All the persistent structures with repetition periods up to 15 steps in the code 20 cellular automaton. The structures shown were found by a systematic method similar to the one used to find all sequences that satisfy the constraints on page 268 .

remarkably poor: instead of steadily evolving to all black or all white, the system quickly gets stuck in a state that contains regions of different colors. … The only configurations that ultimately satisfy the constraints are all white and all black. … The third picture above is a representation of the kind of curve that arises in almost all discrete systems based on constraints.

Yet in almost all such cases the Principle of Computational Equivalence asserts that the systems are in fact universal. … And certainly there are all sorts of situations in which rules are constrained to have behavior that is too simple to support universality. … And what this means is that such rules will typically show the same features as rules chosen at random from all possibilities—with the result that presumably they do in the end exhibit universality in almost all cases where their overall behavior is not obviously simple.

But in all the other cases shown active cells proliferate forever. In case (d), almost all cells are active, and the system operates essentially like a cellular automaton.

So out of all the possible forms, which ones actually occur in real molluscs? The remarkable fact illustrated on the next page is that essentially all of them are found in some kind of mollusc or another. … But what we now see is that in fact all the different forms that are observed are in effect just consequences of the The effects of varying five simple features of the rule for the growth of a mollusc shell: (a) the overall factor by which the size increases in the course of each revolution; (b) the relative amount by which the opening is displaced downward at each revolution; (c) the size of the opening relative to the overall size of the shell; (d) the elongation of the opening; (e) the orientation of elongation in the opening.

In general, if one wants to find a piece of data that has a certain property—either exact or approximate—then one way to do this is just to scan all the pieces of data that one has stored, and test each of them in turn. But even if one does all sorts of parallel processing this approach presumably in the end becomes quite impractical. … And in fact, all that is really necessary is that the hashing procedure generate enough randomness that even though there may be regularities in the original data, the hash codes that are produced still end up being distributed roughly uniformly across all possibilities.

The last digit specifies what color the center cell should be if all its neighbors were white on the previous step, and it too was white. The second-to-last digit specifies what happens if all the neighbors are white, but the center cell itself is black.

So far as I can tell, all possible blocks eventually appear, potentially letting the pattern serve as a kind of directory of all possible computations.

And as a result, at least with respect to any of these methods all we can reasonably say is that the behavior we see seems for practical purposes random. Irreversible Data Compression All the methods of data compression that we discussed in the previous section are set up to be reversible, in the sense that from the encoded version of any piece of data it is always possible to recover every detail of the original. … But with images or sounds it is typically no longer so necessary: for in such cases all that in the end usually matters is that one be able to recover something that looks or sounds right.

And in the picture below, sequences (b), (c) and (d) are all nested. … If one looks at the set of all possible sequences, one can fairly easily calculate the distribution of frequencies for any particular block. … (Note that these substitution systems are the simplest ones that yield equal frequencies of all blocks up to lengths 1, 2 and 3 respectively.)