For small systems, the state-transition graphs of cellular automata can be drawn as a planar diagram (figure 2). To produce these diagrams, a random initial condition is chosen and iterated until it entered a temporal cycle. Each configuration is represented as a point, and the temporal cycle is shown as a ring of points in the center of the diagram. In the next largest concentric ring are shown the preimages of the configurations on the cycle, connected to their image by a line segment. The preimages of these are shown in the next concentric circle, and so on.
It is difficult to draw conclusions concerning the ultimate shape of state- transition graphs from diagrams such as shown in figure 2. The structure of small systems tends to be dominated by features which result from the interaction of number-theoretic properties of the system size with properties of the rule table. Large systems, on the other hand, contain too many states to be conveniently displayed in a single image. Nonetheless, we can see in figure 2 that the structures generated by rule 30 are composed of long, bare trees, and the structures generated by rule 4 are composed almost exclusively of leaves. The other rules show an intermediate degree of branching, with a trend toward greater branching as the sequence passes from rule 30 to rule 4.
Figure: State-Transition Diagrams