[No text on this page] A sequence of totalistic cellular automata with rules that involve only nearest neighbors, but where each cell can have three possible colors.

[No text on this page] Examples of three-color totalistic rules that yield patterns with seemingly random features.

At step 2 in the rule 255 example on the facing page , however, the network has only one loop—representing the fact that at this step the only sequences which can occur with this rule are ones that consist purely of black cells, just as we saw on the previous page . The case of rule 4 is slightly more complicated: at step 2, the possible sequences that can occur are now represented by a network with two nodes. Starting at the right-hand node one can go around the loop to the right any number of times, corresponding to sequences of Four different initial conditions that all lead to the same final state in the rule 4 cellular automaton shown on the previous page .

Perhaps it is because in a simple program it is so easy to see the underlying elements and the rules that govern them. But countless times I have been asked how models based on simple programs can possibly be correct, since even though they may successfully reproduce the behavior of some system, one can plainly see that the system itself does not, for example, actually consist of discrete cells that, say, follow the rules of a cellular automaton. … But in the cellular automaton this effect is just implemented by some rule for certain configurations of cells—and there is no need for the rule to correspond in any way to the detailed dynamics of water molecules.

But which specific cellular automaton rule will any given mollusc use? The pictures at the bottom of the facing page show all the possible symmetrical rules that involve two colors and nearest neighbors. … Rather, it just arises as an inevitable consequence of the basic phenomenon discovered in this book that simple rules will often yield complex behavior.

Indeed, as a first approximation one can imagine that much as in a cellular automaton entities in a market could follow simple rules based on the behavior of other entities. … And one would have to assign fairly complicated rules to each entity—certainly as complicated as the rules in a typical programmed trading system. … But as a very simple idealization of the way that information flows in a market, one can, for example, take each color to be given by a fixed rule that is based on each entity looking at the actions of its neighbors on the previous step.

But I very much doubt that any such obvious symmetry between space and time exists in the fundamental rules for our universe. … And in particular—just as in a system like a cellular automaton—the network can be built up incrementally by starting with certain initial conditions and then applying appropriate underlying rules over and over again. Any such rules can in principle be thought of as providing a set of constraints for the spacetime network.

And the crucial point is that whenever the underlying system is causal invariant the exact same underlying rules will account for what one sees in slices at different angles. And what this means is that in effect the same rules will apply regardless of how fast one is going. … And second, that with causal invariance different slices through a causal network can be produced by the same underlying rules.

As we saw a few sections ago [ 12 , 13 ], many underlying replacement rules end up producing networks that are for example too extensively connected to correspond to ordinary space in any finite number of dimensions. But I suspect that if one has replacement rules that are causal invariant and that in effect successfully maintain a fixed number of dimensions they will almost inevitably lead to behavior that follows something close to the Einstein equations. Probably the situation is somewhat analogous to what we saw with fluid behavior in cellular automata in Chapter 8 —that at least if there are underlying rules whose behavior is complicated enough to generate significant effective randomness, then almost whenever the rules lead to conservation of total particle number and momentum something close to the ordinary Navier–Stokes equation behavior emerges.

It turns out that for most of the kinds of rules used in traditional mathematics, it is in fact fairly easy. But for the more general rules that I discuss in this book it appears to often be extremely difficult. For even though the rules may be simple, the behavior they produce is often highly complex, and shows absolutely no obvious trace of its simple origins.