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.
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)
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.
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)
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.
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).
> But while the length of the newly designed path is easily measurable, the coastline that it follows is not.
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.
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...
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.
https://www.researchgate.net/profile/Ion-Andronache/post/Wha...
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).