The Peace War by Vernor Vinge

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.
    "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 unaesthetic 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