So what does one find if one allows templates that include nodes out to distance two? There are a total of 690 distinct non-trivial such templates—and of these, 681 allow no complete networks to be formed, as in case (g). Six of the remaining templates then again allow an infinite sequence of networks. But there are three templates—shown as cases (h), (i) and (j)—that turn out to allow just single networks. These networks are however rather simple, and indeed the most complicated of them—case (i)—has just 20 nodes, and corresponds to a dodecahedron.
So are there in fact reasonably simple sets of constraints that in the end allow just one highly complex network, or perhaps a family of similar networks? I tend to doubt it. For our experience in Chapter 5 was that even in the much more rigid case of arrays of black and white squares, it was rather difficult to find constraints that would succeed in forcing anything but very simple patterns to occur.
So what does this mean for getting the kind of complexity that we see in our universe? We have not had difficulty in getting remarkable complexity from systems like cellular automata that we have discussed in this book. But such systems work not by being required to satisfy constraints, but instead by just repeatedly applying explicit rules.
So is it in the end sensible to think of the universe as a single structure in spacetime whose form is determined by a set of constraints? Should we really imagine that the complete spacetime history of the universe somehow always exists, and that as time progresses, we are merely exploring different parts of it? Or should we instead think that the universe—more like systems such as cellular automata—explicitly evolves in time, so that at each moment a new state of the universe is in effect created, and the old one is lost?
Models based on traditional mathematical equations—in which space and time appear just as abstract symbolic variables—have never had to make much distinction between these two views. But in trying to understand the ultimate underlying mechanisms of the universe, I believe that one must inevitably distinguish between these views.
And I strongly believe that the second view is the one most likely to provide a meaningful underlying model for our universe. But while this view is closer to our everyday perception of time, it seems to