[No text on this page] Five hundred steps in the evolution of the rule 30 cellular automaton from page 27 . … The asymmetry between the left and right-hand sides is a direct consequence of asymmetry that exists in the particular underlying cellular automaton rule used.

At each step the transformation is done by scanning once from left to right, and applying the rule wherever possible without overlapping. … At each step each boxed region is transformed according to the rule shown. This transformation corresponds to applying the basic Mathematica operation expression /. rule .

But despite fundamental differences like this in underlying rules, the overall behavior produced by systems based on numbers is still very similar to what one sees for example in cellular automata. So just like for the various kinds of programs that we discussed in the previous chapter , the details of underlying rules again do not seem to have a crucial effect on the kinds of behavior that can occur. … An example of a system defined by the following rule: at each step, take the number obtained at that step and write its base 2 digits in reverse order, then add the resulting number to the original one.

It is straightforward to generalize the setup for two-dimensional rules to the three-dimensional case. … Patterns produced by evolution according to a simple two-dimensional cellular automaton rule starting from rows of black cells of various lengths. … The patterns in the picture are obtained by 60 steps of evolution according to this rule.

The system works by partitioning the sequence of cells that exists at each step into pairs, then replacing these pairs by other pairs according to the rule shown. … Like many block cellular automata, the system shown is reversible, since in the rule each pair has a unique predecessor. … It so happens that all but the second of the rules shown here not only conserve the total number of black cells but also turn out to be reversible.

As I have discovered in this book, it is rather easy to generate complex behavior by starting from simple initial conditions and then following simple sets of rules. But the point is that if one starts from some particular piece of behavior there are in general no such simple rules that allow one to go backwards and find out how this behavior can be produced. … Patterns produced by taking a single black cell, then evolving for 50 and 100 steps according to outer totalistic cellular automaton rules 54, 222 and 374.

In setting up rules for network systems, it is convenient to distinguish the two connections that come out of each node. … The pictures on the facing page show examples of evolution obtained with four different choices of underlying rules. … And it turns out that none of the rules I consider here can ever reconnect these pieces again.

Unlike in the picture below, these rules do not reach their final states after one step, but instead just progressively evolve towards these states. … Both rules start on step 1 from random initial conditions in which all sequences of black and white cells are allowed. On subsequent steps, rule 255 allows only sequences containing just black cells, while rule 4 allows sequences that contain both black and white cells, but requires that every black cell be surrounded by white cells.

And this means that rules based on replacements for collections of these clusters will have the property that the causal networks they produce are independent of the updating scheme used. … But beyond the practical problem of displaying what happens, there is actually no fundamental difficulty in setting up rules that can generate non-planarity—and indeed many rules based on the clusters below will for example do this. So in the end, if one manages to find the ultimate rules for the universe, my expectation is that they will give rise to networks that on a small scale look largely random.

But the idea of a cyclic tag system is to make the underlying rule already specify exactly what block can be added at each step. … There are two cases in the rule, and these cases are used on alternate steps, as indicated by the circle icons on the left. … The rule can be summarized just by giving the blocks to be used in each case, as shown in the rule summary.