Let's put all this in perspective by looking at some actual numbers.
To make things simple, let's make the length of our initial segment be one meter.
Then we can look at a table which uses our formula for curve length.
That is, four raised to the level number, divided by three raised to the level number, times one (4^N/3^N)*1.
That is one meter (1 m) here.
You can see that as the level goes up, the curve length also goes up.
Why is that?
Well, the reason is that each time we're dividing the segment length by three,
but we're multiplying the number of segments by four,
so the number of segments is going up faster than the segment length is going down.
So by Level 100, while the segment length is extremely small,
on the order of ten to the minus forty-eighth power (10^(-48)),
that's a decimal place followed by forty-eight zeros before we get to any non-zero digits
but the number of segments has gone up astronomically to the order of ten to the sixty (10^60).
So the actual curve length is 3.1 trillion meters.
What that means in more familiar terms is that the curve length at Level 100 is
three billion kilometers (3 x 10^9 km) or two billion (2 x 10^9) miles.
Now that's just amazing.
Think for a minute what that means.
What that means is that even though we have the whole curve that fits into our meter length ruler,
it's able to squeeze in, via these little nooks and crannies, like we saw on the coastline,
an enormous amount of distance.
This is not Level 100--we really couldn't see the little nooks and crannies.
But at Level 100, the curve would be able to squeeze in about two billion miles within this meter length curve.
That's just astounding.
Of course, we don't see that much being squeezed into fractal structures in nature,
but it gives us some hint as to why nature prefers fractal structures.
It's an extremely efficient way of squeezing in a huge amount of material,
whether it be tree branches, or broccoli florets, or mountain landscapes,
into a small amount of space.
That is, that the curve is what is called "space filling."
There are many other examples of space-filling structures in nature such as
the veins, arteries, and capillaries that make up the blood transport system in the body;
the roots of plants that grow in the ground; and
the structures in the brain.
In all of these examples, a sort of fractal geometry is being used
to optimize the amount of material that can be squeezed into a small amount of space.
We'll hear more about this in the unit on scaling.