Chapter 6: Starting from Randomness

Section 5: Randomness in Class 3 Systems

Density oscillations in rule 73

Although there are always some fluctuations, most rules yield densities that converge more or less uniformly to their final values. One exception is rule 73, which yields densities that continue to oscillate with a period of 3 steps forever. The origin of this phenomenon is that with completely random initial conditions rule 73 evolves to a collection of independent regions, as in the picture below, and many of these regions contain patterns that repeat with period 3. The boundaries between regions come from blocks of even numbers of black cells in the initial conditions, and if one does not allow any such blocks, the density oscillations no longer occur. (See also page 699.)

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