The example at the top is Pascal's triangle, formed by making each number be the sum of the numbers immediately to its left and right on the row above. … For the top picture the pattern is what would be generated by an additive cellular automaton following rule 90; for the bottom picture it is what would be generated by one following rule 150. The numbers in the top picture are binomial coefficients; those in the bottom picture are particular trinomial coefficients.

The objects in the top two pictures correspond to the thick vertical black and gray lines in picture (d) on page 679 . … (Note that all the structures are left-right reversed in rule 110.) … All the localized structures involved in the pictures above were shown individually on page 292 .

In all cases no Turing machines with the same number of states compute the same functions in fewer steps. … The top row of pictures are all scaled to be exactly the same height, even though the initial conditions cannot be chosen to make the number of steps in each case anything more than roughly the same.

Examples of cellular automata which continue to allow all possible sequences of black and white cells at any step in their evolution. … Networks representing possible sequences that can occur in the evolution of the cellular automata at the top of the page , starting from initial conditions in which black cells are only allowed to appear in pairs.

And indeed only some causal networks even yield a reasonable notion of space at all. … But in fact, without updating events, no causal network at all gets built up. And so a system like the one at the top of the next page is about the simplest that can yield something even vaguely reminiscent of ordinary space.

rapidly becomes astronomical, and to test all of them becomes completely infeasible. … One picks some sequence of cells for the right half of the top row, then evolves down the page. … So by sampling a limited number of sequences on the top row, one can often find a second column that then allows columns to the left to be determined, and thus for a candidate key to be found.

In cases (a) and (c) the networks obtained in this way have the property that all connections between nodes go either across or down the page. … And for example the pictures at the top of the facing page show the causal networks for rules (e) and (f) from the previous page —but now with each node numbered to specify the step of mobile automaton evolution from which it was derived. And what we see is that even nodes that are close to the top of the causal network can correspond to events which occur after a large number of steps of mobile automaton evolution.

In a reversible rule, such patterns can grow and shrink, but can never die out completely.

In the top set of pictures, the rule specifies that a cell should become black whenever any of the six neighbors with which it shares a face were black on the step before. … In the top pictures, the limiting shape obtained is a regular octahedron.

For the particular cellular automaton shown here the rule specifies—as in the picture below—that a cell should be black in all cases where it or either of its neighbors were black on the step before. … At the first step the cell in the center is black and all other cells are white. … The top row in each box gives one of the possible combinations of colors for a cell and its immediate neighbors.