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The laws of physics in effect provide a collection of constraints on the structure. And while these laws are traditionally stated in terms of sophisticated mathematical equations, their basic character is similar to the simple constraints on arrays of black and white cells that I discussed at the end of Chapter 5. But now instead of defining constraints just in space, the laws of physics can be thought of as defining constraints on what can happen in both space and time.

Just as for space, it is my strong belief that time is fundamentally discrete. And from the discussion of networks for space in the previous section, one might imagine that perhaps the whole history of the universe in spacetime could be represented by a giant four-dimensional network.

By analogy with the systems at the end of Chapter 5 a simple model would then be that this network is determined by the constraint that around every one of its nodes the overall arrangement of other nodes must match some particular template or set of templates.

Yet much as in Chapter 5 it turns out often not to be especially easy to find out which networks, if any, satisfy specific constraints of this kind. The pictures on the facing page nevertheless show results for quite a few choices of templates—where in each case the dangling connections in a template are taken to go to nodes that are not part of the template itself.

Pictures (a) and (b) show what happens with the two very simplest possible templates—involving just a single node. In case (a), all networks are allowed except for ones in which a node is connected directly to itself. In case (b), only the single network shown is allowed.

With templates that involve nodes out to distance one there are a total of 11 distinct non-trivial cases. And of these, 8 allow no complete networks to be formed, as in picture (e). But there turn out to be three cases—shown as pictures (c), (d) and (f)—in which complete networks can be formed, and in each of these one discovers that a fairly simple infinite set of networks are actually allowed.

In order to have a meaningful model for the universe, however, what must presumably happen is that essentially just one network can satisfy whatever constraints there are, and this one network must then represent all of the complex spacetime history of our universe.

From Stephen Wolfram: A New Kind of Science [citation]