Iterated maps For maps of the form x  a x (1 - x) discussed on page 920 the attractor for small a is a fixed point, then a period 2 limit cycle, then period 4, 8, 16, etc.

In case (a), the fraction of black elements fluctuates around 1/2; in (b) it approaches 3/4; in (d) it fluctuates around near 0.3548, while in (e) and (f) it does not appear to stabilize.

Properties [of logical primitives] Page 813 lists theorems satisfied by each function. {0, 1, 6, 7, 8, 9, 14, 15} are commutative (orderless) so that a ∘ b ＝ b ∘ a , while {0, 6, 8, 9, 10, 12, 14, 15} are associative (flat), so that a ∘ (b ∘ c) ＝ (a ∘ b) ∘ c .

Paperfolding sequences The sequence of up and down creases in a strip of paper that is successively folded in half is given by a substitution system; after t steps the sequence turns out to be NestList[Join[#, {0}, Reverse[1 - #]] &, {0}, t] .

After t steps, the difference in size resulting from the change in initial conditions will be multiplied by approximately 2 λ t —at least until this difference is of order 1.

Implementation [of continuous cellular automata] The state of a continuous cellular automaton at a particular step can be represented by a list of numbers, each lying between 0 and 1. This list can then be updated using CCAEvolveStep[f_, list_List] := Map[f, (RotateLeft[list] + list + RotateRight[list])/3] CCAEvolveList[f_, init_List, t_Integer] := NestList[CCAEvolveStep[f, #] &, init, t] where for the rule on page 157 f is FractionalPart[3#/2] & while for the rule on page 158 it is FractionalPart[# + 1/4] & .

But already the sequence of values for x 2 y - x y 3 or even x (y 2 + 1) seem quite complicated. And for example from the fact that x 2  y 2 + (x y ± 1) has solutions Fibonacci[n] it follows that the positive values of (2 - (x 2 - y 2 - x y) 2 )x are just Fibonacci[n] (achieved when {x, y} is Fibonacci[{n, n - 1}] ).

The plots below show the actual numbers of nodes reached as a function of r for the systems on pages 202 and 203 at steps 1, 10, 20, ..., 200.

Iterative improvement [of constraint satisfaction] The borders of the regions of black and white in the picture shown here essentially follow random walks and annihilate in pairs so that their number decreases with time like 1/ √ t .

With three colors, there are 7 cases to be specified, and 2187 possible rules; with five colors, there are 13 cases to be specified, and 1,220,703,125 possible rules.