SIX

Many things were the same after that night. They were the outward things: Wili worked in the gardens almost as much as before. Even with the gifts of food the visitors had brought, they still needed to work to feed themselves. (Wili's appetite was greater than the others'. It didn't seem to help; he remained as undernourished and stunted as ever.) But in the afternoons and evenings he worked with Naismith's machines.

It turned out the ghost was one of those machines. Jill, the old man called her, was actually an interface program run on a special processor system. She was good, almost like a person. With the projection equipment Naismith had built into the walls of the veranda, she could even appear in open space. Jill was the perfect tutor, infinitely patient but with enough "humanity" to make Wili want to please her. Hour after hour, she flashed language questions at him. It was like some verbal Celest. In a matter of weeks, Wili progressed from being barely literate to having a fair command of technical written English.

At the same time, Naismith began teaching him math. At first Wili was contemptuous of these problems. He could do arithmetic as fast as Naismith. But he discovered that there was more to math than the four basic arithmetic operations. There were roots and transcendental functions; there were the relationships that drove both Celest and the planets.

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 interesting. 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 narrower rectangles. The heights bounced back and forth between 1 and -1.

35

"Well," he said, turning from the display, "what is the pattern? 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 unesthetic 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 minimum 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 narrow 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 differences 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 decomposition questions that took Wili many minutes to solve... until Naismith showed him a little trick - something 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 already 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 himself 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. Naismith 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. Naismith 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 between 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