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Cycles for a sixteen-member ring

The data eventually becomes so voluminous that it seems only to be worthwhile tabulating the cycles into which the rings of length N evolve, ignoring all the transients. Below, we show one particular case, N = 16, in detail. The ring is long enough to show a variety of interesting behaviour, but yet not so long that it cannot still be readily displayed.

The notation describing each cycle means that the period of the cycle is x, but that there are only y distinct phases within the cycle. This could either mean that the original pattern has reflected after y generations and will be completed by running through the mirror images, or that the pattern has been translated by a certain amount. Thus x must be a multiple of y, cycles of the form almost always (but not exclusively) resulting from reflection. Sometimes an even more explicit notation, such as 16:14.7(ref) or is required, in which the length of the ring is shown along with an indication of the direction and distance of displacement. Since reflectivity could be inferred if a displacement were not shown, it is not always indicated.

First, we show the smallest numerical representatives for each of the eight distinct symmetry classes of cycles for a ring of length N=16:

Figure: Rule 22 rings of length 16 have eight symmetry classes of cycles.

Next, the eight figures which follow show a full cycle of evolution for every one of these patterns. In preparing each illustration, every line is copied twice to improve perception of the region which would otherwise be broken up by the boundary.

Harold V. McIntosh