Math CentralQuandaries & Queries


Question from Jalyn, a student:

I'm doing this math project comparing measurements and I want to know the perimeter of the Upper Peninsula of Michigan. Please Help me!


It is a tough question - so it is not a surprise that you are getting stuck!

One of the key observations about the 'perimeter' of a shoreline is that:
- it depends on the scale you measure on.
If you measure with miles, going point to point, then you get one measurement.
If you measure point to point with feet (and convert) it clearly is a bigger measurement, because all the little wiggles are longer than the straight line point to point.
If you measure on a scale of millimeters (and convert) it is even bigger.

It is a fractal. Check out some references on shore line, fractal using something like 'google'.

Walter Whiteley


It is not well defined! If you measure it from a very small scale map you will jump across many bays, etc, and get a shorter figure than if you measure it from a bigger map. And this process keeps on going; even a 1:1000 scale map will not show many little one-meter scale bends where the shoreline detours around rocks, so if you actually walk the shoreline you will walk further than the map suggests. Then get down on your knees and look closely at the land-water boundary, and imagine an ant trying to follow you. It would have to make many detours where you made a single straight step, and walk still farther. Even the ant's track would, under a microscope, not follow every tiny wiggle.

You will need to decide on the scale you want to work at and use (say) a piece of string or a wheel to measure it ignoring deviations below that scale. Any figure you find on the Internet or in an encyclopedia will also involve an assumption about scale, which may or may not be made explicit. Once again: there IS no single "scalefree" answer.

Good Hunting!

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