, and Tithomis, and... every god we ever had lived forever, if he didn't die by violence, and some could heal from that. Some religions say that some part of us lives on after we die."

  "I will go to your library tomorrow," the alien said without enthusiasm. "Do you have no more than legends?"

  "No, but... do other species tell cautionary tales?"

  "I do not understand."

  Gordon said, "Some of our legends say you wouldn't want to live forever. Tithonus, for instance. A goddess gave him the gift of living forever, but she forgot to keep him young. He withered into a lizard.

  Adam and Eve were exiled by God; he was afraid they'd learn the secret of immortality and be as good as Him. Orpheus tried to bring a woman back from the dead. Some of the stories say you can't get immortality, and some say you'd go insane with boredom."

  The alien pondered. "The tale tellers disdain immortality because they cannot have it. Jealousy? Could immortal beings have walked among you once?"

  Gordon laughed. "I doubt it. Was that what made you come to me?"

  "I go to the worlds where many species meet. When I see a creature new to me, then I ask. Sometimes I can sense others like me, who want never to die."

  Gordon looked down past the edge of the balcony, down through the great window at the banded Jovian planet that held this swarm of bubble-worlds in their orbits. He came here every day; small wonder that the alien had picked him out. He came because he would not eat with the others. They thought he was crazy. He thought of them as mayflies, with their attention always on the passing moment, and no thought for the future. He thought of himself as an ambitious mayfly; and he ate alone.

  The alien was saying, "When I was young I looked for the secret among the most advanced species. The great interstellar empires, the makers of artificial worlds, the creatures who mine stars for elements and send ships through the universe seeking ever more knowledge, would build their own immortality. But they die as you and I die. Some races live longer than mine, but they all die."

  "The Ftokteek have a computerized library the size of a small planet," Gordon said. He meant to get there someday, if he lived. "It must know damn near everything."

  The alien answered with a whispery chuckle. "No bigger than a moon is the Ftokteek library. It told me nothing I could use."

  The banded world passed from view.

  "Then I looked among primitives," the alien said, "who live closer to their legends. They die. When I thought to talk to their ghosts, there was nothing, though I used their own techniques. Afterward I searched the vicinities of the black holes and other strange pockets of the universe, hoping that there may be places where entropy reverses itself. I found nothing. I examined the mathematics that describe the universe. I have learned a score of mathematical systems, and none hold any hope of entropy reversal, natural or created."

  Gordon watched stars pass below his feet. He said, "Relativity. We used to think that if you traveled faster than light, time would reverse itself."

  "I know eight systems of traveling faster than light."

  "Eight? What is there besides ours and the Ftokteek drive?"

  "Six others. I rode them all, and always I arrived older. My time runs short. I never examined the quasars, and now I would not live to reach them. What else is left? I have been searching for fourteen thousand years—" The alien didn't notice when Gordon made a peculiar hissing sound. "—in our counting.

  Less in yours, perhaps. Our world huddles closer to a cooler sun than this. Our year is twenty-one million standard seconds."

  "What are you saying? Ours is only thirty-one million—"

  "My present age is three hundred thirty-six point seven billion standard seconds in the Ftokteek counting."

  "Ten thousand Earth years. More!"

  "Far too long. I never mated. None carry my genes. Now none ever will, unless I can grow young again.

  There is little time left."

  "But why?"

  The alien seemed startled. "Because it is not enough. Because I am afraid to die. Are you shortlived, then?"

  "Yes," said Gordon.

  "Well, I have traveled with short-lived companions. They die, I mourn. I need a companion with the strength of youth. My spacecraft is better than any you could command. You may benefit from my research. We breathe a similar air mixture, our bodies use the same chemistry, we search for the same treasure. Will you join my quest?"

  "No."

  "But I sensed that you seek immortality. I am never wrong. Don't you feel it, the certainty that there is a way to thwart entropy, to live forever?"

  "I used to think so," said Gordon.

  In the morning he arranged passage home to Sol system. Ten thousand years wasn't enough... no lifetime was enough, unless you lived it in such a way as to make it enough.

  ***

  A lot of my main characters have been two to three hundred years old and in the prime of health. What the heck, it's science fiction... but it's also a
side effect of my own personality. I want to live a long time. I can't help it.

  ***

  Rotating Cylinders and the Possibility of Global Causality Violation

  This story has a catchy title. I stole it from a mathematics paper by Frank J. Tipler.

  ***

  "Three hundred years we've been at war," said Quifting, "and I have the means to end it. I can destroy the Hallane Regency." He seemed very pleased with himself, and not at all awed at being in the presence of the emperor of seventy worlds.

  The aforementioned emperor said, "That's a neat trick. If you can't pull it off, you can guess what penalties I might impose. None of my generals would dare such a brag."

  "Their tools are not mine." Quifting shifted in a valuable antique massage chair. He was small and round and completely hairless: the style of the nonaristocratic professional. He should have been overawed, and frightened. "I'm a mathematician. Would you agree that a time machine would be a useful weapon of war?"

  "I would," said the emperor. "Or I'd take a faster-than-light starship, if you're offering miracles."

  "I'm offering miracles," said Quifting, "but to the enemy."

  The emperor wondered if Quifting was mad. Mad or not, he was hardly dangerous. The emperor was halfway around the planet from him, on the night side. His side of the meeting room was only a holographic projection, though Quifting wouldn't know that.

  Half a dozen clerks and couriers had allowed this man to reach the emperor's ersatz presence. Why?

  Possibly Quifting had useful suggestions, but not necessarily. Sometimes they let an entertaining madman through, lest the emperor grow bored.

  "It's a very old idea," Quifting said earnestly. "I've traced it back three thousand years, to the era when spaceflight itself was only a dream. I can demonstrate that a massive rotating cylinder, infinite in length, can be circled by closed timelike paths. It sees reasonable that a long but finite—"

  "Wait. I must have missed something."

  "Take a massive cylinder," Quifting said patiently, "and put a rapid spin on it. I can plot a course for a spacecraft that will bring it around the cylinder and back to its starting point in space and time."