The cybernetics movement highlighted the question of what it takes for self-reproduction to occur autonomously, and in 1952 John von Neumann designed an elaborate 2D cellular automaton that would automatically make a copy of its initial configuration of cells (see page 876 ).

And in a sense presenting a particular object found by experiment (such as a cellular automaton whose evolution shows some particular property) can be viewed as a constructive existence proof for such an object.

In the pictures below the liquid is divided into cells, with each cell having a temperature from 0 to 1, corresponding exactly to a continuous cellular automaton of the kind discussed on page 155 .

If instead of an ordinary cellular automaton with a limited number of possible colors one considers a system in which every cell can have any integer value then additivity with respect to ordinary addition becomes just traditional linearity. … And indeed a cellular automaton whose rule is based on Mod[x+y, π ] will show additivity with respect to this operation (see page 922 ). … This assumes, however, that the underlying cellular automaton has discrete cells.

Note (g) for More Cellular Automata…[Cellular automaton] rules based on algebraic systems If the values of cells are taken to be elements of some finite algebraic system, then one can set up a cellular automaton with rule a[t_, i_] := f[a[t - 1, i - 1], a[t - 1, i]] where f is the analog of multiplication for the system (see also page 1094 ).

Binary decision diagrams One can specify a Boolean function of n variables by giving a finite automaton (and thus a network) in which paths exist only for those lists of values for which the function yields True . … For cellular automata with simple behavior, the minimal BDD typically grows linearly on successive steps. … For cellular automata with more complex behavior, it typically grows roughly exponentially.

For 3-input functions, corresponding to elementary cellular automaton rules, 56 of the 256 possibilities turn out to be universal.

Note (e) for Cellular Automata…Numbers of possible [2D cellular automaton] rules The table below gives the total number of 2D rules of various types with two possible colors for each cell.

The 2D cellular automaton used as a model of an idealized gas on page 446 provides an example of a system that can be viewed as conserving a vector quantity.

In a system like a cellular automaton, the same underlying rule is in a sense always applied in exact synchrony to every cell at every step.