in practice. For as we shall see later in this book, it is certainly not that nature fundamentally follows these abstractions.
And I suspect that in fact the current predominance of partial differential equations is in many respects a historical accident—and that had computer technology been developed earlier in the history of mathematics, the situation would probably now be very different.
But particularly before computers, the great attraction of partial differential equations was that at least in simple cases explicit mathematical formulas could be found for their behavior. And this meant that it was possible to work out, for example, the gray level at a particular point in space and time just by evaluating a single mathematical formula, without having in a sense to follow the complete evolution of the partial differential equation.
The pictures on the facing page show three common partial differential equations that have been studied over the years.
The first picture shows the diffusion equation, which can be viewed as a limiting case of the continuous cellular automaton on page 156. Its behavior is always very simple: any initial gray progressively diffuses away, so that in the end only uniform white is left.
The second picture shows the wave equation. And with this equation, the initial lump of gray shown just breaks into two identical pieces which propagate to the left and right without change.
The third picture shows the sine-Gordon equation. This leads to slightly more complicated behavior than the other equations—though the pattern it generates still has a simple repetitive form.
Considering the amount of mathematical work that has been done on partial differential equations, one might have thought that a vast range of different equations would by now have been studied. But in fact almost all the work—at least in one dimension—has concentrated on just the three specific equations on the facing page, together with a few others that are essentially equivalent to them.
And as we have seen, these equations yield only simple behavior.
So is it in fact possible to get more complicated behavior in partial differential equations? The results in this book on other kinds of systems strongly suggest that it should be. But traditional