Region (c) shows what happens when the information corresponding to one element in a block passes through the kind of object produced in region (a). … Region (d) then shows how the object in region (c) comes to an end when the beginning of the block separator from the next step arrives. … Note that even though they begin very differently, regions (d) and (i) end in the same way, reflecting the fact that in both cases the system is ready to handle a new block, whatever that block may be.

The initial condition consists of a block of length 41 inserted between blocks of the background.

The specific rules are based on 2×2 blocks—a two-dimensional generalization of the block cellular automata to be discussed in the next section . For each 2×2 block the configuration of particles is taken to remain the same at a particular step unless there are exactly two particles arranged diagonally within the block, in which case the particles move to the opposite diagonal.

[No text on this page] Responses to a smaller version of the image from page 578 by cells sensitive to all 16 possible 2×2 blocks, as well as their repetitive 3×3 extensions. Patches which appear to have different textures in the original image are seen to contain characteristically different densities of these various blocks.

The patterns can also be viewed as outputs from a single step in the evolution of two-dimensional block cellular automata in which the rules specify that a block becomes dark if it has the arrangement of cells shown, and becomes light otherwise. The comparative sparsity of dark blocks is a consequence of the fact that at any given position a dark block can occur in only one of the 16 cases shown. The absence of any dark blocks in many of the cases shown can be viewed as a reflection of constraints introduced by the construction of the images from one-dimensional cellular automaton rules.

So given a particular elementary cellular automaton one can then ask what other elementary cellular automata it can emulate using blocks up to a certain length. … But at least with blocks up to length 25, rule 30 for example is not able to emulate any non-trivial rules at all. … And in the particular case shown, this is achieved by having blocks 3 cells wide between each input position.

produced must have exactly the same structure whether it is looked at in terms of individual cells or in terms of blocks of cells. … Another example of a rule in which blocks of cells can behave just like individual cells. … A rule that is not additive, but in which blocks of cells can again behave just like individual cells.

So even though individual block frequencies seem to suggest that sequences (d) and (e) are random, the lack of any spread in these frequencies provides evidence that in fact they are not. … But it turns out that many of the obvious quantities one might consider computing are in the end equivalent to various combinations of block frequencies. And perhaps as a result of this, it has sometimes been thought that if one could just compute frequencies of blocks of all lengths one would have a kind of universal test for randomness.

Occurrences of progressively longer blocks in the pattern generated by rule 30 starting from a single black cell. 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 to see the effect of the grid, I show what happens when each of these cellular automata is started from blocks of black cells arranged at three different angles. In all cases the patterns produced follow at least to some extent the orientation of the initial block. … Three different initial conditions, consisting of blocks at three different angles, are shown.