I was thinking about this recently, the way to do is to define a radius, and then imagine rolling a circle of that radius around the outside of the coastline (or around the inside! Define that as well) and then take the length of the equivalent track that never leaves contact with the circle.
So you get a different length depending on the radius you choose, but at least you get an answer.
You could define the radius in a scale-invariant way (proportional to the perimeter of the convex hull of the land mass for example) so that scaling the land mass up/down would also scale our declared coastline length proportionally.
tl;dr - for the same reason as any other coastline or complex border.
Also, it annoys me that the trail in question is advertised as allowing one to walk the entire English coast - but fails to mention Wales and Scotland are in the way (the trail is not contiguous).
This article says that by using a smaller unit of measure, the measured coastline increases.
The concept of dimension in fractals is backed by a similar idea! Take the Koch curve for example, at any iteration it gets longer and its 1-dimensional length loses the usual meaning because it diverges to infinity as you continue iterating. Intuitively the fractal dimension allows you to calculate how fast the measurement increases as the scale to measure it gets smaller.
In a more precise way, for most self-similar fractal made of N copies of itself, each scaled by factor r, the dimension is defined as:
D = log(N)/log(1/r)
That is the article. There’s no article, just a rehash of the coastline paradox. All while missing most interesting parts. The Wikipedia article is a much better exposition https://en.wikipedia.org/wiki/Coastline_paradox).
First, it can’t have an exact length because it’s not a static thing, but a process.
Second, as opposed to fractals that can be zoomed in forever, our measurements seem to hit some limits, so the reality is not quite like fractals in this sense.
Of course, it all makes sense if you think about it. What’s perhaps more interesting is we can’t “really measure” anything absolutely and the whole idea of absolute measures becomes rather tricky once you get to physics. In fact it gets philosophical and disputed and you realize that nothing is quite certain, nor quite agreed upon.
I think Quanta Magazine does a good job making justice to these things though.
I completely understand why measuring the length of coastlines is not possible but surely measuring a trail should be doable quite easily, you could simply use a gps tracker and it would be precise enough.
How on earth can you write an article that practically plagiarizes the title, mention the paradox, and neither mention mandelbrot nor cite the original paper anywhere!?
I guess I could understand why this would be borderline impossible if you did it manually, but surely today with satellite images and computer vision it really shouldn’t be that difficult to agree on a standard unit and then just automate it. Surely just make the scale human at its smallest (meters works and can get converted from there, assuming you have sufficient zoom level data for the coastline) and call it a day - I have no clue why we are discussing atomic fractal calculus approaching the limit as if that's a real problem for agencies trying to give a cogent answer about a particular country's coastline.
The article is not about practical measurements at all. Doing it manually has nothing to do with it. It is explicitly about why the measured length depends on the precision you choose to measure with.
Side anecdote, as a kid growing up I watched a documentary about the coastline problem on the BBC and I started thinking about the paradox and infinity and it made me incredibly scared and unwell and then threw up. Has anyone else ever experienced this?
Bullshit.
Tedious, bullshit.
The only rational deifinition of a coastline is that it would be a rough average measurement, as nothing else is possible.
Very large areas are washed away daily, and in other places the wind and tide deposit a wide variety of material that does in fact become dry land repleet with trees.
The daily shift is likely as much as several miles in storm season.
My advice is to put your maths, away, and go for a walk, and contemplate that which will not conform to any or in fact ALL compuational power that exists.
13 comments
[ 4.4 ms ] story [ 34.5 ms ] threadSo you get a different length depending on the radius you choose, but at least you get an answer.
You could define the radius in a scale-invariant way (proportional to the perimeter of the convex hull of the land mass for example) so that scaling the land mass up/down would also scale our declared coastline length proportionally.
Also, it annoys me that the trail in question is advertised as allowing one to walk the entire English coast - but fails to mention Wales and Scotland are in the way (the trail is not contiguous).
The concept of dimension in fractals is backed by a similar idea! Take the Koch curve for example, at any iteration it gets longer and its 1-dimensional length loses the usual meaning because it diverges to infinity as you continue iterating. Intuitively the fractal dimension allows you to calculate how fast the measurement increases as the scale to measure it gets smaller.
In a more precise way, for most self-similar fractal made of N copies of itself, each scaled by factor r, the dimension is defined as: D = log(N)/log(1/r)
In the case of Koch curve it’s 1.2619...
First, it can’t have an exact length because it’s not a static thing, but a process. Second, as opposed to fractals that can be zoomed in forever, our measurements seem to hit some limits, so the reality is not quite like fractals in this sense.
Of course, it all makes sense if you think about it. What’s perhaps more interesting is we can’t “really measure” anything absolutely and the whole idea of absolute measures becomes rather tricky once you get to physics. In fact it gets philosophical and disputed and you realize that nothing is quite certain, nor quite agreed upon.
I think Quanta Magazine does a good job making justice to these things though.
https://www.researchgate.net/profile/Ion-Andronache/post/Wha...
This is a strange concept as there are two countries that share a land border so how do they manage the “gaps”?