But what if one compares different breeds or species of animals? … Needless to say, just like with leaves and shells, such differences can have effects that are quite dramatic both visually and mechanically—turning, say, an animal that walks on four legs into one that walks on

But one of the crucial discoveries of this book is that even programs with very simple underlying rules can yield great complexity. … But from what I have found so far I am extremely optimistic that by using the ideas of this book the most fundamental problem of physics—and one of the ultimate problems of all of science—may finally be within sight of being solved.

One aspect of the generation of randomness that we have noted several times in earlier chapters is that once significant randomness has been produced in a system, the overall properties of that system tend to become largely independent of the details of its initial conditions. … So this means that if a system generates sufficient randomness, one can think of it as evolving towards a unique equilibrium whose properties are for practical purposes independent of its initial conditions.

One class of systems where some types of complexity were noticed in the 1950s are so-called iterated maps. … Almost all the results that were obtained are still military secrets, but I do not believe that any phenomena like the ones described in this chapter were discovered.

But this seems to be considerably less true when one is dealing with descriptions in which information can be lost. … Before the discoveries in this book, one might have thought that to create anything with a significant level of apparent complexity would necessarily require a procedure which itself had significant complexity.

For among other things, whereas in the process of thinking we routinely manage to retrieve remarkable connections almost instantaneously from memory, we tend to be able to carry out logical reasoning only by laboriously going from one step to the next. … As it happens, however, one notable exception is Mathematica.

But one of the important conclusions from what I have done in this book is that this is far from correct. … Yet if one looks at the types of systems that are actually studied in mathematics they continue even to this day to be far from as general as possible.

Traditional engineering might tend to make one think so. … But one of the results of this book is that in general things need not work like this.

For one step in rule 30, for example, this yields {{1, 0, 0}, {0, 1, 1}, {0, 1, 0}, {0, 0, 1}} , as shown on page 616 . One can think of this as specifying corners that should be colored on an n -dimensional Boolean hypercube. … Given an original DNF list s , this can be done using PI[s, n] : PI[s_, n_] := Union[Flatten[ FixedPointList[f[Last[#], n] &, {{}, s}] 〚 All, 1 〛 , 1]] g[a_, b_] := With[{i = Position[Transpose[{a, b}], {0,1}]}, If[Length[i]  1 && Delete[a, i] === Delete[b, i], {ReplacePart[a, _, i]}, {}]] f[s_, n_] := With[ {w = Flatten[Apply[Outer[g, #1, #2, 1] &, Partition[Table[ Select[s, Count[#, 1]  i &], {i, 0, n}], 2, 1], {1}], 3]}, {Complement[s, w, SameTest  MatchQ], w}] The minimal DNF then consists of a collection of these prime implicants.

To get some idea about the origin of this behavior, one can assume that successive values of n are randomly even and odd with equal probability. … One reason that all sequences do not grow forever is that even with perfect randomness, there will be fluctuations, and occasionally n will reach a low value that makes it get stuck in a repetitive sequence. If one applies the same kind of argument to the standard 3n+1 problem, then one concludes that n should on average decrease by a factor of √ 3 /2 at each step, making it unsurprising that at least in most cases n eventually reaches the value 1.