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...
[1] https://en.wikipedia.org/wiki/Fractal_dimension
[2] https://en.wikipedia.org/wiki/Coastline_paradox> 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.
https://www.researchgate.net/profile/Ion-Andronache/post/Wha...
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.
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).