Properties [of difference patterns] In rule 126, the outer edges of the region of change always expand by exactly one cell per step. The same is true of the right-hand edge in rule 30—though the left-hand edge in this case expands only about 0.2428 cells on average per step. In rule 22, both edges expand about 0.7660 cells on average per step.

The basic idea is to have the cellular automaton produce a pattern that expands and contracts on each side in a way that corresponds to the incrementing and decrementing of the sizes of numbers in the first and second registers of A cellular automaton set up to emulate a sequential substitution system.

And as the table below illustrates, the entries in Pascal's triangle are simply the binomial coefficients that appear when one expands out the powers of 1 + x . … The succession of polynomials above can be obtained by expanding the generating functions 1/(1 - (1 + x) y) and 1/(1 - (1 + x + x 2 ) y) .

And so, for example, in looking at the pictures below it would normally seem much more plausible that rule 254 might have been set up for the purpose of generating a uniformly expanding pattern than that rule 30 might have been. … But while this cellular automaton seems to have little extraneous going on, it operates in a slow and sequential way, and its underlying If the purpose is to generate a uniformly expanding pattern it seems more plausible that the top cellular automaton should have been the one created for this purpose.

The speed at which the region of differences expands in the cellular automaton can thus be thought of as giving a Lyapunov exponent (see page 921 ) that characterizes instability in the system.

With a background consisting of repetitions of the block , insertion of a single initial white cell yields a largely random pattern that expands by one cell per step.

For rather than starting with very specific definitions and then expanding from these, I start from general intuition and then use this to come up with more specific results.

Starting from a single black cell, this rule just yields a uniformly expanding diamond-shaped region of black cells.

The reason for this is presumably that all parts of the universe are expanding—with the local consequence that radiation is more often emitted than absorbed, as evidenced by the fact that the night sky is dark.

There is also a region of repetitive behavior on each side of the pattern; the random part in the middle expands at about 0.766 cells per step—the same speed that we found on page 949 that changes spread in this rule.