by Vernor Vinge
Naismith's machines showed him functions as graphs and related function operations to those pictures. As the days passed, the functions became very specialized and interest-ing. One night, Naismith sat at the controls and caused a string of rectangles of varying width to appear on the screen. They looked like irregular crenellations on some battlement. Below the first plot, the old man produced a second and then a third, each somewhat like the first but with more and nar-rower rectangles. The heights bounced back and forth between 1 and -1.
"Well," he said, turning from the display, "what is the pat-tern? Can you show me the next three plots in this series?" It was a game they had been playing for several days now. Of course, it was all a matter of opinion what really constituted a pattern, and sometimes there was more than one answer that would satisfy a person's taste, but it was amazing how often Wili felt a certain rightness in some answers and an unes-thetic blankness in others. He looked at the screen for several seconds. This was harder than Celest, where he merely cranked on deterministic relationships. Hmmm. The squares got smaller, the heights stayed the same, the mini-mum rectangle width decreased by a factor of two on every new line. He reached out and slid his finger across the screen, sketching the three graphs of his answer.
"Good," said Naismith. "And I think you see how you could make more plots, until the rectangles became so nar-row that you couldn't finger-sketch or even display them properly.
"Now look at this." He drew another row of crenellations, one clearly not in the sequence: The heights were not restricted to 1 and -1. "Write me that as the sum and dif-ferences of the functions we've already plotted. Decompose it into the other functions." Wili scowled at the display; worse than "guess the pattern," this was. Then he saw it: three of the first graph minus four copies of the third graph plus...
His answer was right, but Wili's pride was short-lived, since the old man followed this problem with similar decom-position questions that took Wili many minutes to solve... until Naismith showed him a little trick - some-thing called orthogonal decomposition - that used a peculiar and wonderful property of these graphs, these "walsh waves" he called them. The insight brought a feeling of awe just a little like learning about the moving stars, to know that hidden away in the patterns were realities that might take him days to discover by himself.
Wili spent a week dreaming up other orthogonal families and was disappointed to discover that most of them were al-ready famous - haar waves, trig waves - and that others were special cases of general families known for more than two hundred years. He was ready for Naismith's books now. He dived into them, rushed past the preliminary chapters, pushed himself toward the frontier where any new insights would be beyond the farthest reach of previous explorers.
In the outside world, in the fields and the forest that now were such a small part of his consciousness, summer moved into fall. They worked longer hours, to get what crops remained into storage before the frosts. Even Naismith did his best to help, though the others tried to prevent this. The old man was not weak, but there was an air of physical fragility about him.
From the high end of the bean patch, Wili could see over the pines. The leafy forests had changed color and were a band of orange-red beyond the evergreen. The land along the coast was clouded over, but Wili suspected that the jungle there was still wet and green. Vandenberg Dome seemed to hang in the clouds, as awesome as ever. Wili knew more about it now, and someday he would discover all its secrets. It was simply a matter of asking the right questions - of him-self and of Paul Naismith.
Indoors, in his greater universe, Wili had completed his first pass through functional analysis and now undertook a three-pronged expedition that Naismith had set for him: into finite galois theory, stochastics, and electromagnetics. There was a goal in sight, though (Wili was pleased to see) there was no ultimate end to what could be learned. Nais-mith had a project, and it would be Wili's if he was clever enough.
Wili saw why Naismith was valued and saw the peculiar service he provided to people all over the continent. Nais-mith solved problems. Almost every day the old man was on the phone, sometimes talking to people locally - like Miguel Rosas down in Santa Ynez - but just as often to people in Fremont, or in places so far away that it was night on the screen while still day here in Middle California. He talked to people in English and in Spanish, and in languages that Wili had never heard. He talked to people who were neither Jonques nor Anglos nor blacks.
