Notes

Chapter 6: Starting from Randomness

Section 5: Randomness in Class 3 Systems


Rule 22 [with simple initial conditions]

Randomness is obtained with initial conditions consisting of two black squares 4m positions apart for any m >= 2. The base 2 digit sequences for 19, 25, 37, 39, 41, 45, 47, 51, 57, 61, ... also give initial conditions that yield randomness. Despite its overall randomness there are some regularities in the pattern shown at the bottom of the page. The overall density of black cells is not 1/2 but is instead approximately 0.35, just as for random initial conditions. And if one looks at the center cell in the pattern one finds that it is never black on two successive steps, and the probability for white to follow white is about twice the probability for black to follow white. 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.

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