My recent publication at the Wolfram Demonstrations Project: Cobweb Diagrams of Elementary Cellular Automata

“Each possible state in the evolution of an elementary cellular automaton (ECA) can be uniquely labeled by an integer. For a finite-width ECA, this set is mapped to itself during the ECA evolution. Thus an ECA defines a recurrence relation on a finite set of integers. In fact, this can be applied to map the Cantor set to itself (see Related Links and Details sections).

Like more familiar iterated maps, such as the logistic map, cobweb diagrams show the dynamics of such a map. This Demonstration simultaneously plots cobwebs starting from all possible initial states of a finite ECA to show fast convergence to an invariant pattern. In the case of type 3 rules with high randomness, trajectories completely mix and almost cover the whole space. In other cases the patterns can be quite intricate, reflecting the subtle interplay between various attracting or repelling cycles of evolution.

The opacity control helps to find a suitable “density depth” that captures both frequently and seldomly visited orbits. The buttons “left” and “right” indicate the direction from which the binary ECA state is read. Please wait for an evaluation to complete before varying a control, as the computations take time to finish. Browse the snapshots and bookmarks for interesting patterns. The typical samples shown at the bottom-left under the controls are illustrative; the cobwebs were not built from them.”