And from this it follows for example that even the most circuitous path between two nodes must be of finite length.

• Is there a path shorter than some given length that visits all of some set of points in the plane? … (See page 984 .) • Is there a string of some limited length that solves a correspondence problem?

With distance on a network defined as the length of shortest path between nodes one immediately gets all three of these properties.

The basic idea is to take a lump of dough-like material, and repeatedly to stretch this material to twice its original length, cut it in two, then stack the pieces on top of each other.

At a basic level, one knows that snowflakes are formed when water vapor in a cloud freezes into ice, and that the structure of a given snowflake is determined by the temperature and humidity of the environment in which it grows, and the length of time it spends there.

One effect, to be discussed at greater length in the next section , is essentially just a matter of geometry.

Quite possibly there will sometimes be at least some correspondence between the lengths and angles that appear in the rules for overall growth and for the growth of leaves.

The existence of a fixed length scale at which such processes occur then almost inevitably implies that an embryo must develop in a somewhat hierarchical fashion.

Instead of taking each square to have a color that is chosen completely independently, one can for example take blocks of squares of some given length to have their colors chosen together.

And the crucial point that turns out to be the basis for much of the success of traditional theoretical science is that in fact most standard mathematical functions can be evaluated in a number of steps that is far smaller than the numerical value of their input, and that instead normally grows only slowly with the length of the digit sequence of their input.