Wili had learned enough now to see that these were not nearly as simple as making local calls. Communication be-tween towns along the coast was trivial over the fiber, where almost any bandwidth could be accommodated. For longer distances, such as from Naismith's palace to the coast, it was still relatively easy to have video communication: The coherent radiators on the roof could put out microwave and infrared beams in any direction. On a clear day, when the IR radiator could be used, it was almost as good as a fiber (even with all the tricks Naismith used to disguise their location). But for talking around the curve of the Earth, across forests and rivers where no fiber had been strung and no line of sight existed, it was a different story: Naismith used what he called "short-waves" (which were really in the one- to ten- meter range). These were quite unsuitable for high-fidelity communication. To transmit video-even the wavery black-and-white flat pictures Naismith used in his transcontinental calls - took incredibly clever coding schemes and some real-time adaptation to changing conditions in the upper atmosphere.
The people at the other end brought Naismith problems, and he came back with answers. Not immediately, of course; it often took him weeks, but he eventually thought of some-thing. At least the people at the other end seemed happy. Though it was still unclear to Wili how gratitude on the other side of the continent could help Naismith, he was beginning to understand what had paid for the palace and how Naismith could afford full-scale holo projectors. It was one of these problems that Naismith turned over to his ap-prentice. If he succeeded, they might actually be able to steal pictures off the Authority's snooper satellites.
It wasn't only people that appeared on the screens.
One evening shortly after the first snowfall of the season, Wili came in from the stable to find Naismith watching what appeared to be an empty patch of snow-covered ground. The picture jerked every few seconds, as if the camera were held by a drunkard. Wili sat down beside the old man. His stomach was more upset than usual and the swinging of the picture did nothing to help the situation -but his curiosity gave him no rest. The camera suddenly swung up to eye level and looked through the pine trees at a house, barely visible in the evening gloom. Wili gasped - it was the build-ing they were sitting in.
Naismith turned from the screen and smiled. "It's a deer, I think. South of the house. I've been following her for the last couple of nights." It took Wili a second to realize he was referring to what was holding the camera. Wili tried to im-agine how anyone could catch a deer and strap a camera on it. Naismith must have noticed his puzzlement. "Just a second." He rummaged through a nearby drawer and handed Wili a tiny brown ball. "That's a camera like the one on the critter. It's wide enough so I have resolution about as good as the human eye. And I can shift the decoding parameters so it will 'look' in different directions without the deer's having to move.
`Jill, move the look axis, will you?"
"Right, Paul." The view slid upward till they were looking into overhanging branches and then down the other side. Wili and Naismith saw a scrawny back and part of a furry ear.
Wili looked at the object Paul had placed in his hand. The "camera" was only three or four millimeters across. It felt warm and almost sticky in Wili's hand. It was a far cry from the lensed contraptions he had seen in Jonque villas. So you just stick them to the fur, true?" said Wili.
Naismith shook his head. "Even easier than that. I can get these in hundred lots from the Greens in Norcross. I scatter them through the forest, on branches and such. All sorts of animals pick them up. It provides just a l
ittle extra security. The hills are safer than they were years ago, but there are still a few bandits."
"Um." If Naismith had weapons to match his senses, the manor was better protected than any castle in Los Angeles. "This would be greater protection if you could have people watching all the views all the time."
Naismith smiled, and Wili thought of Jill. He knew enough now to see that the program could be made to do just that.
Wili watched for more than an hour as Naismith showed him scenes from a number of cameras, including one from a bird. That gave the same sweeping view he imagined could be seen from Peace Authority aircraft.
When at last he went to his room, Wili sat for a long while looking out the garret window at the snow-covered trees, looking at what he had just seen with godlike clarity from dozens of other eyes. Finally he stood up, trying to ignore the cramp in his gut that had become so persistent these last few weeks. He removed his clothes from the closet and lay them on the bed, then inspected every square centimeter with his eyes and fingers. His favorite jacket and his usual work pant both had tiny brown balls stuck to cuffs or seams. Wili removed them; they looked so innocuous in the room's pale lamplight.
He put them in a dresser drawer and returned his clothes to the closet.
