For even though this pattern is produced by a simple one-dimensional cellular automaton rule, and even though one can see by eye that it contains at least some small-scale regularities, none of the schemes we have discussed up till now have succeeded in compressing it at all.

The only pattern that is known to be obtainable by evolving down the page according to a simple local rule is (j), which corresponds to the rule 60 elementary cellular automaton.

Note (b) for More Cellular Automata…Rule equivalences [for cellular automata] The table below gives basic equivalences between elementary cellular automaton rules.

Numbers of reversible [cellular automaton] rules For k = 2 , r = 1 , there are 6 reversible rules, as shown on page 436 . … For k = 4 , r = 1 , some of the reversible rules can be constructed from the second-order cellular automata below.

Related [texture perception] models Rather than requiring particular templates to be matched, one can consider applying arbitrary cellular automaton rules.

Formulas [and computational irreducibility] It is always in principle possible to build up some kind of formula for the outcome of any process of evolution, say of a cellular automaton (see page 618 ).

Special Initial Conditions We have seen that cellular automata such as rule 30 generate seemingly random behavior when they are started both from random initial conditions and from simple ones. … Examples of special initial conditions that make the rule 30 cellular automaton yield simple repetitive behavior. … Finding initial conditions that make cellular automata yield behavior with certain repetition periods is closely related to the problem of satisfying constraints discussed on page 210 .

Diffusion equation In an appropriate limit the density distribution for cellular automaton (d) appears to satisfy the usual diffusion equation ∂ t f[x, t]  c ∂ xx f[x, t] discussed on page 163 .

Cellular automata in which domains of repetitive behavior form, but in which walls typically remain forever between these domains. A cellular automaton (rule 184) in which domains quickly combine to make the whole system repetitive in space.

An extended version of the picture on the facing page , in which the reversibility of the underlying cellular automaton is more clearly manifest.