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_paradoxFirst, 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.
This is a strange concept as there are two countries that share a land border so how do they manage the “gaps”?
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
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.
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.
> But while the length of the newly designed path is easily measurable, the coastline that it follows is not.
https://www.researchgate.net/profile/Ion-Andronache/post/Wha...
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.
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?
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).
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.
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.
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.