## Monday, May 3, 2010

### It's impossible to say at this time

I suppose the title of this posting must be explained. We tend to think that math can give us concrete, specific answers to well-asked questions. Sometimes it can, but not always.

Here's an example - how long is the coastline nearest your house? Start at one border of your state or country, and go across (up/down) to the other border. What is the length?

You can't say. It depends. It depends on the tide. It depends on the length of your ruler. It depends on the size of the grains of sand or rocks you chose to include or exclude. It depends on how you treat rivers and estuaries and other coastal features. It depends on your definition of littoral.

Take a pair of dividers and step your way along a map of the coastline (as this man is doing in the etching from the Gutenberg library). You will find that the length of the coast is inversely related to the gap between the pointers of your dividers. To be more specific, as the distance between the pointers approaches zero, the length of your coastline approaches infinity. L.F. Richardson was the first to clearly state this problem - about 60 years ago.

How can this be? Can reality be changed? Do the dimensions really shift as we measure? No. Reality isn't changing, only the way that we measure it!

Here are a couple maps of England, Scotland and Wales that I found in Wikimedia commons. The first is measured with a 50km ruler and the second with a 100km ruler.  I've counted the segments to save you the trouble.

Measuring the coastline with a 50km ruler gives an estimated coastline of 3425km, while measuring with a 100km ruler results in a total of 2750km.

How can you use this in real life? I suppose it could be applicable to waistlines too.

If someone rudely asks your waist size, you might say "It's impossible to say at this time ..." and go on to explain the coastline paradox.