mathematical methods give very little guidance about how to find such behavior. Indeed, it seems that the best approach is essentially just to search through many different partial differential equations, looking for ones that turn out to show complex behavior.

But an immediate difficulty is that there is no obvious way to sample possible partial differential equations. In discrete systems such as cellular automata there are always a discrete set of possible rules. But in partial differential equations any mathematical formula can appear.

Nevertheless, by representing formulas as symbolic expressions with discrete sets of possible components, one can devise at least some schemes for sampling partial differential equations.

But even given a particular partial differential equation, there is no guarantee that the equation will yield self-consistent results. Indeed, for a very large fraction of randomly chosen partial differential equations what one finds is that after just a small amount of time, the gray level one gets either becomes infinitely large or starts to vary infinitely quickly in space or time. And whenever such phenomena occur, the original equation can no longer be used to determine future behavior.

But despite these difficulties I was eventually able to find the partial differential equations shown on the next two pages [165, 166].

The mathematical statement of these equations is fairly simple. But as the pictures show, their behavior is highly complex.

Indeed, strangely enough, even though the underlying equations are continuous, the patterns they produce seem to involve patches that have a somewhat discrete structure.

But the main point that the pictures on the next two pages [165, 166] make is that the kind of complex behavior that we have seen in this book is in no way restricted to systems that are based on discrete elements. It is certainly much easier to find and to study such behavior in these discrete systems, but from what we have learned in this section, we now know that the same kind of behavior can also occur in completely continuous systems such as partial differential equations.