Why it’s impossible to measure England’s coastline

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

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.

Relevant 3blue1brown (20 min)

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

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.

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.

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.

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

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.

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