And this means that the rule—or at least its behavior—will necessarily seem to us unfamiliar and abstract. … But I believe that it will be essentially impossible to find such a formulation without already knowing the rule. And as a result, my guess is that the only realistic way to find the rule in the first place will be to start from some very straightforward representation, and then just to search through large numbers of possible rules in this representation.

So what about more complicated cellular automaton rules? … The facing page shows a representative sequence of such rules. … Interpreting the sequence of new colors as a sequence of base 3 digits, one can assign a code number to each totalistic rule.

The resulting rules can be run both forwards and backwards. … Patterns produced by the rule must exhibit the same time reversal symmetry, as shown on the left. The specific rule used here is based on taking elementary rule 214, then adding the specification that the new color of a cell should be inverted whenever the cell was black two steps back.

For it could still be that the particular rules that appear are somehow specially selected to be ones that are not universal. … But when there are no constraints that force simple overall behavior, my guess is that most rules that appear in nature can be viewed as being selected in no special way—save perhaps for the fact that the structure of the rules themselves tends to be fairly simple. And what this means is that such rules will typically show the same features as rules chosen at random from all possibilities—with the result that presumably they do in the end exhibit universality in almost all cases where their overall behavior is not obviously simple.

What about rules that have more than two possible colors for each cell? It turns out that there is a general way of emulating such rules by using rules that have just two colors but a larger number of neighbors. … The same basic scheme can be used for rules with any number of colors.

The picture below shows what happens in rule 45. … So although this means that the particular type of approach we used to demonstrate the universality of rule 110 cannot immediately be used for rule 30 or rule 45, it certainly does not mean that these rules are not in the end universal. … Note the appearance of a slanted version of the nested pattern from rule 90.

In rule 22 the low initial density has no long-term effect. But in rule 90 its effect continues forever. The reason for this difference is that in rule 22 the randomness we see is intrinsically generated by the evolution of the system, while in rule 90 it comes from randomness in the initial conditions.

[No text on this page] Details of how the universal cellular automaton emulates rule 90. The only difference in initial conditions from the picture on the previous page is that each block now encodes rule 90 instead of rule 254.

At the end of the last section I mentioned rule 54 as another elementary cellular automaton besides rule 110 that might be class 4. The pictures below show examples of the typical behavior of rule 54. Two views of the evolution of rule 54 from typical random initial conditions.

[No text on this page] Details of how the universal cellular automaton emulates rule 30. Once again, the only difference in initial conditions from the facing page is that each block now encodes rule 30 instead of rule 90.