Some direct calculations of interactions between vortices have been done in the context of the Navier–Stokes equations, but the cellular automaton approach of page 378 seems to provide essentially the first reliable global results.

Whenever a general statement about a system like a Turing machine or a cellular automaton is undecidable, at least some instances of that statement encoded in an axiom system must be unprovable.

Given an integer a for which IntegerDigits[a, 2] gives the cell values for a cellular automaton, a single step of evolution according say to rule 30 is given by BitXor[a, 2 BitOr[a, 2a]] where (see page 871 ) BitXor[x, y]  BitOr[x, y] - BitAnd[x, y] and a is assumed to be padded with 0's at each end. … Note that it is potentially somewhat easier to construct Diophantine equations to emulate register machines—or arithmetic systems from page 673 —than to emulate cellular automata, but exactly the same basic methods can be used.

As on page 333 , it is common for randomness to add robustness—as for example in cellular automaton fluids, or in saccadic eye movements in biology.

When I started studying cellular automata in the early 1980s I was quickly struck by the difficulty of finding formulas for their behavior. … But the evolution of a cellular automaton was immediately reminiscent of other computational processes—leading me by 1984 to formulate explicitly the concept of computational irreducibility.

I suspect, however, that in fact the most important source of randomness in most cases will instead be the phenomenon of intrinsic randomness generation that I first discovered in systems like the rule 30 cellular automaton.