How can a finite area like Great Britain have an infinite length edge?
In: Mathematics
The more detailed a map of a shoreline is, the more intricacies of said shoreline are visible, and the longer it is when measured.
The paradox says that since the shoreline gets longer the more detailed your measurement is, then an infinitely detailed measurement would mean an infinitely long shoreline.
It’s a measure of *exactness*, and larger lines/units of measurement giving general, less exact “true” measure. The [wiki](https://en.wikipedia.org/wiki/Coastline_paradox) article actually does a great job of describing it already.
How *exactingly* do we measure the coastline–and what units do we use (kilometers/miles, meters/yards? Centimeters/inches)?
With a surveyor’s device, from mile to mile (or kilometer to kilometer)? It’s one length.
Do we send a guy with a tape measure (or one of those wheeled distance-measuring devices), and tell him, at some arbitrary point of the day every day, walk out from *this point*, and “hug the coastline” (which is *always* changing due to tidal forces and erosion, *natch*), and measure it *that way*? You’ll get *another* total.
Do we send someone out with a *flexible* tape measure that you use to measure fabric–the kind that you can roll up? So they can “hug the coastline” (ha!) You’d get an even larger, different total length.
Like Pi, depending on the level of exactness you demand, it could be a never-ending total. Add to this that the *thing you’re measuring* is changing, *while you try to measure it*.
It turns out it’s quite easy to define a shape that has finite area but infinite perimeter, by giving it finer and finer details at smaller scales (shapes that have finer details at smaller scales like this are called “fractals”). A simple example is the Koch snowflake.
A coastline works a little bit like this. If you calculate the length of a coastline from a world map, you will miss out lots of bays and inlets. If you use a larger scale map, you will take those into account and get a larger value (often much larger) but you will still miss some smaller features. However, once you get down to the smallest length scales, the coastline becomes quite hard to define because of tides and waves. So it’s not exactly a fractal, but it works a little bit like one.
One thing to remember is that in real life this doesn’t really go to infinity because at smalls scale waves are constantly modifying what counts as the coast line and even if you froze time, you wouldn’t be able to go below their scale and any fractals would disappear.
The reason it’s a “thing” is because maps got more and more precise and they needed an explanation for why they kept measuring things longer and longer. But you’re right that it’s an arbitrary distinction that would apply to anything. Of course there can always be a more exact measurement that would involve smaller and smaller incremental differences.
Beyond the immediate need for an explanation from mapmakers and mathematicians, a coastline is also just a good example to use because people can easily conceptualize a coastline as a fractal shape.
Imagine you and a friend are standing on a beach. You and that friend want to measure how long the beach is.
Both of you decide to measure the beach in terms of stride lengths; that is, you move your foot forwards. You agree on a rule that as each of you moves on the beach, your right foot must get wet and your left foot remain dry.
You decide to move forward 20 centimetres with every step. Your friend moves forward 10 centimetres with every step.
You will notice that your friend has to do much more walking to keep their feet on the edge than you do. (*See* [*this*](https://www.researchgate.net/profile/Terry-Marks-Tarlow/publication/268803624/figure/fig3/AS:667862427521034@1536242219962/The-Coastline-Paradox-illustrates-that-shorter-measuring-units-create-longer-coastlines.ppm) *for a graphical illustration*).
As a result, your friend measures a longer distance than you. Who is right? As it turns out, there is no clear answer.
**The shorter the units of measurement, the more small features matter in the length, and thus the longer the beach appears to be**.
I think you’re getting hung up on the infinite part, at some point the laws of physics would prevent us from measuring any further but conceptually we can always divide a number in half. It could be physically impossible to measure 1/100,000,000,000,000,000,000,000 mm but it doesn’t mean it can exist on paper.
I guess you could describe the area as how much paint it would require to “fill” a shape while perimeter would be the effort it takes to “draw” said shape. Say you had a square that was 10×10 meters. Now say you cut out a 1×1 meter square from one side and pasted it to another. Your area remains 100 m, you have not technically added anything that wasn’t already there, but your perimeter has increased from 40 m to 44 m. You can do this indefinitely with smaller squares but the area will always remain the same.
When measuring a coastline, the little nooks and crannies can drastically increase the perimeter of your coast, sometimes increasing it multiple fold. Obviously in real life, you can only measure to a certain degree, but “coastline paradox” probably stuck better than “fractal curve paradox”.
Coastlines are like a fractal curve, the length of a fractal curve is infinite.
Of course in the real world it’s not infinite, just like a drop of water doesn’t have a singularity in the moment in drops off from the surface of a faucet. But the length of a coastline is an absurdly large number when you count the distance between all atoms on a coastline.
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