The course of evolution of any finite automaton can be traced through a diagram constructed for the purpose, or such an equivalent representation as the connectivity matrix of the diagram. Features of interest surely include:
The first item is the only one of importance for the operation of an automaton after an extremely long time, but it is still useful to understand the short term behaviour, particularly as it affects the choice among different long term alternatives. Furthermore, what is a long term for a ring of a hundred cells is still very short term for a ring of a thousand cells.
The technique of tracing out the full evolution of a ring of length N yields all the configurations in an infinite ring which repeat themselves after translation through a distance N, whatever their period of repetition in time; in other words all the patterns with a given spatial periodicity. In the process of observing the results one learns to recognize certain patterns and to be able to predict their periodicities. It would be nice to have a procedure which would yield all the configurations with a given period, irrespective of the length of the ring on which they might occur, beginning with a method for finding all the still lifes - patterns which never change with time.