Cellular automaton fluids A large number of technical issues can be studied in connection with cellular automaton fluids. … The model can be viewed as a block cellular automaton of the type discussed on page 460 , but on a 2D hexagonal grid. In general a block cellular automaton works by making replacements for overlapping blocks of cells on alternating steps.

PDE approximations [to cellular automata] Cellular automaton (d) in the main text can be viewed as minimal discrete approximations to the diffusion equation. … The cellular automaton itself uses in effect a distributed representation of the density.

More Cellular Automata…[No text on this page] Nine thousand steps in the evolution of the three-color totalistic cellular automaton with code number 1599.

Built-in cellular automaton function Versions of Mathematica subsequent to the release of this book will include a very general function for cellular automaton evolution.

One straightforward way to generate collections of systems that will inevitably exhibit conserved quantities is to work not with ordinary cellular automata but instead with block cellular automata. The basic idea of a block cellular automaton is illustrated at the top of the next page . … Examples of cellular automata with next-nearest-neighbor rules whose evolution conserves the total number of black cells.

The Threshold of Universality in Cellular Automata…But it does once again support the idea that almost any cellular automaton whose behavior seems to us complex can be made to do computations that are in a sense as sophisticated as one wants. And this suggests that such cellular automata will in the end turn out to be universal—with the result that out of the 256 elementary rules one expects that perhaps as many as 27 will in fact be universal. Summaries of how various underlying cellular automata do in emulating a single step in the evolution of each of the 256 possible elementary cellular automata using the scheme from the facing page with blocks of successively greater widths.

But explicit construction, based on correspondence with one-dimensional cellular automata, leads to the example shown at the top of the facing page : a system with 56 allowed templates in which the only pattern satisfying the constraint is a complex and largely random one, derived from the rule 30 cellular automaton. … The system shown was specifically constructed in correspondence with the rule 60 elementary one-dimensional cellular automaton.

And if one runs the cellular automaton for more steps, as in the picture below, then a rather intricate pattern emerges. … So of the three cellular automata that we have seen so far, all ultimately yield patterns that are highly regular: the first a simple uniform pattern, the second a repetitive pattern, and the third an intricate but still nested pattern. … The picture shows five hundred steps of cellular automaton evolution.

As one example, I discovered the classification scheme for cellular automata with random initial conditions described at the beginning of Chapter 6 when I first looked at large numbers of different cellular automata together on high-resolution graphics displays. … For a computer display, like a cellular automaton, consists of a regular array of discrete cells or pixels. … And as a more significant example, one might have imagined that the simple structure of cellular automaton programs would make it straightforward to foresee their behavior.

So as an example of all this consider cellular automata that achieve the purpose of doubling the width of the pattern given in their input. Case (a) in the picture on the next page is a cellular automaton one might construct for this purpose by using ideas from traditional engineering. But while this cellular automaton seems to have little extraneous going on, it operates in a slow and sequential way, and its underlying If the purpose is to generate a uniformly expanding pattern it seems more plausible that the top cellular automaton should have been the one created for this purpose.