So what this suggests is that even if in some idealized sense a system in nature might be expected to satisfy certain constraints, it is likely that in practice the system will actually not have a way to come even close to doing this.

So do all sufficiently large networks somehow correspond to ordinary space in a certain number of dimensions?

For certainly we do not notice any kind of active cell visiting different places in the universe in sequence.

But the particular kinds of systems I have discussed for both strings and networks in the past few sections [ 10 , 11 , 12 ] do have a certain locality, in that each individual replacement they make involves only a few nearby elements.

But in packing as much as possible into these notes I have often been unable to do this.

In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.

But almost certainly these structures have nothing to do with life, and are instead formed by ordinary precipitation of minerals.

So given, say, an ordinary piece of rock in which there is all sorts of complicated electron motion this may in a fundamental sense be doing no less than some system of the future constructed with nanotechnology to implement operations of human thinking.

This program always halts, yet it does not correspond to any possible value of i —even though universality implies that any program should be encodable by a single integer i .

And thus for example even though it may be possible to establish by a finite computation that a particular system halts, it will often be impossible to do the same for the negation of this statement.