If nodes can have more than three connections, then they will often be able to evolve to have any number of connections—in which case one must give what is in effect an infinite set of rules to specify what to do for each number of connections.

Monocotyledons—of which palms and grasses are two examples—typically have only one primary site of growth, and thus do not exhibit repeated branching. … Note that in the vast majority of cases, branches do not lie in a plane; often they are instead arranged in a spiral, as discussed on page 408 .

It turns out that this particular system always evolves to a fixed configuration, but for initial conditions of size n can take roughly n iterated powers of 2 (or 2 2 2… ) to do so.

A dot signifies that the rule does not change the color of the center cell in the neighborhood.

And to do so it was necessary to abandon concrete intuition, and instead to rely on abstract mathematical theorems.

In 164 t[x] does not increase with the number of digits n in x , in 295 it increases linearly, in 27 quadratically, and in 6 exponentially. … Among the 2,985,984 Turing machines with s = 3 , k = 2 , at least 2,550,972 sometimes halt, and about 1,271,304 always do. … Among the 2 32 Turing machines with s = 4 , k = 2 about 80% at least sometimes halt, and about 16% always do.

But as discussed in the main text I strongly suspect that in fact there are no such global constraints, and instead almost all reasonable values of parameters from the simple model used do actually occur in real molluscs. … Most of these models have far more parameters than the simple one used here, and by varying these parameters it is almost always possible to get forms that probably do not correspond to real shells. And presumably the reason for this is just that such models represent processes that do not occur in the growth of actual molluscs.

Statistical methods [for texture analysis] Even though they do not appear to correspond to how the human visual system works, statistical methods are often used in trying to discriminate textures automatically.

But in fact, much as in the case of time travel, such connections do not represent additional observable effects; they simply imply consistency conditions, in this case between universes whose paths converge.

For most types of systems (such as Turing machines) such non-deterministic versions do not ultimately allow any greater range of computations to be performed than deterministic ones.