A typical example of the first approach is the Ising model for spin systems in which relative probabilities of sequences are found by multiplying together the results of applying a simple function to blocks of nearby elements in the sequence.

Each repeating block of digits typically seems quite random, and has properties such as all possible subblocks of digits up to a certain length appearing (see page 1084 ).

Each block of Turing machines yields the same output for a given input.

Thus, for example, with the rule {{1, 0}  {1, 1, -1}, {1, 1}  {2, 1, 1}, {2, 0}  {1, 0, -1}, {2, 1}  {1,0,1}} the head moves to the right whenever the initial condition consists of odd-length blocks of 1's separated by single 0's; otherwise it stays in a fixed region.

The picture below shows the digit sequences of successive numbers in base -2; the row j from the bottom turns out to consist of alternating black and white blocks of length 2 j .

1D cellular automata In a cellular automaton with k colors and r neighbors, configurations that are left invariant after t steps of evolution according to the cellular automaton rule are exactly the ones which contain only those length 2r + 1 blocks in which the center cell is the same before and after the evolution.

In each block the second entry is the rule obtained by interchanging black and white, the third entry is the rule obtained by interchanging left and right, and the fourth entry the rule obtained by applying both operations.

The block cellular automata from previous pages [ 461 , 462 ] started from initial conditions containing regions of different density.

At each step the leftmost ball in the trough is released, and if this ball is black (as determined, for example, by size) a mechanism causes a new block of balls to be added at the right-hand end of the trough.

One way to achieve this is to break the array into 2 n × 2 n blocks, then successively to fill in pixels in each block until the appropriate gray level is reached, as in the pictures below, in an order given for example by Nest[ Flatten2D[{{4 # + 0, 4 # + 2}, {4 # + 3, 4 # + 1}}] &, {{0}}, n] An alternative to this so-called ordered dither approach is the Floyd–Steinberg or error-diffusion method invented in 1976.