At least in simple cases, the basic idea—as illustrated in the pictures below—is to consider systems whose elements do not just lie along a one-dimensional line, but instead are arranged for example on a two-dimensional grid.

Examples of cellular automata that do allow information to be transmitted over large distances, but only in very restricted ways.

And indeed, in my experience the single most common mistake in doing computer experiments is to look at systems that are vastly more complicated than is necessary. … And this is often a humbling experience, for it is common to find that the system does in fact manage to produce interesting behavior—but just in a way that one was not imaginative enough to guess.

So what could I do? … So I thought that perhaps I could do something similar in natural science: that there might be some appropriate primitives that I could find that would successfully capture a vast range of natural phenomena.

In the early 1980s my single greatest impediment had been the practical difficulty of doing computer experiments using the various rather low-level tools that were available. But by 1986 I had realized that with a number of new ideas I had it would be possible to build a single coherent system for doing all kinds of technical computing.

But I do not think so. … But, unfortunately perhaps, I do not think that this conclusion necessarily follows.

So what does one find if one allows templates that include nodes out to distance two? … So what does this mean for getting the kind of complexity that we see in our universe?

So what does this mean for our notion of time? … But if one identifies time with position down the page, the presence of connections that go up as well as down the page implies that in some sense time does not always progress in the same direction.

From our experience with everyday objects we might think that it should somehow be possible to do this without affecting the electron. … And indeed there usually have to be a huge number of particles doing more or less the same thing before we successfully register it.

So can one do better at maintaining the quality of the image? … But most other images typically require more forms to be included—and thus do not allow such high levels of compression.