Thus for example, however many times one runs a rule 30 cellular automaton, starting with a single black cell, the behavior one gets will always be exactly the same. … As an example, one can consider allowing each cell to be not just black or white, but any shade of gray, as in the continuous cellular automata we discussed on page 155 .

But despite their reversibility, the rules still manage to produce the kinds of complex behavior that we have seen in cellular automata and many other systems throughout this book. … The picture at the top of the next page shows an example based on a reversible cellular automaton of the type discussed in the previous section .

The patterns can also be viewed as outputs from a single step in the evolution of two-dimensional block cellular automata in which the rules specify that a block becomes dark if it has the arrangement of cells shown, and becomes light otherwise. … The absence of any dark blocks in many of the cases shown can be viewed as a reflection of constraints introduced by the construction of the images from one-dimensional cellular automaton rules.

The Phenomenon of Universality In the previous section we saw that it is possible to get cellular automata to perform some fairly sophisticated computations. But for each specific computation we wanted to do, we always set up a cellular automaton with a different set of underlying rules.

But is case (b) really the minimal cellular automaton that achieves the purpose of doubling its input? … After all, Examples of cellular automata that can be viewed as achieving the purpose of doubling the width of the pattern given in their input.

As it turns out, the first substitution system shown works almost exactly like a cellular automaton. … So in the end the patterns we obtain can look just as random as what we have seen in systems like cellular automata.

Note that processes such as cellular automaton evolution do not yield networks whose properties are particularly close to those of purely random ones.

Implementation [of phyllotaxis model] It is convenient to consider a line of discrete cells, much as in a continuous cellular automaton.

Note (e) for The Threshold of Universality in Cellular Automata…[Cellular automaton] rule emulations The network below shows which quiescent symmetric elementary rules can emulate which with blocks of length 8 or less. … In any 1D cellular automaton the color of a particular cell can always be determined from the colors t steps back of a block of 2 r t + 1 cells (compare pages 605 and 960 ).

General features of phase transitions To reproduce the Ising model, a cellular automaton must have several special properties. … And with the constraint of reversibility, it turns out that it is impossible to get a non-trivial phase transition in any 1D system with the kind of short-range interactions that exist in a cellular automaton. … And from studying phase transitions in cellular automata, it does not seem that an interpretation in terms of symmetry is particularly useful.