[No text on this page] The universal cellular automaton emulating one step in the evolution of the rule shown above, which involves next-nearest as well as nearest-neighbor cells. The rule now covers a total of 32 cases, corresponding to the possible arrangements of colors of a cell and its nearest and next-nearest neighbors. The picture shows the evolution of five cells according to the rule shown, with each cell now being represented by a block of 70 cells in the universal cellular automaton.

[No text on this page] Sequential substitution systems that emulate cellular automata with rules 90 and 30. … Tag systems that emulate the rule 90 and rule 30 cellular automata.

Even early in antiquity attempts were presumably made to see whether simple abstract rules could reproduce the behavior of natural systems. But so far as one can tell the only types of rules that were tried were ones associated with standard geometry and arithmetic. … And from seeing the sophistication of these rules there began to develop an implicit belief that in almost no important cases would simpler rules be useful in reproducing the behavior of natural systems.

There are now altogether five variables, but at least for rules like rules 254 and 90 the individual terms end up not depending on most of these variables. Boolean expression representations of the rules for three elementary cellular automata. The first row shows the original cellular automaton rules.

[No text on this page] Examples of rules with three colors that achieve the purpose of doubling the width of the pattern given in their input. These examples are taken from the 4277 found in effect by searching exhaustively all 7,625,597,484,987 possible rules with three colors. In most cases the number of steps to generate the final pattern increases roughly linearly with the width of the input—although in the case of the fourth-to-last rule on the second row it is 2(n 2 -n+1) for width n .

Rule (b) gives the so-called Thue–Morse sequence, which we will encounter many times in this book. Rule (c) is related to the Fibonacci sequence. Rule (d) gives a version of the Cantor set.

And with this notation, what the rule does is to specify how f[n] should be calculated from previous numbers in the sequence. … The table below gives results obtained with a few specific rules. … With all the rules shown here, successive elements either increase smoothly or fluctuate in a purely repetitive way.

For even programs with some of the very simplest possible rules yield highly complex behavior, while programs with fairly complicated rules often yield only rather simple behavior. … If one just looks at a rule in its raw form, it is usually almost impossible to tell much about the overall behavior it will produce. … Despite the similarity of their rules, the overall behavior of these cellular automata differs considerably.

The first set of pictures below show an example, based on the rule 184 cellular automaton. … As an example, the second picture below shows the rule 110 cellular automaton evolving from random initial conditions. The picture The generation of a nested pattern by rule 184 starting from random initial conditions.

Having more complicated underlying rules has not, it seems, led to much greater complexity in overall behavior. … Examples of three-color totalistic rules that yield patterns which attain a certain size, then repeat forever. The maximum repetition period is found to be 78 steps, and is achieved by the rule with code number 1329.