[No text on this page] Another example of a cellular automaton that produces a nested pattern even from random initial conditions.

[No text on this page] A cellular automaton whose behavior seems neither highly regular nor completely random.

And indeed, what is perhaps most bizarre about the pictures is just how little trace they ultimately show of the simplicity of the underlying cellular automaton rule that was used to produce them. One might think, for example, that the fact that all the cells in a cellular automaton follow exactly the same rule would mean that in pictures like the last few pages [ 32 , 33 , 34 , 35 , 36 , 37 , 38 ] all cells would somehow obviously be doing the same thing. … Looking just at the original cellular automaton rule one would have no realistic way to foresee all of this.

But in fact, in the fifteen years or so since I first emphasized the importance of cellular automata all sorts of traditional mathematical work has actually been done on them. … So for example when presented with the 256 elementary cellular automaton patterns shown on page 55 mathematicians in my experience have two common responses: either to single out specific patterns that have a simple repetitive or perhaps nested form, or to generalize and look not at individual patterns, but rather at aggregate properties obtained say by evolving from all possible initial conditions. … And my results An axiom system for statements about the rule 110 cellular automaton.

States of matter [and cellular automaton classes] As suggested by pages 944 and 1193 , working out whether a particular substance at a particular temperature will be a solid, liquid or gas may in fact be computationally comparable in difficulty to working out what class of behavior a particular cellular automaton will exhibit.

Continuous Cellular Automata…And what we see is that despite the presence of continuous gray levels, the behavior that is produced exhibits the same kind of complexity that we have seen in many ordinary cellular automata and other systems with discrete underlying elements. A continuous cellular automaton with a slightly more complicated rule.

Implementing cellular automata It is convenient to represent the state of a cellular automaton at each step by a list such as {0, 0, 1, 0, 0} , where 0 corresponds to a white cell and 1 to a black cell. … First, cellular automaton rules are always defined to use the old values of neighbors in determining the new value of any particular cell. … Another tricky point in cellular automaton programs concerns boundary conditions.

Deducing cellular automaton rules Given a complete cellular automaton pattern it is easy to deduce the rule which produced it just by identifying examples of places where each element in the rule was used, as in the picture below.

The first example corresponds to elementary cellular automaton rule 60. Note that any cellular automaton rule can be reproduced by some appropriate combination of bitwise and arithmetic operations.

Block cellular automata With a rule of the form {{1, 1}  {1, 1}, {1, 0}  {1, 0}, {0, 1}  {0, 0}, {0, 0}  {0, 1}} the evolution of a block cellular automaton with blocks of size n can be implemented using BCAEvolveList[{n_Integer, rule_}, init_, t_] := FoldList[BCAStep[{n, rule}, #1, #2]&, init, Range[t]] /; Mod[Length[init], n]  0 BCAStep[{n_, rule_}, a_, d_] := RotateRight[ Flatten[Partition[RotateLeft[a, d], n]/.rule], d] Starting with a single black cell, none of the k = 2 , n = 2 block cellular automata generate anything beyond simple nested patterns. … In general, a block cellular automaton is reversible only if its rule simply permutes the k n possible blocks. Compressing each block into a single cell, and n steps into one, any block cellular automaton with k colors and block size n can be translated directly into an ordinary cellular automaton with k n colors and range r = n/2 .