The total number of combinator expressions of successively greater sizes is {2, 4, 16, 80, 448, 2688, 16896, 109824, …} (or in general 2 n Binomial[2n - 2, n - 1]/n ; see page 897 ). … Of combinator expressions up to size 6 all evolve to fixed points, in at most {1, 1, 2, 3, 4, 7} steps respectively (compare case (a)); the largest fixed points have sizes {1, 2, 3, 4, 6, 10} (compare case (b)). … For s[s][k][s[s[s][k]]][k] (case (k)) the size at step t - 7 is given by h[1] = h[2] = h[3] = 12 h[t_] := If[Mod[t, 4]  2, 2, 1] (h[Ceiling[t/2] - 1] + t) + {3, 5, -7, -1} 〚 Mod[t, 4] + 1 〛 Examples with similar behavior are s[s[s][k]][s][s[s][k]] , s[s[s]][s][s[s][k]][k] and s[s[s][s]][s][s[s][k]] .

[PDEs involving] different powers The equations ∂ tt u[t, x]  ∂ xx u[t, x] + (1 - u[t,x] n )(1 + a u[t, x]) with n = 4 , 6 , 8 , etc. appear to show similar behavior to the n = 2 equation in the main text.

Symbolic systems [emulating cellular automata] Given the rules for an elementary cellular automaton in the form used on page 867 (with {0, 0, 0}  0 ), the following will construct a symbolic system which emulates it: Flatten[{Array[(p[x_][#1][#2][#3]  p[x[{##} /. rules]][#2][#3]) &, {2, 2, 2}, 0] /. {0  p, 1  q}, {r[x_]  p[r[p][p]][x], p[x_][p][p][r]  x[p][p][r]}}] The initial condition for the symbolic system is given by Fold[#1[#2] &, r[p][p], init /. {0  p, 1  q}][p][p][r] Step t in the cellular automaton corresponds to step t (t + Length[init] + 3) in the symbolic system.

On row (a) of page 415 the parameter a varies from 1.05 to 1.65, while on row (b) b varies from 0 to 6. The complete surface of the shell is obtained by varying both t and θ in a t {Cos[t] (1 + c (Cos[e] Cos[ θ ] + d Sin[e] Sin[ θ ])), Sin[t] (1 + c (Cos[e] Cos[ θ ] + d Sin[e] Sin[ θ ])), b + c (Cos[ θ ] Sin[e] - d Cos[e] Sin[ θ ])} where c varies from 0.4 to 1.6 on row (c), d from 1 to 4 on row (d) and e from 0 to 1.2 on row (e).

In case (c), it is nested—the size of the network at step t is related to the number of 1's in the base 2 digit sequence of t .

And so for example all rules that lie in the first two columns on page 232 can be shown to be unable ever to produce anything besides class 1 or class 2 behavior. … The top row of rules all have class 1 behavior. … The first rule can be either class 2 or class 4, the second class 3 or 4, the third class 2 or 3 and the fourth class 1, 2 or 3.

The point is that class 1 and 2 systems rapidly settle down to states in which there is essentially no further activity. … The underlying rules in such systems involve a parameter that can vary smoothly from 0 to 1. … The answer is that there are normally some stretches of class 1 or 2 behavior, and some stretches of class 3 behavior.

And in Mathematica—ever since it was first released— Random[Integer] has generated 0's and 1's using exactly the rule 30 cellular automaton. … One consequence of this, as discussed on page 259 , is that the sequence of 0's and 1's that is generated must then eventually repeat. … Another issue is that if one always ran the cellular automaton from page 315 with the particular initial condition shown there, then one would always get exactly the same sequence of 0's and 1's.

Earlier in this chapter we saw that if a network is going to correspond to ordinary space in some number of dimensions d , then this means that by going r connections from any given node one must reach about r d - 1 nodes. … And in general the leading correction to the number of nodes reached turns out to be proportional to the curvature multiplied by r d + 1 . … But in fact the leading correction to the number of nodes reached is always quite simple: it is just proportional to what is called the Ricci scalar curvature, multiplied by r d + 1 .

With r string pairs and n = StringLength[StringJoin[p]] there are 2 n Binomial[n - 1, 2r - 1] possible constraints (assuming no strings of zero length), each being related to at most 8 r! others by straightforward symmetries (or altogether 4 n - 1 for given n ). … From looking at the structure of the individual pairs one can see that if there is a solution it must begin with pair 1 or pair 3, and end with pair 1.