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.

Self-avoiding [random] walks Any walk where the probabilities for a given step depend only on a fixed number of preceding steps gives the same kind of limiting Gaussian distribution. … If one adds individual steps at random then in 2D one typically gets stuck after perhaps a few tens of steps. … (In d ≤ 4 dimensions the exponent is close to the Flory mean field theory value 3/(2 + d) ; for d > 4 the results are the same as without self-avoidance.)

Features of the [patterning] model The model is a totalistic 2D cellular automaton, as discussed on page 927 . It shows class 2 behavior in which information propagates only over limited distances, so that except when the total size of the system is comparable to the range of the rule, boundary conditions are not crucial.

The examples in this chapter suggest that if the rules for a particular system are sufficiently simple, then the system will only ever exhibit purely repetitive behavior.

And indeed in the course of this chapter we have seen that in every single one of the general kinds of systems that we have discussed, it ultimately takes only very simple rules to produce behavior of great complexity.

Indeed, even though the only operation used was just simple multiplication, the final pattern obtained in this case is highly complex. … Starting with 1, the successive numbers that one obtains in this way are 1, 3/2 = 1.5 , 9/4 = 2.25 , 27/8 = 3.375 , 81/16 = 5.0625 , 243/32 = 7.59375 , 729/64 = 11.390625 , ... … Multiplication by 2 turns out to correspond just to shifting all digits in base 2 one position to the left, so that the overall pattern produced in this case is very simple.

But since essentially the only methods of analysis available were ones from traditional mathematics, very little of the complex behavior of typical biological systems was successfully captured.

But for the most part it has used traditional mathematical methods, and as a result has only been able to investigate the formation of fairly simple structures.

And as in an electric circuit, the properties of the system depend only on the way in which the nodes are connected together, and not on any specific layout for the nodes that may happen to be used.

In the vast majority of cases, simple repetitive patterns, or mixtures of such patterns, are the only ones that are needed.