He lay awake for many minutes, thinking about a place and time he had resolved never to dwell on again. What could a hovel in Glendora have in common with a palace in the mountains? Nothing. Everything. There had been safety there. There had been Uncle Sylvester. He had learned there, too - arithmetic and a little reading. Before the Jon-ques, before the Ndelante -it had been a child's paradise, a time lost forever.
Wili quietly got up and slipped the cameras back into his clothing. Maybe not lost forever.
SEVEN
January passed, an almost uninterrupted snowstorm. The winds coming off Vandenberg brought ever-higher drifts that eventually reached the mansion's second storey and would have totally blocked the entrances if not for the heroic efforts of Bill and Irma. The pain in Wili's middle became constant, intense. Winters had always been bad for him, but this one was worse than ever before, and the others eventual-ly became aware of it. He could not suppress the occasional grimace, the faint groan. He was always hungry, always eating-and yet losing weight.
But there was great good, too. He was beyond the fron-tiers of Naismith's books! Paul claimed that no previous human had insight on the coding problem that he had at-tacked! Wili didn't need Naismith's machines now; the images in his mind were so much more complete. He sat in the living room for hours-through most of his waking time - almost unaware of the outside world, almost unaware of his pain, dreaming of the problem and his schemes for its defeat. All existence was groups and graphs and endless combinatorical refinements on the decryption scheme he hoped would break the problem.
But when he ate and even when he slept, the pain levered itself back into his soul.
It was Irma, not Wili, who noticed that the paler skin on his palms had a yellow cast beneath the brown. She sat beside him at the dining table, holding his small hands in her large, calloused ones. Wili bristled at her touch. He was here to eat, not to be inspected. But Paul stood behind her.
"And the nails look discolored, too." She reached across to one of Wili's yellowed fingernails and gave it a gentle tug. Without sound or pain, the nail came away at its root. Wili stared stupidly for a second, then jerked his hand back with a shriek. Pain was one thing; this was the nightmare of a body slowly dismembering itself. For an instant terror blotted out his gutpain the way mathematics had done before.
They moved him to a basement room, where he could be warm all the time. Wili found himself in bed most of each day. His only view of the outside, of the cloudswept purity of Vandenberg, was via the holo. The mountain snows were too deep to pass travelers; there would be no doctors. But Nais-mith moved cameras and high-bandwidth equipment into the room, and once when Wili was not lost in dreaming, he saw that someone from far away was looking on, was being interrogated by Naismith. The old man seemed very angry.
Wili reached out to touch his sleeve. "It will be all right, Uncle Syl - Paul. This problem I have always had and worst in the winters. I will be okay in the spring."
Naismith smiled and nodded, then turned away.
But Wili was not delirious in any normal sense. During the long hours an average patient would have lain staring at the ceiling or watching the holo and trying to ignore his pain, Wili dreamed on and on about the communications problem that had resisted his manifold efforts all these weeks. When the others were absent, there was still Jill, taking notes, ready to call for help; she was more real than any of them. It was hard to imagine that her voice and pretty face had ever seemed threatening.
In a sense, he had already solved the problem, but his scheme was too slow; he needed n*log(n) time for this ap-plication. He was far beyond the tools provided by his brief, intense education. Something new, something clever was needed, and by the One True God he would find it!
And when the solution did come it was like a sun rising on a clear morning, which was appropriate since this was the first clear day in almost a month. Bill brought him up to ground level to sit in the sunlight before the newly cleared windows. The sky was not just clear, but an intense blue. The snow was piled deep, a blinding white. Icicles grew down from every edge and corner, dripping tiny diamonds in the warm light.
Wili had been dictating to Jill for nearly an hour when the old man came down for breakfast. He took one look over Wili's shoulder and then grabbed his reader, saying not a word to Wili or anyone else. Naismith paused many times, his eyes half closed in concentration. He was about a third of the way through when Wili finished. He looked up when Wili stopped talking, "You got it?"
