Indeed, it turns out that almost all the traditional mathematical models that have been used in physics and other areas of science are ultimately based on partial differential equations. Thus, for example, Maxwell's equations for electromagnetism, Einstein's equations for gravity, Schrödinger's equation for quantum mechanics and the Hodgkin–Huxley equation for the electrochemistry of nerve cells are all examples of partial differential equations.

And it turns out that one of these is related to the method used since the late 1940s for generating random numbers in almost all practical computer systems. … It has then often been assumed that having maximal repetition period will somehow imply maximum randomness in all aspects of the sequence one gets.

But the whole point is that all any model is supposed to do—whether it is a cellular automaton, a differential equation, or anything else—is to provide an abstract representation of effects that are important in determining the behavior of a system. … And the only way to achieve this is to pick out only certain effects that are important, and to ignore all others.

But in general the details of all these rules will no doubt depend on very specific characteristics of individual plants. The distance before a new stem appears is, for example, probably determined by the rates of production and diffusion of plant hormones and related substances, and these rates will inevitably depend both on the thickness and mechanical structure of the stem, as well as on all kinds of biochemical properties of the plant.

Rather, they are usually made up of a collection of separate identifiable parts, like heads, tails, legs, eyes and so on, all with their own specific structure. … So how do all these parts get produced?

To be at all realistic one would have to set up an elaborate network to represent the flow of information between different entities. … One can imagine all sorts of schemes by which such colors could be updated.

For if all these features were somehow explicitly and separately included, the rule would necessarily have to be very complicated to fit them all in.

Space as a Network In the last section I argued that if the ultimate model of physics is to be as simple as possible, then one should expect that all the features of our universe must at some level emerge purely from properties of space. … And in addition, nothing fundamental is lost by requiring that all the nodes in a network have exactly the same total number of connections to other nodes.

In case (a), all networks are allowed except for ones in which a node is connected directly to itself. … 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.

But one of the features of a cellular automaton is that it is set up to update all of its cells together, as if at each tick of some global clock. … At first it may seem bizarre, but one possibility that I believe is ultimately not too far from correct is that the universe might work not like a cellular automaton in which all cells get updated at once, but instead like a mobile automaton or Turing machine, in which just a single cell gets updated at each step.