In each case single cells are encoded as blocks of cells, and all distinct such encodings with blocks up to length 20 are shown.

The pictures below show how this works: on alternate steps the arrangement of blocks in rule 126 corresponds exactly to the arrangement of individual cells in rule 90. … The initial conditions that are used consist of blocks of cells where each block contains either two black cells or two white cells. If one looks only on every other step, then the blocks behave exactly like individual cells in rule 90.

Each cell in rule 254 is represented by a block of 20 cells in the universal cellular automaton. Each of these blocks encodes both the color of the cell it represents, and the rule for updating this color.

If one looks at the set of all possible sequences, one can fairly easily calculate the distribution of frequencies for any particular block. And from this distribution one can tell with Statistics of block frequencies for various sequences. In each case the frequency of a particular block is represented by gray level, with results for blocks of successively greater lengths being shown on successive rows as indicated on the bottom left.

And once past, the stripe continues to the right, finally adding the block it represents to the end of the sequence. … The collections of lines coming in from the left represent the blocks that can be added at successive steps. The beginning of each block is indicated by a dashed line, while the elements within the block are indicated by solid black and gray lines.

In the upper block of pictures, every cell is chosen to be black or white with equal probability on the two successive first steps. In the lower block of pictures, only the center cell is taken to be black on these steps.

As the pictures below suggest, it is usually quite easy to see if an image is purely repetitive—even in cases where the block that repeats is fairly large. … Note that in a pattern generated by repeating one particular block, there will normally be other blocks that occur with other alignments. Page 215 shows patterns obtained in systems based on constraints in which one effectively requires that only certain blocks or sets of blocks occur.

In each case the initial conditions consist of a fixed block of cells that is repeated over and over again. … To get period 11, a block that contains 275 cells is required.

The constraints require that concatenating in some order the blocks shown should yield identical upper and lower strings. … When the constraints involve more than two blocks there seems in general to be no upper limit on how long a string one may need to consider to tell whether the constraints can be satisfied. Pictures (a), (b), (h) and (j) show the longest minimal strings needed for any of the 4096, 16384, 65536 and 262144 constraints involving blocks with totals of 7, 8, 9 and 10 elements.

And in this case the best model is again straightforward to find: it simply takes the probabilities for different blocks to be equal to the frequencies with which these blocks occur in the data. If one does not decide in advance how long the blocks are going to be, however, then things can become more complicated. For in such a case one can always just make up an extreme model in which only one very long block is allowed, with this block being precisely the sequence that is observed in the data.