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.
Mathematically, this is based on the assumption of the infinite and the continuous.
However, quantum mechanics tells us that the world should be finite and discrete. Therefore, measuring the coastline of England naturally has nothing to do with limits. Of course, in practice, infinite measurement precision (not to mention the uncertainty principle) is impossible, so obtaining the "accurate" length of England's coastline—at least within the current framework of physics—is impossible, but that is another issue.
The meaning of "coastline" breaks down far before you reach quantum levels. I challenge you to go to the beach and pick a square inch where the "coastline" splits it half and half, this part England, that part the sea.
This is also ignoring the time component, where individual fluctuations in water will change the measurement, tides will as well on a daily and monthly cycle, and as years go by erosion and global climate change will shrink the land and grow the sea.
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?
I had a nightmare one night after studying Economic Philosophy. The Thesis, Antithesis, Synthesis struck a subconscious fear. My dream was lightning bolts splitting things up.
The Wales Coast Path[0] links up with it, so (aside from the small gap in Cumbria), you can walk all the way round if you start and end at the Scottish border.
There’s also the Hadrian’s Wall Path[1] that allows you to link back into a loop new the Scottish border, though it misses out a section of the English coast in the North East.
A section? All of Northumberland, as Wallsend is in Tyne and Wear.
The task of walking (or measuring) the whole of England's coastline is made a bit more difficult by the existence of various islands classed as part of England.
e.g. Isle of Wight, St. Mary's Island, Coquet Island, Lindisfarne, the other Farne islands, etc.
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.
I think the idea is to show we don't know the exact lengths of any paths that aren't constructed from our handful of mathematically known curves and must approximate using them instead.
If you measure with GPS coordinates, you still run into the same problem. The number of points plotted onto a curve affects the result, and then you are possibly also adding more error than you'd have compared to tracing from aerial photos.
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!?
This kind of comment always strikes me as the textual equivalent of the Youtube thumbnail with the big yellow characters and multiple people making an O face and pointing.
The article mentions who observed it in history first, Lewis Fry Richardson, and then links to an encyclopedia article about it which describes Mandelbrot. On top of that, Mandelbrot explicitly named his paper that because he is referencing Richardson.
"Practically plagiarizes the title" because it references the same original referent that Mandelbrot was referencing is really stretching it, mate. Honestly. It would be like calling a book review of Wuthering Heights by Bronte "practically plagiarizing" Fennell's 2026 movie by the same name.
That’s fair, the “practically plagiarize” comment was admittedly harsh and quickly written.It’s obviously an allusion to the paper.
Nevertheless, referencing the LFR work and merely including a link where the paper is discussed feels a little like beating around the bush. The primary article doesn’t seem to use the words “self-similar” nor “fractal” - if you’re only interested in LFR work, why reference Mandelbrot at all?
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.
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).
unless they have edited they do not say Wales and Scotland are England, they complain the trail is not contiguous, which to be fair might also be implied by the fact that Wales and Scotland are not England but I guess I could see a scenario that allowed made the trail contiguous, which would still allow you to walk the English coast, but also of course allow you to walk some other parts that are not on the coast.
I would assume now that they have added the Welsh path to this one that it is now "technically" contiguous, though not yet circular around the whole British mainland until someone does something about Scotland.
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.
It's in no way a meaningful solution. If you're settling for a resolution, you don't need a ball-rolling analogy. We already know the length of a given coastline at given resolutions (ignoring the constant changing of the coastline itself). What's practically not feasible is getting every country on earth to agree on the right resolutions. And that's for good reasons, because the desired accuracy depends on many factors, some situational and harder to quantify than just size of the enclosed land mass.
You don't need anyone else to agree on the resolution.
You can just pick one when you are doing some work that requires knowing the length of the coastline.
I wasn't trying to say that we should all agree on a universal definition and use that for everything? That would be insane. I was just providing a way to get a stable answer for the length of the perimeter of a fractal area.
Why would it be insane? We have globally accepted answers for the area of each country (modulo territorial disputes, geological changes and similar). One would expect the same thing to be possible for the circumference. So to most people it will be a surprise that it is, in fact, impossible. It is mathematically impossible because the problem is underdetermined, and it is practically impossible because agreeing internationally on how to fully determine the mathematical problem seems unrealistic.
Not a bad idea - one issue would be when the circle approaches a 'narrow' section that widens out again. If too big to fit into the gap, the circle method would simply not count any of this as land. I think it would be unreliable compared to moving along the coastline in fixed increments (IE one-mile increments or one-foot increments, depending on your goal)
Plank's length is an ok answer, but coast line reaches a steady state way before that. Nature only has approximate fractals.
Way before plank length you'll get the surface and line energies of the material interfaces dominating the total energy. Those tend to force very smooth and very discreet lengths.
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.
The article spends a lot of time acting like this is some intractable problem and that its intractability is the reason there is so much discrepancy between countries and agencies.
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)
In the case of Koch curve it’s 1.2619...
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.
Indeed, the Wikipedia page on fractal dimension[1] uses the coastline paradox[2] as a leading example.
Mathematically, this is based on the assumption of the infinite and the continuous. However, quantum mechanics tells us that the world should be finite and discrete. Therefore, measuring the coastline of England naturally has nothing to do with limits. Of course, in practice, infinite measurement precision (not to mention the uncertainty principle) is impossible, so obtaining the "accurate" length of England's coastline—at least within the current framework of physics—is impossible, but that is another issue.
The meaning of "coastline" breaks down far before you reach quantum levels. I challenge you to go to the beach and pick a square inch where the "coastline" splits it half and half, this part England, that part the sea.
