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consists of all configurations in which black cells occur only when they are surrounded on each side by at least one white cell.

The picture below shows that for any particular configuration of this kind, there are in general many different initial conditions that can lead to it. In a mechanical analogy each possible final configuration is like the lowest point in a basin—and a ball started anywhere in the basin will then always roll to that lowest point.

For one-dimensional cellular automata, it turns out that there is a rather compact way to summarize all the possible sequences of black and white cells that can occur at any given step in their evolution.

The basic idea is to construct a network in which each such sequence of black and white cells corresponds to a possible path.

In the pictures at the top of the facing page, the first network in each case represents random initial conditions in which any possible sequence of black and white cells can occur. Starting from the node in the middle, one can go around either the left or the right loop in the network any number of times in any order—representing the fact that black and white cells can appear any number of times in any order.

At step 2 in the rule 255 example on the facing page, however, the network has only one loop—representing the fact that at this step the only sequences which can occur with this rule are ones that consist purely of black cells, just as we saw on the previous page.

The case of rule 4 is slightly more complicated: at step 2, the possible sequences that can occur are now represented by a network with two nodes. Starting at the right-hand node one can go around the loop to the right any number of times, corresponding to sequences of

Captions on this page:

Four different initial conditions that all lead to the same final state in the rule 4 cellular automaton shown on the previous page. The final state can be thought of as one of the possible attractors for the evolution of the cellular automaton; the initial conditions shown then represent different elements in the basin of attraction for this attractor.

From Stephen Wolfram: A New Kind of Science [citation]