Wili nodded, grinning. "Sure, and in n*log(n) time, too." He glanced at Naismith's reader. "You're still looking at the filter setting up. The real trick isn't for a hundred more lines." He scanned forward. Naismith looked at it for a long time, finally nodded. "I, I think I see. I'll have to study it, but I think... My little Ramanujan. How do you feel?"
"Great," filled with elation, "but tired. The pain has been less these last days, I think. Who is Ramanujan?"
"Twentieth-century mathematician. An Indian. There are a lot of similarities: You both started out without much for-mal education. You are both very, very good."
Wili smiled, the warmth of the sun barely matching what he felt. These were the first words of real praise he had heard from Naismith. He resolved to look up everything on file about this Ramanujan.... His mind drifted, freed from the fixation of the last weeks. Through the pines, he could see the sun on Vandenberg. There were so many mysteries left to master....
EIGHT
Naismith made some phone calls the next day. The first was to Miguel Rosas at the SYP Company. Rosas was under-sheriff to Sy Wentz, but the Tinkers around Vandenberg hired him for almost all their police operations.
The cop's dark face seemed a touch pale after he watched Naismith's video replay. "Okay," he finally said, "who was Ramanujan?"
Naismith felt the tears coming back to his eyes. "That was a bad slip; now the boy is sure to look him up. Ramanujan was everything I told Wili: a really brilliant fellow, without much college education." This wouldn't impress Mike, Nais-mith knew. There were no colleges now, just apprenticeships. "He was invited to England to work with some of the best number theorists of the time. He got TB, died young."
...Oh. I get the connection, Paul. But I hope you don't think that bringing Wili into the mountains did anything to hurt him."
"His problem is worse during winters, and our winters are fierce compared to L.A.'s. This has pushed him over the edge."
"Bull! It may have aggravated his problem, but he got bet-ter food here and more of it. Face it, Paul. This sort of wasting just gets worse and worse. You've seen it before."
"More than you!" That and the more acute diseases of the plague years had come close to destroying mankind. Then Nais
mith brought himself up short, remembering Miguel's two little sisters. Three orphans from Arizona they had been, but only one survived. Every winter, the girls had sickened again. When they died, their bodies were near-skeletons. The young cop had seen more of it than most in his genera-tion.
"Listen, Mike, we've got to do something. Two or three years is the most he has. But hell, even before the War a good pharmaceutical lab could have cured this sort of thing. We were on the verge of cracking DNA coding and -
"Even then, Paul? Where do you think the plagues came from? That's not just Peace Authority jive. We know the Peace is almost as scared of bioresearch as they are that some-one might find the secret of their bobbles. They bobbled Yakima a few years ago just because one of the their agents found a recombination analyzer in the city hospital. That's ten thousand people asphyxiated because of a silly antique. Face it: The bastards who started the plagues are forty years dead-and good riddance."
Naismith sighed. His conscience was going to hurt him on this - a little matter of protecting your customers. "You're wrong, Mike. I have business with lots of people. I have a good idea what most of them do."
Rosas' head snapped up. "Bioscience labs, even in our time?"
"Yes. At least three, perhaps ten. I can't be sure, since of course they don't admit to it. And there's only one whose location is certain."
'Jesus, Paul, how can you deal with such vermin?"
Naismith shrugged. "The Peace Authority is the real enemy. In spite of what you say, it's only their word that the bioscience people caused the plagues, trying to win back for their governments what all the armies could not. I know the Peace," he stopped for a moment, remembering treachery that had been a personal, secret thing for fifty years.
"I've tried to convince you tech people: The Authority can't tolerate you. You follow their laws: You don't make high-density power sources, don't make vehicles or experiment with nucleonics or biology. But if the Authority knew what was going on within the rules... You must have heard about the NCC: I showed conclusively that the Peace is beginning to catch on to us. They are beginning to understand how far we have gone without big power sources and universities and old-style capital industry. They are beginning to realize how far our electronics is ahead of their best. When they see us clearly, they'll step on us the way they have on all opposition, and we're going to have to fight."