And in fact it turns out to be quite common for there to exist special initial conditions for one cellular automaton that make it behave just like some other cellular automaton.

So just what set of structures does the code 20 cellular automaton ultimately support? … Persistent structures found by testing the first twenty-five billion possible initial conditions for the code 20 cellular automaton shown on the previous page.

The picture below shows the rule 30 cellular automaton in which I first identified this mechanism for randomness. … As we have discussed before, traditional intuition makes it hard to believe that such complexity could arise from such a simple The rule 30 cellular automaton from page 27 that was the first example I found of intrinsic randomness generation.

Homogenous growth from a single point is one straightforward way that uniformity in space can be produced, here illustrated in a mobile automaton and a cellular automaton. … Class 1 cellular automata that exhibit evolution to a uniform state, as discussed in Chapter 6 .

Relation to 2D cellular automata The kind of constraints discussed are exactly those that must be satisfied by configurations that remain unchanged in the evolution of a 2D cellular automaton. The argument for this is similar to the one on pages 941 and 954 for 1D cellular automata. The point is that of the 32 5-cell neighborhoods involved in the 2D cellular automaton rule, only some subset will have the property that the center cell remains unchanged after applying the rule.

Note (c) for Emulating Cellular Automata with Other Systems…Tag systems [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 , the following will construct a tag system which emulates it: CAToTS[rules_] := {2, {{s[x_], s[y_]}  {d[x, y], d[x, y]}, {d[w_, x_], d[y_, z_]}  {s[{w, x, y} /. rules], s[{x, y, z} /. rules]}, {s[x_], d[y_, z_]}  {s[0], s[0], s[{0, y, z} /. rules]}, {d[x_, y_], s[z_]}  {s[{x, y, 0} /. rules], s[0], s[0]}}} The initial condition for the tag system that corresponds to a single black cell in the cellular automaton is {s[0], s[0], s[1], s[0], s[0]} . Given a list of all steps in the evolution of the tag system, Cases[list, {__s}] picks out successive steps in the cellular automaton evolution.

Symmetry [of discrete space] A system like a cellular automaton that consists of a large number of identical cells must in effect be arranged like a crystal, and therefore must exhibit one of the limited number of possible crystal symmetries in any particular dimension, as discussed on page 929 . And even a generalized cellular automaton constructed say on a Penrose tiling still turns out to have a discrete spatial symmetry.

Sensitivity to Initial Conditions In the previous section we identified four basic classes of cellular automata by looking at the overall appearance of patterns they produce. … The pictures below show the effect of changing the initial color of a single cell in a typical cellular automaton from each of the four classes of cellular automata identified in the previous section . … For as the facing page shows, any change that is made The effect of changing the color of a single cell in the initial conditions for typical cellular automata from each of the four classes identified in the previous section .

Continuous Cellular Automata…And to address this question, what I will do in this section is to consider a generalization of cellular automata in which each cell is not just black or white, but instead can have any of a continuous range of possible levels of gray. One can update the gray level of each cell by using rules that are in a sense a cross between the totalistic cellular automaton rules that we discussed at the beginning of the last chapter and the iterated maps that we just discussed in the previous section . … A continuous cellular automaton in which each cell can have any level of gray between white (0) and black (1).

But when I studied more detailed properties of cellular automata, what I found was that most of these properties were closely correlated with the classes that I had already identified. Indeed, in trying to predict detailed properties of a particular cellular automaton, it was often enough just to know what class the cellular automaton was in.