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Conceptually superluminal configurations may be somewhat easier to describe than those of lower velocity. For convenience, consider a configuration, which is supposed to have a single phase which shifts by a distance of 2 in each generation. Its evolution would have the symbolic form
From this it is evident that a, b, and c are arbitrary, but that necessarily
The point is that any seed sequence a, b, c determines the entire sequence of cells from some point on, but only eight different binary sequences of three cells are possible. The original seed need not repeat itself, but some seed must repeat within at least eight cells. Thus only eight strings need to be examined to ascertain all the possible configurations; for Rule 22 we would find:
These configurations are not very interesting, because they all have simpler descriptions; but they all fulfill the technical description, and are moreover the only configurations which do so.
If one wanted to find configurations, the seed would have length 4, and in general a configuration would have a seed of length p+d and the superluminal configuration would have length less than and have to be manifest within that many cycles of iteration of the above procedure.
The following superluminal configurations (single phase, shift no longer than 12) exist for Rule 22:
Subluminal configurations can be found by the same procedure, but since the displaced seed overlaps its own neighborhood, consistency requirements must be met. Thus superluminal configurations always exist even if they are degenerate or trivial, but subluminal configurations may fail the additional requirement. In any event, the de Bruijn diagram yields a more systematic procedure for all configurations, and the foregoing discussion is mainly useful an an existence theorem.