samples just the first cell in every 14×7 block of cells, making each domain of repetitive behavior stand out as having a uniform color. … In the second picture, each element shown represents a 2×2 block of original cells.

[No text on this page] Examples of tag systems in which at each step two elements are removed from the beginning of the sequence and then, based on what these elements are, a specified block of new elements is added to the end of the sequence.

In the lattice version in physics one typically considers what happens to averages over all possible configurations of a system if one does a so-called blocking transformation that replaces blocks of elements by individual elements. And what one finds is that in certain cases—notably in connection with nesting at critical points associated with phase transitions (see page 981 )—certain averages turn out to be the same as one would get if one did no blocking but just changed parameters ("coupling constants") in the underlying rules that specify the weighting of different configurations. … What I do in the main text can be thought of as carrying out blocking transformations on cellular automata.

But what about rules that have replacements involving blocks of more than one element? … And one way to guarantee this is if the blocks involved in replacements can never overlap.

For it means that there will be certain blocks—say corresponding to words like "the" in English—that occur much more often than others in the original message. And since such blocks must be encrypted in the same way whenever they occur at the same point in the repetition period of the encrypting sequence they will lead to occasional repeats in the encrypted message—with the spacing of such repeats always being some multiple of the repetition period.

The picture on the facing page shows for several different underlying rules which of the 256 possible elementary rules can successfully be emulated with successively wider blocks.

Entropy estimates [for sequences] Fitting the number of distinct blocks of length b to the form k h b for large b the quantity h gives the so-called topological entropy of the system. The so-called measure entropy is given as discussed on page 959 by the limit of -Sum[p i Log[k, p i ], {i, k b }]/b where the p i are the probabilities for the blocks.

The boundaries between regions come from blocks of even numbers of black cells in the initial conditions, and if one does not allow any such blocks, the density oscillations no longer occur.

The picture below shows how this works for a simple block cellular automaton. … Note that the rules can be thought of as replacements such as "A>" for blocks of length 4 with 4 colors.

Other conserved quantities The conserved quantities discussed so far can all be thought of as taking values assigned to blocks of different kinds in a given state and then just adding them up as ordinary numbers. … One can also consider combining values of blocks by the multiplication operation in a group—and seeing whether the conjugacy class of the result is conserved.