Indeed, the rules for this cellular automaton are in some respects much simpler than for even a rather basic linear congruential generator. … The point is that unlike the rule 30 cellular automaton that we discussed above, linear congruential generators are readily amenable to detailed mathematical analysis. … But the fact that simple cellular automaton rules are sufficient to give rise to intrinsic randomness generation suggests that in reality it is rather easy for this mechanism to occur.

One can capture this basic effect by having a cellular automaton with rules in which cells become black if they have exactly one black neighbor, but stay white whenever they have more than one black neighbor. The pictures on the facing page show a sequence of steps in the evolution of such a cellular automaton. … From looking at the behavior of the cellular automaton, one can immediately make various predictions about snowflakes.

But as a rough approximation one can perhaps assume that each element of a solid is either displaced or not, and that the displacements of neighboring elements interact by some definite rule—say a simple cellular automaton rule. … A very simple cellular automaton model for fracture. At each step, the color of each cell, which roughly represents the displacement of an element of the solid, is updated according to a cellular automaton rule.

Cellular Automata…[No text on this page] Stages in the evolution of the cellular automaton from the facing page , starting with an initial condition consisting of a row of 11 black cells.

The picture below shows the first few persistent structures found in the code 1329 cellular automaton from the bottom of page 282 . The smallest structures are stationary, but at initial condition 916 a structure is found that moves—all much the same as in the two other class 4 cellular automata that we have just discussed. … Persistent structures in the code 1329 cellular automaton shown on page 282 .

The Notion of Reversibility At any particular step in the evolution of a system like a cellular automaton the underlying rule for the system tells one how to proceed to the next step. … In the first cellular automaton shown below it is also straightforward to do this. … But the second cellular automaton works differently, and does not allow one to go backwards.

Yet looking at the cellular automaton on the previous page there are clearly at least some regularities in the pattern it produces—like the diagonal stripes on the left. … But the remarkable fact is that none of these methods seem to reveal any real regularities whatsoever in the rule 30 cellular automaton sequence. … Yet starting with a simple initial condition and then applying a simple cellular automaton rule constitutes a simple

But the whole point is that all any model is supposed to do—whether it is a cellular automaton, a differential equation, or anything else—is to provide an abstract representation of effects that are important in determining the behavior of a system. … Thus, for example, a cellular automaton can readily be set up to represent the effect of an inhibition on growth at points on the surface of a snowflake where new material has recently been added. But in the cellular automaton this effect is just implemented by some rule for certain configurations of cells—and there is no need for the rule to correspond in any way to the detailed dynamics of water molecules.

More Cellular Automata…Beyond randomness, the last example in the previous chapter was rule 110: a cellular automaton whose behavior becomes partitioned into a complex mixture of regular and irregular parts. … So what about more complicated cellular automaton rules? … Example of a totalistic cellular automaton with three possible colors for each cell.

Cellular Automata…So what about three-dimensional cellular automata? … But particularly on a printed page it is fairly difficult to display the evolution of a three-dimensional cellular automaton in a way that can readily be assimilated. … Patterns produced by evolution according to a simple two-dimensional cellular automaton rule starting from rows of black cells of various lengths.