Chapter 6: Starting from Randomness

Section 1: The Emergence of Order

Properties of [initially random cellular automaton] patterns

For a random initial condition, the average density of black cells is exactly 1/2. For rule 126, the density after many steps is still 1/2. For rule 22, it is approximately 0.35095. For rule 30 and rule 150 it is exactly 1/2, while for rule 182 it is 3/4. And insofar as rule 110 converges to a definite density, the density is 4/7. (See page 953 for a method of estimating these densities.)

Even after many steps, individual lines in the patterns produced by rules 30 and 150 remain in general completely random. But in rule 126, black cells always tend to appear in pairs, while in rule 182, every white cell tends to be surrounded by black ones. And in rule 22, there are more complicated conditions involving blocks of 4 cells.

The density of triangles of size n goes roughly like 2-n for rules 126, 30 (see also page 871), 150 and 182 and roughly like 1.3-n for rule 22.

In the algebraic representation discussed on page 869, rule 22 is Mod[p + q + r + p q r, 2], rule 126 is Mod[(p + q)(q + r) + (p + r), 2], rule 150 is Mod[p + q + r, 2] and rule 182 is Mod[p r (1 + q) + (p + q + r), 2].

Image Source Notebooks:

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