But it turns out that by using a slightly different setup one can construct systems whose operation is in some ways even simpler. In an ordinary tag system, one does not know in advance which of several possible blocks will be added at each step. … But if one looks at the fluctuations in this growth, as in the plots on the next page , then one finds that these fluctuations are in many respects random.

If one does this, then the typical experience is that in any particular representation, some class of numbers will have simple forms. … And from this it seems appropriate to conclude that numbers generated by simple mathematical operations are often in some intrinsic sense complex, independent of the particular representation that one uses to look at them.

In the second set of pictures, the rule specifies that a cell should become black only when exactly one of its six neighbors was black on the step before. … In the bottom pictures, it is a nested pattern analogous to the two-dimensional one on page 171 .

one step in the evolution of every single one of the 256 possible elementary cellular automata.

And so far as one can tell, almost all these kinds of numbers also have apparently random digit sequences. … To find Sqrt[n] one starts by setting r=n and s=0 . Then at each step one applies the rule {r, s} -> If[r >= s+1, {4(r–s–1), 2(s+2)}, {4r, 2s}] .

Substitution Systems and Fractals One-dimensional substitution systems of the kind we discussed on page 82 can be thought of as working by progressively subdividing each element they contain into several smaller elements. One can construct two-dimensional substitution systems that work in essentially the same way, as shown in the pictures below. … Indeed, just like patterns produced by one-dimensional substitution systems on page 83 , all the patterns shown here ultimately have a simple nested structure.

The picture below shows the structures one finds by explicitly testing the first two billion possible initial conditions for the code 357 cellular automaton from page 282 . … But with all the first million initial conditions, only one other structure is produced, and this structure is again one that does not move.

So out of all the possible forms, which ones actually occur in real molluscs? … If one just saw a single mollusc shell, one might well think that its elaborate form must have been carefully crafted by some long process of natural selection.

Indeed, if one looks carefully, one can see that every pattern just consists of a collection of identical nested pieces. … The nested structure becomes even clearer if one represents elements not as boxes, but instead as branches on a tree.

So how can one find out what these structures are for a particular cellular automaton? One approach is just to try each possible initial condition in turn, looking to see whether it leads to a new persistent structure. … But going on to initial condition 195, we again find a more complicated structure—this time one that repeats only every 22 steps.