Next time you go to measure your waistline, just remember: have you really put on weight, or is it just the coastline paradox?
Of course, if you are using tape to measure your weight, you are using the wrong tool for the job.
Why?
Coastline paradox
101–110 of 111 posts
Re: Coastline paradox
#102Can you get a decent approximation by measuring displacement? Dump a bucket of water in the ocean, observe the level of sea rise, that tells you the volume of the island, and indirectly the area of the edge.
Consider a short portion of shoreline in Fourier space. By keeping the "DC" term constant (i.e. the integral), you have complete freedom to change all other terms. By increasing high frequency terms the length of shoreline can tend to infinity without changing the area of the island.
Re: Coastline paradox
#103I’ve always found this argument very cool (and obviously true in a mathematical sense) but not really ”practically” relevant. Like, the only reason you would be interested to know the length of a coastline is because you want to know how far a ship would need to travel, and that is pretty well-defined. Or, like, if you wanted to do a naval blockade, you would need to know how many ships you’d need to cover the entire…
what if you want to take a hike along the beach and you want to estimate how long it will take to walk?
This is well defined as your stride and ability to micro correct determine the distance
Re: Coastline paradox
#104Earlier quoted context omitted.
Not always. The boundary between VA and MD (states in the USA) is the Virginia shoreline of the Potomac River. Which has caused oddities over the years… At one time, gambling and liquor were illegal in VA, so casinos set up boats at the low tide line. Customers parked in VA and walked onto the casino barges from the Va side, but they were technically in MD. You need to follow MD fishing regulations even from VA shore…
My uncle once argued a case in the U.S. Supreme Court regarding Indiana & Kentucky's border, which was initially established in the middle of the river. The Ohio River has gradually shifted south since the border was established, so some parts of Kentucky are now on the "wrong" side of the river.
The former capital of Illinois is on the wrong side of the Mississippi nowadays
Re: Coastline paradox
#105Why don’t the people in charge of measuring just agree on some unit of resolution?
Re: Coastline paradox
#106Are there any definitions of length, or specifically perimeter length, that are more intuitive for comparison purposes, and practical, and actually used, perhaps separating smoothed path length from path roughness?
There's convex hull which is how you measure your body with a measuring tape. The tape straightens itself over any recesses like where your belly button is.
Re: Coastline paradox
#107I think I've heard this in reference to the difficulty of software estimation. The project details and potential problems are unknowns until you get to that specific part of the project. Even though theoretically you can see the "shape" and rough size of the project when zoomed out.
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Re: Coastline paradox
#108Earlier quoted context omitted.
Your impression is exactly right. The difference lies in something called smoothness . Loosely speaking, a smooth curve is one where if you zoom in far enough at any point, the curve starts to look like a straight line. You might have to zoom in a very long way, but as long as you eventually end up with something that looks straight, it counts as smooth, even if the overall shape is very wiggly. Loosely speaking, smo…
Great explanatory writing, thanks. > In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a single value -- incorporating more detail would change the length, but only by tinier and tinier amounts. Can you say something bout the exponential curve in this context? IIRC whatever level we zoom to we see the same curve, neither converging nor diverging. Is th…
I think you might be remembering something else -- an exponential curve is proportional to its own rate of change. Exponential curves are smooth in the way I described -- if you zoom in far enough, they look straight.
The part about being proportional to its own rate of change does make exponential curves very important in calculus (and thus in science in general). Sine waves also have this property (although in a more complicated way). This is why you see exponential decay and sinusoidal oscillation so much in physics.
Re: Coastline paradox
#109Earlier quoted context omitted.
The Wikipedia article states it's not really fractal.
It's not really a fractal in the sense that a fractal is a mathematical object, and a coastline lives in the real world. As your ruler gets smaller, it also becomes impossible to define what even is the coastline anymore - is it the wetted sandline? On a sufficiently small scale, where is the interface between land and sea? Thus at sufficiently small scale it does break down, but the fractal nature over a very broad…
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Re: Coastline paradox
#110I understand why the measurement of a coastline changes depending on how you measure it. I do not understand why it is important to measure coastlines. ¯\_(ツ)_/¯
Note that it can apply to borders too. > the Portuguese reported their measured border with Spain to be 987 km, but the Spanish reported it as 1214 km.
ok so why does it matter for borders?
the location of the border seems important, but not the length.