The pictures on the next page show typical examples based on cellular automata that exhibit repetitive and nested behavior. In the patterns on the left the color of each cell at any given step is in effect found by tracing the explicit evolution of the cellular automaton up to that step. … These procedures are again based on cellular automata.

But in a system like a cellular automaton the typical reason for it is just that in the end effects never spread beyond a limited region, as in the examples shown in the first set of pictures below. … Examples of behavior in mobile automata and cellular automata that remains localized to a limited region and thus always eventually repeats. … Cellular automata in which a repetitive pattern in both space and time is generated by evolution from a simple seed.

Thus, for example, cellular automata probably already have too rigid a built-in notion of space. For a defining feature of cellular automata is that their cells are always arranged in a rigid array in space. … The particular rule shown here is the elementary cellular automaton with rule number 94, and with initial condition .

[No text on this page] A larger example of the cellular automaton system shown on the previous page .

History [of second-order cellular automata] The concept of getting reversibility in a cellular automaton by having a second-order rule was apparently first suggested by Edward Fredkin around 1970 in the context of 2D systems—on the basis of an analogy with second-order differential equations in physics.

So what about systems like cellular automata that have definite rules for evolution? … Such a system has many features immediately reminiscent of a cellular automaton.

Note (c) for More Cellular Automata…Special [cellular automaton] rules Rule 51: complement; rule 170: left shift; rule 204: identity; rule 240: right shift.

Models with discrete elements were already considered in the 1960s, and in 1977 James Greenberg and Stuart Hastings introduced a simple 2D cellular automaton with three colors. The pictures below show what is probably the most complex feature of this cellular automaton and related systems: the formation of spiral waves.

And as an example of a simple approach to modelling this, one can consider having a collection of discrete eddies that occur at discrete positions in the fluid, and interact through simple cellular automaton rules. … A cellular automaton (rule 225) whose behavior is reminiscent of turbulent fluid flow.

Indeed, as a first approximation one can imagine that much as in a cellular automaton entities in a market could follow simple rules based on the behavior of other entities. … And so as a minimal idealization one can for example try viewing a market as being like a simple one-dimensional cellular automaton.