How would you explain the "coastline paradox" to a non-mathematician?
It seems to trip some people out when I tell them coastlines are not possible to measure.
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Answers ( 2 )
This is a classic "a picture is worth a thousand words" situation, so I'd point them to something like this:
https://www.sketchplanations.com/post/183170271181/the-coastline-paradox-this-is-the-fascinating
The key point is that the more closely you try to measure a coastline, the longer the length turns out to be. The reason is that if you lose a long ruler to do the measuring -- think, like, a mile long -- then you have to approximate each mile by skipping over all the little "nooks and crannies" and curves and bends and whatnot, because your ruler is still straight. If you were to use a meter stick, you'd get a much more crookedy measure over that same mile, and crookedy always means "longer than a straight line". And if you use a foot rule, it'd be crookeder still, because a foot is shorter than a meter.
And you can go right down the rabbit hole, always finding more and more little bends and turns that need to be measured. In the real world, you meet a limit when you say "I don't have a ruler short enough to measure around this grain of sand!", but in mathematics, there's no limit to how small a length can be -- and thus, how long the perimeter of an object (it's "coastline") can be.
The point is ultimately that you can fit an arbitrary amount of arclength into a finite amount of area.
A comparable fact which people seem more used to accepting is the fact that there are something like a hundred thousand miles of blood vessels in the human body, despite the fact that the average human only takes up a couple of cubic feet. (In metric: over a hundred thousand kilometers of blood vessels in a body that takes up a tiny fraction of a cubic meter.)
In that case we're talking about packing a bunch of arclength into a small volume, rather than a small area, but the principle is the same; a constraint on the available 2- or 3-dimensional space doesn't constrain the available amount of 1-dimensional space.
The fractal nature of a coastline isn't of much importance (except maybe to someone trying to measure one), but the general principle that you can pack a lot of wiggle into a small amount of space gets used all over the place in nature. It's why lungs have a lot of complicated branching structures inside them, and why the endoplasmic reticulum inside a cell consists of a bunch of folded layers.