Given these models the only way to find out what they do will usually be just to run them.

So are there computations that take still longer to do?

But how do such problems compare to each other?

Yet despite all this, we do not in our everyday experience typically have much difficulty telling living systems from non-living ones.

Foremost among them is that I have lived at the moment in history when technology has first made it possible to do the kinds of things I have done. … For while building Mathematica has taken a considerable amount of my time, I would without it as a tool never have been able to do the vast majority of what is now in this book. … Part of what has allowed me to do this is reading an immense number of books, articles and websites.

Isotropy [in lattice systems] Any pattern grown from a single cell according to rules that do not distinguish different directions on a lattice must show the same symmetry as the lattice. … And at least in the case of ordinary random walks, they do, so that for example, the ratio averaged over all possible walks of n = 4 tensor components after t steps on a square lattice is β = 3 + 2/(t - 1) , converging to the isotropic value 3, and the ratio of n = 6 components is 5 - 4/(t - 1) + 32/(3t - 4) .

In the late 1950s Berni Alder and Thomas Wainwright began to do computer simulations of idealized molecular dynamics of 2D hard spheres—mainly to investigate transitions between solids, liquids and gases. … And in 1984, as part of work I was doing on massively parallel computing, I resolved to develop a practical approach to fluid mechanics based on cellular automata.

But just what such numerical results actually have to do with detailed solutions to the Navier–Stokes equations is not clear. … One of the key advantages of my cellular automaton approach to fluids is precisely that it does not require any such approximations.

If these are small enough then it makes sense to do a perturbation expansion in which one approximates field configurations in terms of a succession of arrangements of ordinary waves—as in Feynman diagrams. But just as one cannot expect to capture fully turbulent fluid flow in terms of a few simple waves, so in general as soon as there is substantial nonlinearity it will no longer be sufficient just to do perturbation expansions. … But while there is no problem in doing this at a formal mathematical level—and indeed the expressions one gets from Feynman diagrams can always be analytically continued in this way—what general correspondence there is for actual physical processes is far from clear.

The pictures on pages 179 – 181 show one rule, for example, that does not.