Is the Coastline of Britain Infinite?

Date: 08/17/99 at 05:30:16
From: Puzzled
Subject: Coastline of Britain is infinite... Can't be!

According to fractal god Benoit Mandelbrot, the coastline of Britain 
is infinite. His theory is this:

Draw the simplest 2-dimensional shape of Britain, a triangle. This 
would be roughly the perimeter of Britain, but only very roughly. To 
make it more accurate, we'd have to increase the number of vertices. 
The more vertices, the more accurate. The more vertices, the longer 
the perimeter, since the perimeter has to increase every time a new 
vertex is added. Exactly circumscribing the coastline of Britain would 
mean encircling every rock, every tidal pool, and every pebble that 
happens to be on the coastline of Britain. It is quite impossible to 
do that; therefore the coastline of Britain is infinite.

I agree that encircling every pebble in the coastline is very hard, 
but not impossible. It might take a ling time, and no one would 
want to waste time doing that, but it is possible. There is not an 
infinite number of vertices, so the perimeter of the coastline cannot 
be infinite. 

But Mandelbrot is a fractal genius. Is his theory just too deep for 
normal people to understand?

Puzzled

Date: 08/17/99 at 12:58:15
From: Doctor Peterson
Subject: Re: Coastline of Britain is infinite... Can't be!

Hi, Puzzled.

A fractal is really only a mathematical concept, not something in real 
life, just as a point is something that doesn't exist in the real 
world (it has no size) but it can be visualized by imagining a very 
small dot. 

The coastline of an island is fractal-like, because when we look 
closer and closer we see smaller and smaller irregularities. If it 
stopped at the size of pebbles, you would be right: it would be 
entirely possible (though not feasible) to measure the whole coast. 
But if you use a microscope, you'll see irregularities in the pebble's 
surface, and with a yet smaller "ruler" you would have to go around 
each irregularity. Eventually you'd have to measure around each atom, 
and then around each electron. We don't know how far down you can 
actually go in size, so at this point math takes over from reality, 
and we imagine that the irregularities continue forever. If the 
irregularities follow the right kind of pattern, this means the 
"coastline" would be infinite. (It's not really true that it has to be 
infinite just because it has an infinite number of vertices; a circle 
can be thought of as a polygon with an infinite number of vertices, 
and it has a perfectly finite circumference.)

Mandelbrot is not really making a statement about the coast of 
Britain, but about the nature of fractals, using the coast as a 
familiar picture to illustrate what he is talking about. He's really 
talking about certain imaginary fractal curves that look like a 
coastline, but whose properties can be determined exactly by math, 
rather than needing actual measurement.

So you're right in what you said, but he's also right, because he's 
gone a step beyond the real world.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/