Implementation [of branching model] It is convenient to represent the positions of all tips by complex numbers. … And after n steps the positions of all tips generated are given simply by Nest[Flatten[Outer[Times, 1 + #, b]] &, {0}, n]

Now collect and rank all the "words" delimited by spaces that are formed. … If all k letters have equal probabilities, there will be many words with equal frequency, so the distribution will contain steps, as in the second picture below. … If all letter probabilities are equal, then words will simply be ranked by length, with all k m words of length m occurring with frequency p m .

Much as in the previous section, even if paths do not converge for every possible string, it can still be true that paths converge for all strings that are actually generated from a particular initial string. … Note that a rule such as {"A"  "B", "A"  "C", "B"  "A", "B"  "D"} exhibits convergence for all paths that have diverged for only one step, but not for all those that have diverged for longer. In general it is formally undecidable whether a particular multiway system will eventually exhibit convergence of all paths.

These examples are taken from the 4277 found in effect by searching exhaustively all 7,625,597,484,987 possible rules with three colors.

In case (c), alternate steps in the leftmost column (which in all cyclic tag systems determines the overall behavior) have the same nested form as the third neighbor-independent substitution system shown on page 83 .

Non-standard arithmetic Goodstein's result from page 1163 is true for all ordinary integers. … For any set of objects that satisfy the axioms of arithmetic must include all finite ordinary integers, since each of these can be reached just by using Δ repeatedly. And the axioms then turn out to imply that any additional objects must be larger than all these integers—and must therefore be infinite.

Continuum limits [of networks] For all everyday purposes a region in a network with enough nodes and an appropriate pattern of connections can act just like ordinary continuous space. … And in general, there is no reason to expect that all properties of the system (notably for example the existence of particles) will be preserved by taking such a limit. … Yet as I will discuss on pages 534 and 1050 even at such scales it is far from straightforward to see how all the various well-studied properties of ordinary continuous space (as embodied for example in the theory of manifolds) can emerge from discrete underlying networks.

For even though it is intricate, one can see that it actually consists of many nested triangular pieces that all have exactly the same form. … So of the three cellular automata that we have seen so far, all ultimately yield patterns that are highly regular: the first a simple uniform pattern, the second a repetitive pattern, and the third an intricate but still nested pattern.

Nesting in rule 184 is easiest to see when the initial conditions contain exactly equal numbers of black and white cells, so that the numbers of left and right stripes exactly balance, and all stripes eventually annihilate. … The initial condition used has exactly equal numbers of black and white cells, causing all the stripes eventually to annihilate.

The overall limiting pattern will be finite so long as Abs[c] < 1 for all elements of b . After n steps the total length of all stems is given by Apply[Plus, Abs[b]] n .