When Martin Gardner first announced Conway's game of * Life*
, there was
much experimentation with simple designs, some of the simplest
consisting of just a single row, column, or diagonal of live cells.
Columns tended to grow shorter and fatter until they formed a
diamondlike configuration which in one remarkable case---a column of
fifteen cells---retreated back into a column and became periodic. It
was one of the first complex oscillators to be discovered.

However, instead of forming diamonds, very long columns exhibited a rather curious behaviour not unlike a binary counter. That is, except for the ends, a single column survived while two new live columns flanking it were born. The next generation saw the birth of still another pair of flanking columns, but the central three could no longer survive, leaving the new pair of columns isolated by a gap of three cells.

Then in the fourth generation two triple columns separated by a vacant column produced a single pair of columns separated by a distance of seven cells. In subsequent generations these columns repeated the behaviour already established until a collision occurred between the flanks expanding into the central region, This time a single pair of columns separated by a gap of fifteen cells was left in the ninth generation. Continuing this sequence on through further generations inevitably suggests a binary interpretation of the evolution.

Infinitely long constant columns reduce a two dimensional automaton to one
dimension; all the cells in any given column behave alike, leaving all the
significant information to be gleaned from a row acting as a cross section.
Such configurations were called ``ripples'' in Wainwright's newsletter.
Adaptation of Conway's rule shows that Wolfram's Rule 22 for a
automaton gives the appropriate description of the result, which might
reasonably be referred to as * Life in one dimension.*

A great advantage of working with one dimensional cellular automata is that their time evolution can be shown on a two dimensional chart, whereas the evolution of a two dimensional structure would require a third dimension. While not impossible to show, there is too much information involved to keep the presentation from becoming extremely cluttered. Even so, an infinite, or even a very long, line is hard to manage. A systematic study could easily start with short lines, folded around to form a ring, avoiding end cells whose rules of evolution would differ from the interior cells. For short enough rings, the complete evolution of all possible configurations can be calculated.

E-mail:mcintosh@servidor.unam.mx