Chapter 6: Starting from Randomness

Section 3: Sensitivity to Initial Conditions

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 motion of the right-hand edge in rule 30 can be understood by noting that with this rule the color of a particular cell will always change if the color of the cell to its left is changed on the previous step (see page 601). Nothing as simple is true for the left-hand edge, and indeed this seems to execute an essentially random walk—with an average motion of about 0.2428 cells per step. Note that in the approximation that the colors of all cells in the pattern are assumed completely independent and random there should be motion by 0.25 cells per step. Curiously, as discussed on page 871, the region of non-repetitive behavior in evolution from a single black cell according to rule 30 seems to grow at a similar but not identical rate of about 0.252 cells per step. (For rule 45, the left-hand edge of the difference pattern moves about 0.1724 cells per step; for rule 54 both edges move about 0.553 cells per step.)

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