Irreversibility and the Second Law of Thermodynamics All the evidence we have from particle physics and elsewhere suggests that at a fundamental level the laws of physics are precisely reversible. … The picture at the top of the next page shows an example based on a reversible cellular automaton of the type discussed in the previous section .

And indeed, if one takes the patterns from successive steps and stacks them on top of each other to form a three-dimensional object, as in the picture below, then this object has a very regular nested structure.

And indeed when it comes to large-scale fluid behavior, it seems to be true that in almost all cases there is no discernible difference between what happens with different detailed initial conditions. … Presumably there do exist situations in which there is some kind of delicate balance—say of whether the first eddy is shed at the top or bottom of an object—and in which small changes in initial conditions can have a substantial effect.

But it turns out that this very same angle also shows up in all sorts of other features of plants, as shown in the pictures at the top of the facing page .

Picture (e) shows the updating events and stripes from the top of picture (d), with the updating events now explicitly numbered. … For the point is that an observer who is part of the mobile automaton will never see all the individual steps in this evolution.

And at a fundamental level all that matters for such transformations is the structure of expressions. … But by making various details explicit I have been able to put all axiom systems in forms that can be used almost directly in Mathematica. … First, the definitions at the top of page 774 must be used to expand out various pieces of notation.

For any process like the one in the picture above must occur on top of a background of apparently random small-scale rearrangements of the network. … In general, the equations of quantum field theory seem to imply that there can be all sorts of complicated configurations in the field, even in the absence of actual particles.

And indeed it turns out that over the course of the past few decades a rather large number of such problems have in fact all been found to be so-called NP-complete. … The top row in the picture on the facing page shows the first few of the exponentially many possible paths obtained by making successive sequences of choices in a particular non-deterministic Turing machine.

The first stage, illustrated in the top picture on the next page , is to get a cyclic tag system to emulate an ordinary tag system with the property that its rules depend only on the very first element that appears at each step.

Convection When there is a temperature difference between the top and bottom of a fluid, hot fluid tends to rise, and cold fluid then comes down again.