Rule (a) above yields a causal network that is purely repetitive and thus yields no meaningful notion of space. Rules (b), (c) and (d) yield causal networks that in effect grow roughly linearly with time. In rule (f) the causal network grows exponentially, while in rule (e) the causal network also grows quite rapidly, though its overall growth properties are not clear.

On each page the underlying rules for the universal cellular automaton are exactly the same. But on the first page , the initial conditions are set up so as to make the universal cellular automaton emulate rule 254, while on the second page they are set up to make it emulate rule 90, and on the third page rule 30. … The rules for the universal cellular automaton.

But despite this sensitivity at the level of details, the point is that any system like rule 22 or rule 30 yields patterns whose overall properties depend very little on the form of the initial conditions that are given. … So one may wonder whether there are in fact any initial conditions that make rule 30 behave in a simple way. … At left is a representation of rule 30.

means is that any simplicity observed in known physical laws may have little connection with simplicity in the underlying rule. Indeed, it turns out that simple overall laws can emerge almost regardless of underlying rules. … There will certainly be questions about why it is this particular rule, and not another one.

But knowing that a system like rule 110 is universal, one now suspects that this threshold is remarkably easy to reach. … So in retrospect the results of Chapter 3 should already have suggested that simple underlying rules such as rule 110 might be able to achieve universality. … But as we saw in Chapter 6 , such behavior is by no means unique to rule 110.

So what kinds of rules show class 4 behavior? Among the 256 so-called elementary cellular automata that allow only two possible colors for each cell and depend only on nearest neighbors, the only clear immediate example is rule 110—together with rules 124, 137 and 193 obtained by trivially reversing left and right or black and white. … As we will see in the next section , one possibility is rule 54.

The first program contains no rules for changing the color of a cell with any neighborhood. Mutations in successive programs add rules for changing the colors of cells with specific neighborhoods, or modify these rules. … The cellular automata shown here all have 3 possible colors and nearest-neighbor rules.

But what about rules that have replacements involving blocks of more than one element? Can such rules still have the necessary properties? The pictures below show two examples of rules that do.

But so how do cellular automata with all these different rules behave? … The rules are numbered according to the scheme described above. … As indicated, the rules can conveniently be numbered from 0 to 255.

The crucial ingredients that are needed for complex behavior are, it seems, already present in systems with very simple rules, and as a result, nothing fundamentally new typically happens when the rules are made more complex. … And this means, for example, that even with highly complex rules, very simple behavior still often occurs. … The details of the underlying rules for a specific system can certainly affect the details of the behavior it produces.