there might be a general phenomenon whereby simple rules like those for computing π could produce complex results.

In the early 1900s various explicit examples were constructed in several areas of mathematics in which simple rules were repeatedly applied to numbers, sequences or geometrical patterns. And sometimes nested or fractal behavior was seen. And in a few cases substantially more complex behavior was also seen. But the very complexity of this behavior was usually taken to show that it could not be relevant for real mathematical work—and could only be of recreational interest.

When electronic computers began to be used in the 1940s, there were many more opportunities for the phenomenon of complexity to be seen. And indeed, looking back, significant complexity probably did occur in many scientific calculations. But these calculations were almost always based on traditional mathematical models, and since previous analyses of these models had not revealed complexity, it tended to be assumed that any complexity in the computer calculations was just a spurious consequence of the approximations used in them.

One class of systems where some types of complexity were noticed in the 1950s are so-called iterated maps. But as I will discuss on page 149, the traditional mathematics that was used to analyze such systems ended up concentrating only on certain specific features, and completely missed the main phenomenon discovered in this chapter.

It is often useful in practical computing to produce sequences of numbers that seem random. And starting in the 1940s, several simple procedures for generating such sequences were invented. But perhaps because these procedures always seemed quite ad hoc, no general conclusions about randomness and complexity were drawn from them.

Along similar lines, systems not unlike the cellular automata discussed in this chapter were studied in the late 1950s for generating random sequences to be used in cryptography. Almost all the results that were obtained are still military secrets, but I do not believe that any phenomena like the ones described in this chapter were discovered.

And in general, within the context of mainstream science, the standard intuition that had been developed made it very difficult for anyone to imagine that it would be worth studying the behavior of the