In case (h) the number of steps grows so rapidly that only two peaks are seen in the plot.

The first multiway system actually generates all strings in the end (not least since it yields the lemmas -> and -> )—and in fact strings of length n > 2 appear after at most 2n+7 steps. The second multiway system generates only the n+1 strings where black comes before white—and all of these strings appear after at most 7n steps. The third multiway system generates a complicated collection of strings; the numbers of lengths up to 8 are 1, 2, 4, 8, 14, 22, 34, 45.

Then the point is that because the rules depend only on the color of a particular branch, and not on the colors of any neighboring branches, the subtrees that are generated from all the branches of the same color must have exactly the same structure, as in the pictures below.

Self-assembly Given elements (such as pieces of molecules) that fit together only when certain specified constraints are satisfied it is fairly straightforward to force, say, cellular automaton patterns to be generated, as on page 221 .

Spacetime symmetric rules With k = 2 and the neighborhoods shown here, only the additive rules 90R, 105R, 150R and 165R are space-time symmetric.

Most of the programs involve only built-in Mathematica functions, and so can be run in Mathematica without setting up any further definitions. (Many programs nevertheless contain variables that need to be assigned their values before the programs are run—as can be done for example with Block[{k = 2}, program] . … Only occasionally do the programs significantly sacrifice efficiency for elegance.)

(Only the shortest connection from each theorem is shown explicitly.) … (Explicit annotations giving theorems used in proofs were apparently added to editions of Euclid only in the past few centuries; the picture below extends the usual annotations in a few cases.) The theorem with the longest proof is the one that states that there are only five Platonic solids.

2D mobile automata Mobile automata can be generalized just like Turing machines. Even in the simplest case, however, with only four neighbors involved there are already (4k) k 5 possible rules, or nearly 10 29 even for k = 2 .

Systems of Limited Size and Class 2 Behavior…With n cells there are thus 2 n possible states. … Of the elementary rules, however, only rule 45 seems to yield periods that always stay close to the maximum of 2 n . … So how does all of this relate to class 2 behavior?

And indeed at initial condition 187 we see a considerably more complicated structure, that instead of staying still moves systematically to the right, repeating its basic form only every 9 steps. … But going on to initial condition 195, we again find a more complicated structure—this time one that repeats only every 22 steps.