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Is the Coastline of Britain Infinite?

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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
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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/
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Associated Topics:
High School Fractals

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