And in fact, in general the simple cellular automaton shown below seems remarkably successful at reproducing all sorts of obvious features of snowflake growth. … The evolution of a cellular automaton in which each cell on a hexagonal grid becomes black whenever exactly one of its neighbors was black on the step before.

So what about other class 4 cellular automata—like the ones I showed at the beginning of this section ? … All the persistent structures with repetition periods up to 15 steps in the code 20 cellular automaton.

So what about other class 3 cellular automata? … Nevertheless, the pictures on the facing page demonstrate that if one uses initial conditions that are slightly different—though still simple—then one can still see randomness in the behavior of this particular cellular automaton. Patterns produced by the rule 22 cellular automaton starting from random initial conditions and from an initial condition containing a single black cell.

[No text on this page] An example of a reversible cellular automaton whose evolution supports localized structures.

[No text on this page] Emulating the rule 110 cellular automaton using combinators.

Relation to 1D cellular automata A picture that shows the evolution of a 1D cellular automaton can be thought of as a 2D array of cells in which the color of each cell satisfies a constraint that relates it to the cells above according to the cellular automaton rule. … Below this line there would then be a unique pattern corresponding to the application of the cellular automaton rule. … And now it is always possible to construct a repetitive pattern which satisfies the constraints simply by finding repetitive behavior in the evolution of the cellular automaton from a spatially repetitive initial condition.

[No text on this page] Complex behavior in the rule 110 cellular automaton starting from a random initial condition.

It turns out that such discrete transitions are fairly rare among one-dimensional cellular automata, but in two and more dimensions A one-dimensional cellular automaton that shows a discrete change in behavior when the properties of its initial conditions are continuously changed. … The underlying rule for the cellular automaton shown takes the new color of a cell to be the color of its right neighbor if the cell is black and its left neighbor if the cell is white.

The behavior of a simple two-dimensional cellular automaton that emulates an ideal gas of particles. … The underlying rules for the cellular automaton used here are reversible, and conserve the total number of particles. The specific rules are based on 2×2 blocks—a two-dimensional generalization of the block cellular automata to be discussed in the next section .

Nearby cellular automaton rules In a range r cellular automaton the new color of a particular cell depends only on cells at most a distance r away. One can make an equivalent cellular automaton of larger range by having a rule in which cells at distance more than r have no effect. … With larger and larger ranges one can then construct closer approximations to continuous sequences of cellular automata.