The basic answer, much as we saw in one-dimensional substitution systems on page 85 , is some form of interaction between different elements—so that the replacement for a particular element at a given step can depend not only on the characteristics of that element itself, but also on the characteristics of other neighboring elements.

But the crucial realization that led me to develop the new kind of science in this book is that there is in fact no reason to think that systems like those we see in nature should follow only such traditional mathematical rules.

But while we have discussed a whole range of different kinds of underlying rules, we have for the most part considered only the simplest possible initial conditions—so that for example we have usually started with just a single black cell.

But the crucial point that I discovered only some time later is that random behavior can also occur even when there is no randomness in initial conditions.

(In general, the minimum may only be local.) … Case (c) shows the negatively curved surface z=x 2 -y 2 , (d) a paraboloid z=x 2 +y 2 , and (e,f) z=1/(r+δ) —a rough analog of curvature in space produced by a sphere of mass.

So can one then achieve something intermediate in rule 73—in which information is transmitted, but only in a controlled way?

And in fact, by using the universality of rule 110 it turns out to be possible to come up with the vastly simpler universal Turing machine shown below—with just 2 states and 5 possible colors. … The machine has 2 states and 5 possible colors. … The compressed picture is made by keeping only the steps indicated at which the head is further to the right than ever before.

With 3 states and 2 colors it turns out that with blank initial conditions all of the 2,985,984 possible Turing machines of this type quickly evolve to produce simple repetitive or nested behavior. … And from other results earlier in this chapter it seems likely that in fact one would tend to see universality even somewhat earlier—after going through only perhaps just ten or twenty rules. … So this means Examples of 3-state 2-color Turing machines which behave for a while in slightly complicated ways.

The pictures on the right below show Sin[1/2 π a[t, n]] 2 for these functions (equivalent to Mod[a[t, n], 2] for integer a[t, n] ). The discrete results on the left can be obtained by sampling only where integer grid lines cross. … The presence of poles in quantities such as GegenbauerC[1/2, -t, -1/2] leads to essential singularities in the rightmost picture below.

If the elements of list correspond to values of a polynomial of degree n at successive integers, then Nest[d, list, n + 1] will contain only zeros. … The pictures below show the results with k = 2 (rule 60) for (a) Fibonacci[n] , (b) Thue–Morse sequence, (c) Fibonacci substitution system, (d) (Prime[n] - 1)/2 , (e) digits of π .