Length of a Coastline

Question

When B. Mandelbrot's typical example of measuring the length of a coastline is referenced, they mention how at every scale the length increases. In pure mathematics, I can imagine this quite well-- the iterating function goes on forever (infinitely) and there are no restrictions to how small it can get. Do they mean it increases asymptotically? Would the "true" length then be the limit?

In practice, would this scaling factor not be so well defined? Can it be true that a coastline is always getting longer depending on how small of a measuring stick you use? I understand that these topics are frequently explained to non-mathematicians and a lot of the available literature is geared toward them, but that makes it more difficult to grasp. Please use however advanced language you need in order to make the discussion precise.

Answer 1

Maybe the best-looking example of this is the Koch snowflake:

The iteration does indeed go on forever, but there is no limit to the length of the curve! If you look carefully, the snowflake's perimeter increases by a factor of $\frac{4}{3}$ each iteration, so it tends to infinity.

Don't think of the size of the measuring stick. Think instead of errors in measurement of the length; at each size scale, you have some "imprecision" in your measurement of the curve. As you increase the precision of your measurement, "zooming in," you more accurately approximate the length of the curve, and the length of this rectification may tend to $\infty$.Here's a picture of a precision-increasing iteration:

For other examples of this, go to Google Maps, start in orbit and slowly zoom into some nice piece of coastline like the northwestern coast of Norway. In practice, of course, you find that when you zoom in sufficiently far, objects like coastlines cease to display fractal behavior, but fractals are still beautiful math.

Answer 2

There is no asymptotic limit to the length of a typical coastline.

At macroscopic scales (down to a metre or so), you can check this empirically, which is exactly what Mandelbrot did in his celebrated paper How Long is the Coast of Britain.

At microscopic scales, the phenomenon persists. Of course, in practice, it's not so easy to measure the length of a coastline at sub-millimetre scales $-$ the sea won't keep still, for instance. So to make things more precise, let us "freeze" the sea at a certain time, and imagine measuring the length of a beach more and more precisely. You can see that in the limit, this would involve inspecting each grain of sand with a microscope and following its contours more and more closely.

At sub-atomic scales, the problem kind of disappears, because the boundary between wet sea and dry land ceases to be well-defined. But this doesn't refute the principal.