But to develop the new kind of science that I describe in this book I have had no choice but to take several large steps at once, and in doing so I have mostly ended up having to start from scratch—with new ideas and new methods that ultimately depend very little on what has gone before. … And while I hope that all the effort I have put into presentation in this book will make it easier for others, I do not expect it to be a quick process.

In case (a), the fraction of black elements fluctuates around 1/2; in (b) it approaches 3/4; in (d) it fluctuates around near 0.3548, while in (e) and (f) it does not appear to stabilize.

But in an actual physical system one does not expect to be able to find values of amplitudes directly. For according to the standard formalism of quantum theory all amplitudes do is to determine probabilities for particular outcomes of measurements. … It does appear that only modest precision is needed for the initial amplitudes.

. • 1700s and 1800s: The digits of π and other transcendental numbers are seen to exhibit apparent randomness (see page 136 ), but the idea of thinking about this randomness as coming from the process of calculation does not arise. • 1800s: The distribution of primes is studied extensively—but mostly its regularities, rather than its irregularities, are considered. … (See page 1099 .) • Late 1950s: Berni Alder and Thomas Wainwright do computer simulations of dynamics of hard sphere idealized molecules, but concentrate on large-scale features that do not show complexity. … They discover various examples (such as "munching foos") that produce nested behavior (see page 871 ), but do not go further. • 1962: Marvin Minsky and others study many simple Turing machines, but do not go far enough to discover the complex behavior shown on page 81 . • 1963: Edward Lorenz simulates a differential equation that shows complex behavior (see page 971 ), but concentrates on its lack of periodicity and sensitive dependence on initial conditions. • Mid-1960s: Simulations of random Boolean networks are done (see page 936 ), but concentrate on simple average properties. • 1970: John Conway introduces the Game of Life 2D cellular automaton (see above ). • 1971: Michael Paterson considers a class of simple 2D Turing machines that he calls worms and that exhibit complicated behavior (see page 930 ). • 1973: I look at some 2D cellular automata, but force the rules to have properties that prevent complex behavior (see page 864 ). • Mid-1970s: Benoit Mandelbrot develops the idea of fractals (see page 934 ), and emphasizes the importance of computer graphics in studying complex forms. • Mid-1970s: Tommaso Toffoli simulates all 4096 2D cellular automata of the simplest type, but studies mainly just their stabilization from random initial conditions. • Late 1970s: Douglas Hofstadter studies a recursive sequence with complicated behavior (see page 907 ), but does not take it far enough to conclude much. • 1979: Benoit Mandelbrot discovers the Mandelbrot set (see page 934 ) but concentrates on its nested structure, not its overall complexity. • 1981: I begin to study 1D cellular automata, and generate a small picture analogous to the one of rule 30 on page 27 , but fail to study it. • 1984: I make a detailed study of rule 30, and begin to understand the significance of it and systems like it.

And what this suggests is that it makes no more or less sense to talk about the meaning of phenomena in our universe as it does to talk about the meaning of phenomena in the digit sequence of π .

(Attempts are sometimes made to detect sensitive dependence directly by watching whether a system can do different things after it appears to return to almost exactly the same state. But the problem is that it is hard to be sure that the system really is in the same state—and that there are not all sorts of large differences that do not happen to have been observed.)

The basic mistake is usually to make the implicit assumption that computation must be done in some rather specific way—that does not happen to be consistent with the way we have for example seen that it can be done in rule 110.

But division by 2 just does the opposite of multiplication by 2, so in base 2 it simply shifts all digits one position to the right.

Yet the picture itself does not at first appear to be at all reversible.

The pictures on the next page show what happens if one does this first with images of cellular automata and then with images of letters.