But as soon as one perturbs such initial conditions, one normally seems to get only complicated and seemingly random behavior, as in the top row of pictures in the second image. … Note that these localized structures always move one cell to the right at each step—making it impossible for them to interact in non-trivial ways.

One starts by setting r equal to p . Then at each step, one compares the values of 2r and q . … And indeed, if one looks for example at square roots the story is completely different.

But as I will discuss at length in Chapter 7 one must realize that on its own this cannot explain why randomness—or complexity—should occur in any particular case. … For if one thinks about numbers The effect of making a small change in the initial conditions for the shift map—shown as case (d) on pages 150 and 151 . … The second picture shows what happens if one changes the size of the number in this initial condition by just one part in a billion billion.

When we looked at one-dimensional Turing machines earlier in this book, we found that it was possible for them to exhibit complex behavior, but that such behavior was rather rare. … But in fact what we find is that the situation is remarkably similar to one dimension. … Page 186 shows one example where the behavior seems in many respects completely random.

In general, however, one can set up network systems that have rules in which different operations are performed at different nodes, depending on the local structure of the network near each node. One simple scheme for doing this is based on looking at the two connections that come out of each node, and then performing one operation if these two connections lead to the same node, and another if the connections lead to different nodes. … But as soon as one allows dependence on slightly longer-range features of the network, much more complicated behavior immediately Evolution of network systems whose rules involve the addition of new nodes.

But one other example is rule 150—as illustrated in the first set of pictures below. … And in fact one can show that any rule that is additive will be able to emulate itself and will thus yield nested patterns. … Ultimately, however, additive rules are not the only ones that can emulate themselves.

At any point one can follow the arrow to the left to get a black cell, but the form of the network implies that this black cell must always be followed by at least one white cell. … Unlike in the picture below, these rules do not reach their final states after one step, but instead just progressively evolve towards these states. … In rule 128, for example, the fact that regions of black shrink by one cell on each side at each step means that any region of black that exists after t steps must have at least t white cells on either side of it.

Indeed, often all that one knows is the sequence of prices at which trades are executed. … One can always make the underlying system more complicated—say by having a network of cells, or by allowing different cells to have different and perhaps changing rules. But although this will make it more difficult to recognize definite rules even if one looks at the complete behavior of every element in the system, it does not affect the basic point that there is randomness that can intrinsically be generated by the evolution of the system.

But in picture (c) cells not updated on a given step are merged together, yielding vertical stripes of color that extend from one updating event to another. … And as picture (d) begins to emphasize, one can think of these stripes as indicating what causal relationships or connections exist between updating events. … For rather than having a picture based on successive individual steps of evolution, one can instead form a network of the various causal relationships between updating events, with each updating event being a node in this network, and each stripe being a connection from one node to another.

But if one thinks about actually using such formulas one might at first wonder what good they really are. For if one was to work out the value of a power m t by explicitly performing t multiplications, this would be very similar to explicitly following t steps of cellular automaton evolution. But the point is that because of certain mathematical features of powers it turns out to be possible—as indicated in the table below—to find m t with many fewer than t operations; indeed, one or two operations for every base 2 digit in t is always for example sufficient.