This is also ignoring the time component, where individual fluctuations in water will change the measurement, tides will as well on a daily and monthly cycle, and as years go by erosion and global climate change will shrink the land and grow the sea.
Relevant 3blue1brown (20 min)
https://www.youtube.com/watch?v=gB9n2gHsHN4
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?
I had a nightmare one night after studying Economic Philosophy. The Thesis, Antithesis, Synthesis struck a subconscious fear. My dream was lightning bolts splitting things up.
> A new hiking trail will soon allow travellers to walk around England's entire coast
This is a strange concept as there are two countries that share a land border so how do they manage the “gaps”?
The Wales Coast Path[0] links up with it, so (aside from the small gap in Cumbria), you can walk all the way round if you start and end at the Scottish border.
There’s also the Hadrian’s Wall Path[1] that allows you to link back into a loop new the Scottish border, though it misses out a section of the English coast in the North East.
[0] https://en.wikipedia.org/wiki/Wales_Coast_Path [1] https://en.wikipedia.org/wiki/Hadrian%27s_Wall_Path
A section? All of Northumberland, as Wallsend is in Tyne and Wear.
The task of walking (or measuring) the whole of England's coastline is made a bit more difficult by the existence of various islands classed as part of England.
e.g. Isle of Wight, St. Mary's Island, Coquet Island, Lindisfarne, the other Farne islands, etc.
Similar to Hadrian's Wall, there's the Offa's Dyke Path which roughly follows the Welsh border all the way up.
Three countries - England, Scotland, Wales.
> there are two countries that share a land border [ with England ]
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.
I think the idea is to show we don't know the exact lengths of any paths that aren't constructed from our handful of mathematically known curves and must approximate using them instead.
If you measure with GPS coordinates, you still run into the same problem. The number of points plotted onto a curve affects the result, and then you are possibly also adding more error than you'd have compared to tracing from aerial photos.
Beginning of the 3rd paragraph agrees with you:
> But while the length of the newly designed path is easily measurable, the coastline that it follows is not.
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!?
https://www.researchgate.net/profile/Ion-Andronache/post/Wha...
This kind of comment always strikes me as the textual equivalent of the Youtube thumbnail with the big yellow characters and multiple people making an O face and pointing.
The article mentions who observed it in history first, Lewis Fry Richardson, and then links to an encyclopedia article about it which describes Mandelbrot. On top of that, Mandelbrot explicitly named his paper that because he is referencing Richardson.
"Practically plagiarizes the title" because it references the same original referent that Mandelbrot was referencing is really stretching it, mate. Honestly. It would be like calling a book review of Wuthering Heights by Bronte "practically plagiarizing" Fennell's 2026 movie by the same name.
That’s fair, the “practically plagiarize” comment was admittedly harsh and quickly written.It’s obviously an allusion to the paper.
Nevertheless, referencing the LFR work and merely including a link where the paper is discussed feels a little like beating around the bush. The primary article doesn’t seem to use the words “self-similar” nor “fractal” - if you’re only interested in LFR work, why reference Mandelbrot at all?
It is the BBC. They are so busy with propaganda, that any act of thinking, hurts.
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.
At which spot in time? High tide, low tide, king tide? The coastline is never even static over a single daylight period.
[dead]
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).
Assuming they specifically do advertise that you can walk the entire English coast, then it is accurate, as Wales and Scotland are not England.
unless they have edited they do not say Wales and Scotland are England, they complain the trail is not contiguous, which to be fair might also be implied by the fact that Wales and Scotland are not England but I guess I could see a scenario that allowed made the trail contiguous, which would still allow you to walk the English coast, but also of course allow you to walk some other parts that are not on the coast.
I would assume now that they have added the Welsh path to this one that it is now "technically" contiguous, though not yet circular around the whole British mainland until someone does something about Scotland.
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.
It's in no way a meaningful solution. If you're settling for a resolution, you don't need a ball-rolling analogy. We already know the length of a given coastline at given resolutions (ignoring the constant changing of the coastline itself). What's practically not feasible is getting every country on earth to agree on the right resolutions. And that's for good reasons, because the desired accuracy depends on many factors, some situational and harder to quantify than just size of the enclosed land mass.
You don't need anyone else to agree on the resolution.
You can just pick one when you are doing some work that requires knowing the length of the coastline.
I wasn't trying to say that we should all agree on a universal definition and use that for everything? That would be insane. I was just providing a way to get a stable answer for the length of the perimeter of a fractal area.
Why would it be insane? We have globally accepted answers for the area of each country (modulo territorial disputes, geological changes and similar). One would expect the same thing to be possible for the circumference. So to most people it will be a surprise that it is, in fact, impossible. It is mathematically impossible because the problem is underdetermined, and it is practically impossible because agreeing internationally on how to fully determine the mathematical problem seems unrealistic.
Not a bad idea - one issue would be when the circle approaches a 'narrow' section that widens out again. If too big to fit into the gap, the circle method would simply not count any of this as land. I think it would be unreliable compared to moving along the coastline in fixed increments (IE one-mile increments or one-foot increments, depending on your goal)
https://en.wikipedia.org/wiki/Roulette_(curve)
A radius of plank length is the only true answer
Plank's length is an ok answer, but coast line reaches a steady state way before that. Nature only has approximate fractals.
Way before plank length you'll get the surface and line energies of the material interfaces dominating the total energy. Those tend to force very smooth and very discreet lengths.
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.
The article spends a lot of time acting like this is some intractable problem and that its intractability is the reason there is so much discrepancy between countries and agencies.