Chapter 7: Mechanisms in Programs and Nature

Section 6: The Phenomenon of Continuity

Domains [in code 976]

Some of the effective rules for interfaces between black and white domains are easy to state. Given a flat interface, the layer of cells immediately on either side of this interface behaves like the rule 150 1D cellular automaton. On an infinitely long interface, protrusions of cells with one color into a domain of the opposite color get progressively smaller, eventually leaving only a certain pattern of cells in the layer immediately on one side of the interface. 90° corners in an otherwise flat interface effectively act like reflective boundary conditions for the layer of cells on top of the interface.

The phenomenon of domains illustrated here is also found in various 2D cellular automata with 4-neighbor rather than 8-neighbor rules. One example is totalistic code 52, which is a direct analog in the 4-neighbor case of the rule illustrated here. Other examples are outer totalistic codes 111, 293, 295 and 920. The domain boundaries in these cases, however, are not as clear as for the 8-neighbor totalistic rule with code 976 that is shown here.

Image Source Notebooks:

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