By the 1960s undecidability was being found in all sorts of systems, but most of the examples were too complicated to seem of much relevance in practical mathematics or computing.

But at intermediate times one will see all sorts of potentially dramatic gullies that reflect the pattern of drainage, and the formation of a whole tree of streams and rivers.

In the late 1960s there began to be all sorts of simulations of differential equations with complicated behavior, first mainly on analog computers, and later on digital computers. … By the early 1980s at least indirect signs of chaos in this sense (see note below ) had been seen in all sorts of mechanical, electrical, fluid and other systems, and there emerged a widespread conviction that such chaos must be the source of all important randomness in nature.

In this book such offset lists are always taken to be in the order given by Sort , so that for range r rules in d dimensions the order is the same as Flatten[Array[List, Table[2r + 1, {d}], -r], d - 1] .

Particularly over the past few decades all sorts of examples of "1/ f noise" have been identified with α ≃ 1 , including flicker noise in resistors, semiconductor devices and vacuum tubes, as well as thunderstorms, earthquake and sunspot activity, heartbeat intervals, road traffic density and some DNA sequences.

In supersymmetric models—and string theory—there are typically also all sorts of other types of particles, assumed to have high masses since they have not been observed.

Ulam systems Having formulated the system around 1960, Stanislaw Ulam and collaborators (see page 877 ) in 1967 simulated 120 steps of the process shown below, with black cells after t steps occurring at positions Map[First, First[Nest[UStep[p[q[r[#1], #2]] &, {{1, 0}, {0, 1}, {-1, 0}, {0, -1}}, #] &, ({#, #} &)[{{{0, 0}, {0, 0}}}], t]]] UStep[f_, os_, {a_, b_}] := {Join[a, #], #} &[f[Flatten[ Outer[{#1 + #2, #1} &, Map[First, b], os, 1], 1], a]] r[c_]:= Map[First, Select[Split[Sort[c], First[#1]  First[#2] &], Length[#]  1 &]] q[c_, a_] := Select[c, Apply[And, Map[Function[u, qq[#1, u, a]], a]] &] p[c_]:= Select[c, Apply[And, Map[Function[u, pp[#1, u]], c]] &] pp[{x_, u_}, {y_, v_}] := Max[Abs[x - y]] > 1 || u  v qq[{x_, u_}, {y_, v_}, a_] := x  y || Max[Abs[x - y]] > 1 || u  y || First[Cases[a, {u, z_}  z]]  y These rules are fairly complicated, and involve more history than ordinary cellular automata.

But with the advent of widespread computer simulations in the 1980s it became clear that all sorts of features normally associated with life were actually rather easy to obtain.

(Sometimes it is also possible to recognize microscopic features characteristic of particular kinds of use or wear—and it is conceivable that in the future analysis of trillions of atomic-scale features could reveal all sorts of details of the history of an object.)

An example is the algorithm of Anatolii Karatsuba from 1961 for finding products of n -digit numbers (with n = 2 s ) by operating on their digits in the nested pattern of page 608 (see also page 1093 ) according to First[f[IntegerDigits[x, 2, n], IntegerDigits[y, 2, n], n/2]] f[x_, y_, n_] := If[n < 1, x y, g[Partition[x, n], Partition[y, n], n]] g[{x1_, x0_}, {y1_, y0_}, n_] := With[{z1 = f[x1, y1, n/2], z0 = f[x0, y0, n/2]}, z1 2 2n + (f[x0 + x1, y0 + y1, n/2] - z1 - z0)2 n + z0] Other examples include the fast Fourier transform (page 1074 ) and related algorithms for ListConvolve , the quicksort algorithm for Sort , and many algorithms in fields such as computational geometry.