One way that we can now show this is to demonstrate that combinators can emulate rule 110. … And in fact wherever one looks, the threshold for universality seems to be much lower than one would ever have imagined. … A combinator expression that corresponds to the operation of doing one step of rule 110 evolution.

Yet so far in this book all the systems we have discussed have effectively been limited to just one dimension. The purpose of this chapter , therefore, is to see how much of a difference it makes to allow more than one dimension. … Examples of simple arrangements of elements in one, two and three dimensions.

A mollusc shell, like a one-dimensional cellular automaton, in effect grows one line at a time, with new shell material being produced by a lip of soft tissue at the edge of the animal inside the shell. … And given this, the simplest hypothesis in a sense is that the new state of the element is determined from the previous state of its neighbors—just as in a one-dimensional cellular automaton. Examples of patterns produced by the evolution of each of the simplest possible symmetrical one-dimensional cellular automaton rules, starting from a random initial condition.

And indeed in pictures (c) and (d) one can already see the formation of a pair of eddies, just as in one of the pictures on page 377 . So what happens if one increases the speed of the flow? Does one see the same kinds of phenomena as on page 377 ?

Traditional science tends to suggest that allowing more than one dimension will have very important consequences. Indeed, it turns out that many of the phenomena that have been most studied in traditional science simply do not occur in just one dimension. … It could be that in going beyond one dimension the character of the behavior that we would see would immediately change.

But when one looks at actual systems in nature, it turns out that one often sees discrete behavior—so that, for example, the coat of a zebra has discrete black and white stripes, not continuous shades of gray. … If one takes some water and continuously increases its temperature, then for a while nothing much happens. … One might think that if the changes one makes are always continuous, then effects would be correspondingly continuous.

If one looks at air passages or small blood vessels in higher animals then the patterns of branching one sees look similar to those in plants. … One might have thought that, like a stem in a plant, a horn would grow by adding material at its tip. … These pictures can be viewed as one-dimensional analogs of those on the facing page .

But how can one tell if this is so? … And if one finds that a value computed from a particular sequence lies close to the average for all possible sequences then one can take this as evidence that the sequence is indeed random. But if one finds that the value lies far from the average then one can take this as evidence that the sequence is not random.

If one looks at recent work in number theory, most of it tends to be based on rather sophisticated methods that do not obviously depend only on the normal axioms of arithmetic. And for example the elaborate proof of Fermat's Last Theorem that has been developed may make at least some use of axioms that come from fields like set theory and go beyond the normal ones for arithmetic. But so long as one stays within, say, the standard axiom systems of mathematics on pages 773 and 774 , and does not in effect just end up implicitly adding as an axiom whatever result one is trying to prove, my strong suspicion is that one will ultimately never be able to go much further than one can purely with the normal axioms of arithmetic.

On a two-dimensional grid one can certainly imagine snaking backwards and forwards or spiralling outwards to scan all the elements. But as soon as one defines any particular order for elements—however they may be laid out—this in effect reduces one to dealing with a one-dimensional system.