Why it’s impossible to measure England’s coastline

BBiraIgnacio about 5 hours ago 20 commentsRead Article on bbc.com

ZH version is available. Content is displayed in original English for accuracy.

https://www.researchgate.net/profile/Ion-Andronache/post/Wha...

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Lucasoato•about 4 hours ago

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...

interroboink•about 4 hours ago

Indeed, the Wikipedia page on fractal dimension[1] uses the coastline paradox[2] as a leading example.

  [1] https://en.wikipedia.org/wiki/Fractal_dimension
  [2] https://en.wikipedia.org/wiki/Coastline_paradox

linzhangrun•about 2 hours ago

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.

colechristensen•about 1 hour ago

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.

anonymars•about 2 hours ago

Relevant 3blue1brown (20 min)

https://www.youtube.com/watch?v=gB9n2gHsHN4

hleszek•about 3 hours ago

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.

ozyschmozy•about 3 hours ago

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.

sublinear•about 3 hours ago

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.

sota_pop•about 3 hours ago

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...

arjie•about 2 hours ago

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.

avazhi•about 3 hours ago

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.

alistairSH•about 5 hours ago

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).

kelnos•about 4 hours ago

Assuming they specifically do advertise that you can walk the entire English coast, then it is accurate, as Wales and Scotland are not England.

bryanrasmussen•about 1 hour ago

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.

jstanley•about 5 hours ago

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.

c7b•about 5 hours ago

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.

yathern•about 4 hours ago

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)

sublinear•about 2 hours ago

https://en.wikipedia.org/wiki/Roulette_(curve)

colkassad•about 5 hours ago

A radius of plank length is the only true answer

sjshdhs•44 minutes ago

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.