A common example—to be discussed further two sections from now —involves taking, say, a sequence of black and white cells, and then counting the frequency with which each color and each block of colors occurs. … But despite some confusion in the past it is certainly not true that just checking equality of frequencies of blocks of colors—even arbitrarily long ones—is sufficient to ensure that no regularities at all exist.

The idea is to pick a set of pairs of upper and lower blocks, and then to ask whether there is any sequence of such pairs that satisfies the constraint that the upper and lower strings formed end up being in exact correspondence. … But as soon as there are more than two pairs things become much more complicated, and as the pictures on the facing page demonstrate, even with very short blocks remarkably long and seemingly quite random sequences can be required in order to satisfy the constraints.

[2D substitution systems with] non-white backgrounds The pictures below show substitution systems in which white squares are replaced by blocks which contain black squares.

(Each period doubling turns out to occur exactly when a diagonal in the pattern eventually becomes a white stripe, and the diagonal to its left has an odd number of black cells in each repeating block.) … All possible blocks appear to occur eventually (see page 725 ). The probability for a block of n adjacent white cells (corresponding to a row in a white triangle) seems quite accurately to approach 2 -n , with the first length 10 such block occurring at step 67 and the first length 20 one occurring at step 515.

The standard way to do this is to set up an error-correcting code in which blocks of m original data elements are represented by a codeword of length n that in effect includes some redundant elements. … Defining PM[s_] := IntegerDigits[Range[2 s - 1], 2, s] blocks of data of length m can be encoded with Join[data, Mod[data . Select[PM[s], Count[#, 1] > 1 &], 2]] while blocks of length n (and at most one error) can be decoded with Drop[(If[#  0, data, MapAt[1 - # &, data, #]] &)[ FromDigits[Mod[data .

Multidimensional multiway systems As a generalization of multiway systems based on 1D strings one can consider systems in which rules operate on arbitrary blocks of elements in an array in any number of dimensions.

Region (a) shows a block separator—corresponding to a dashed line in picture (d) on page 679 —hitting the single black element in the sequence that exists at the first step. Because the element hit is black, an object must be produced that allows information from the block at this step to pass through.

Most often the tests are applied not directly to sequences of 0's and 1's, but instead say to numbers obtained from blocks of 8 elements. A typical collection of tests described by Donald Knuth in 1968 includes: (1) frequency or equidistribution test (possible elements should occur with equal frequency); (2) serial test (pairs of elements should be equally likely to be in descending and ascending order); (3) gap test (runs of elements all greater or less than some fixed value should have lengths that follow a binomial distribution); (4) poker test (blocks corresponding to possible poker hands should occur with appropriate frequencies); (5) coupon collector's test (runs before complete sets of values are found should have lengths that follow a definite distribution); (6) permutation test (in blocks of elements possible orderings of values should occur equally often); (7) runs up test (runs of monotonically increasing elements should have lengths that follow a definite distribution); (8) maximum-of-t test (maximum values in blocks of elements should follow a power-law distribution). With appropriate values of parameters, these tests in practice tend to be at least somewhat independent, although in principle, if sufficient data were available, they could all be subsumed into basic block frequency and run-length tests.

Note that after just a few steps, the sequences produced always seem to consist of white elements followed by black, with possibly one block of black in the white region. Without this additional block of black, only the first case in the rule can ever apply.

{a_, b_, c_}  {-a, -b, c} The resulting structure is a cubic array of blocks with each block containing 8 nodes.