Ripples

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Ripples refers to a class of agars which are truly one dimensional, in that all the cells along parallel lines are in the same state. In practice the only alternatives are orthogonal lines evolving according to Wolfram's (2,1) Rule 22 and diagonal lines following his (2,2) hexadecimal Rule 01140238, those being the corresponding one-dimensional cross sections of the Life B3/S23 rule.

Periodic orthogonal ripples

Two familiar orthogonal ripples are

  • Venetian blinds, a period 2 agar alternating two cell thick strips of live and dead cells, and
  • Zebra stripes, a still life agar whose stripes are only one cell thick.
orthogonal still life ripples
orthogonal vanishing ripples

Other, more complicated, ripples can be discovered by constructing de Bruijn diagrams for Rule 22. Although their complexity grows rapidly by shift and by period, the smallest diagrams give an idea of what to expect.

These first two images easily show the full diagram since there are only four semi neighborhoods, all of length two. The self link to 00 frequently occurs, since the vacuum meets almost all specifications. Linked to other nodes, the patterns generated by the diagram are freestanding on one side, the other, or both according to the direction of the links. Furthermore, selective omission of poorly connected links or nodes will abbreviate the diagram, enhancing its legibility.

orthogonal period 2 ripples

For still lifes, only those links are retained in the diagram for which the full neighborhood evolves to its central cell. As shown at the right, above, the node 3 is isolated and cannot participate in a static figure; recall that two parallel live bars in Life kill each other. The other loop generates the Zebra stripe agar (or ripple).

To find ripples of higher periodicity, it is only necessary to form the diagram corresponding to the larger neighbourhoods required; for period two they would be five cells long, giving sixteen four cell partial neighborhoods and an evolution appropriate to a (2,2) rule.

The figure at the right shows the three mutually connected components of the period 2 diagram: the zero component, the holdover component from period 1, and the only new result, which generates the Venetian blinds agar.

It is interesting, that aside from zero, there is no orthogonal ripple of period three. Historically this caused puzzlement, but it follows immediately from examining the diagram.

Vanishing orthogonal ripples

Instant vanishing is another interesting property of Life patterns. Accordingly, the diagram at the left holds just those links whose neighborhoods evolve to the constant, 0. This time, node 3 is a participant which can repeat more than once. Such intervals can only be separared by single zeroes; otherwise the live bars would leave fringes behind as they disappear.


fat component of orthogonal ripples vanishing in two steps
thin component of orthogonal ripples vanishing in two steps

Just as still lifes carry over patterns whose periods have divisors, so do the evolutions into constants. Thus the three components of vanishment carry over from period 1, although they are neither enlarged nor do additional components appear. The zero component remains, there is a fat component, in which thick bars collapse. and a thin component in which very slim bars fill gaps, followeb by an overall collapse.

Periodic diagonal ripples and vanishers

In contrast to the multitude of orthogonal ripples, there are relatively few diagonal ripples.

DIADB.GIF

Besides the vacuum, which is technically a ripple, inspection of the de Bruijn diagram for still lifes reveals just two basic ripples. One consists of a single file; if broken it serves as a fuse, the progenitor of a large variety of snakes and fuses (e.g. : ship maker.) The other, more stable, consists of a double file, the progenitor of tubs and barges when broken.

All nontrivial cycles must contain nodes 1 and 8, guaranteeing a separation of at least three cells between any two stripes, although they may otherwise be arbitrarily mixed.

de Bruijn diagram for instant vanishing
four instantly vanishing samples

The de Bruijn diagram for instant vanishing shows the consequences of placing nonempty ripples too close together. A checkerboard pattern, as seen in the loop (5,10), cannot survive; nor a can a wide variety of others - for instance, long intervals of live cells.

Shifting ripples

Beyond still lifes and instant vanishers, there is a wide variety of evolution possible for ripples; shifting and periodicity for example. This can all be systematically investigated through programs dedicated to one-dimensional cellular automata, mainly because the relevant de Bruijn diagrams are much smaller. The diagram below illustrates the point for diagonal ripples which shift to the left after two generations. As usual, the vacuum occupies its own component; shifting patterns cannot be finite. Rather only possessing a few cyclic connected components, at least five interlinked cycles can be seen:

  • A: separated strands which are far enough apart to qualify as still lifes, yet close enough to meet the shifting condition.
  • B: the even phase of strands close enough to oscillate with period 2.
  • C: the matching odd phase, whose strands are alternately separated by two and four cells.
  • D: the two designs can be mixed provided they are bridged by adequate connecting links.
  • E paired with D; provision must be made for the two phases of oscillating ripples.
de Bruijn diagram for diagonal ripples shifting left in 4 generations
some small non-reentrant cycles taken from the diagram

The examples depend on consolidated nodes, arising from chains where there is, except for the ends, exactly one incoming link and one outgoing link per node. Here there are just thirteen of these nodes, which sets an upper bound on the minimum length of the spatial period of the agar. That would be a Hamiltonian path; usually the cycles are much shorter. For characterizing the ripple, listing the minimal cycles usually suffices.