it has a limited application to the length of coastlines - the paradox is about getting the measurement infinitely small, which has NO applications in Joe Lunchbucket’s real world. If you’re worried about measuring the coastline of Norway, you’re not going to use measurements infinitely small.
Velocity part right, there is a finite upper bound for the increase in length.
There’s a finite number of fractal layers. Theres a 10km strip (as crow flies measured.) of coast with the longest true distance somewhere out there.
Velociity wrong to say that the largest increase is 10% … 50 times ?? 100 times ? it might be rather large… even if you measured the coast line with a tolerance of 1cm , you will get a far shorter distance than if you measure it with a tolerance of 1 mm,. its impossible, and poorly defined, to measure the shape of molecules, atoms, so there’s going to be a measurable minimum scale to work on , which prevents this length dilation occurring infinitely, in practice.
So the original point could do with some clarification.
are we dealing with actual measuring things, which neccessarily is limitted by as much as we know of subatomic particles ?
Or are we considering the each subatomic particle may be made of parts, and so the layers of fractals never ends ?
Ah, I see where you go wrong: That’s not the actual paradox even if the OP thought it was. The paradox is that there is no single answer to “how long is the coastline of”.
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length.
Coastline paradox - Wikipedia
It’s a matter of pride for some countries to have longer coasts and beaches. It does have applicability in how many boats you can beach or anchor off the coast, or how many beachgoers can be laid out at a time.
@MrDibble - you triggered a new way of looking at things for me.
So, as a Chemical Engineer, we often have to analyze/characterize catalysts by looking at the surface area (both outside and inside pores ) and the porosity of catalysts. The surface area is usually measured by adsorbing Nitrogen into the pores, but the measurement changes if bigger molecule like Argon or a smaller molecule like Helium is used.
I never put 2 and 2 together until your post. That it is indeed fractal nature changing the surface area measurement depending on the size of the molecule.
There’s two effects that mean theres no one number for “length of the coast”.
One is the random methology of deducing what the “best path” is, even if you limit yourself to a maximum distance from the coast of 100 metres… it would be tedious to define this.
Imagine reading the distance off a map with a measuring wheel system. well some people turn faster when they over land, than when they over water. Some turn faster when they are cross from water to land then when they cross of land to water. There’s various errors where they are not allowing the maximum tolerable distance at some place for no good reason.
If you physically lay a 10 metre stick down, and flip it end to end , well there’s 10000 mm where you might lay it at first… so there’s 10000 different distances, if you could somehow precisely position the stick to any of those 10000 mm’s.
The other is the fractal effect, where you see more to measure
I think the lack of definition is a systematic measurement error.
The coast paradox is the reason we don’t bother trying to work on definitions,and we make regular shapes or simple shapes like ovals and circles, when we do want a reasonably accurate length… rather than defining how to measure random shapes, we work with “well behaved” shapes in practice.
If it were just a matter of measurement error, then there wouldn’t be a problem. Everything has measurement error, and so you just devise a method that has small enough measurement error for the application you’re working on. At worst, occasionally someone needs to devise a more precise method, for when they need less error. And if, by chance, you happen to have a really precise measurement, but you only need a rough, coarse measurement, well, that’s just fine.
But the fractal effect is nothing like that. Different scales lead to completely different measurements. You can’t use a measurement designed around how many boats you can dock to determine how many beachgoers can lie down, nor can you use the beachgoer measurement for the docked boats. They’ll be completely different numbers. Nor will either of them tell you how far your Coast Guard boats will need to go, to patrol the entire coast.
So you can have an ever-increasing number of ever-smaller boats at your ever-more-fractal docks. 
Point 3 is what I am most interested in - so are you saying it is possible for it to appear fractal when going from large scales to smaller scales, but can be mathematically modeled to a finite length at small scales? Are there any natural phenomena in general where things appear to be fractals, but actually aren’t when examined closely enough? Like is it possible that you get increasingly longer coastline lengths from 10km → 1km → 100m → 10m, but then diminishing returns from 10m → 1m → 10cm → 1cm and minimal differences from 1cm → 1mm or smaller, so that it does start to approach a finite length as the scale gets smaller instead of going to infinity? Wondering if the change in “fractional dimensionality” is something that can be plotted - I tried reading the Wikipedia article on fractal dimensions and unfortunately it is beyond my comprehension.
I think it can still be described as fractal even if the fractal dimension is less than 1?
If you measure the 2-dimensional area of a straight line segment, you get zero. If you measure the 0-dimensional count of how many points are in the segment, you get an infinite number. One way of looking at a “fractal” is that the Hausdorff dimension, where the measure changes from infinity to zero, is not an integer. Though some curves may have an integral Hausdorff dimension and still be reasonably described as “fractal”, so I am not proposing that as a formal definition.
All natural fractals fall into that category, because there’s always some real smallest scale beyond which the fractal nature breaks down. If nothing else, the atomic scale, but they usually break down long before that.
Two of my favorite fractals, the dragon curve and the Sierpinski tetrahedron, both have dimension of exactly 2. Though the boundary of the dragon curve has dimension of about 1.52 .
You misunderstand my confusion.
In the physical world, can you fit an infinite anything into a finite anything?
I would be more restrictive. In the physical world is there an infinite anything? I would suggest there is not.
There is handwaving metaphysical invocation in infinities, but nothing that suggests an actual mathematical infinity has any meaning.
I think this is the real sticking point.
There’s no definition of anything that I’m aware of that states that an infinite one dimensional object can’t fit in a finite three dimensional space. What definition are you referring to?
It started in the OP:
That doesn’t answer my question.
You said, “Explain how Norway, a place of finite size on a planet of finite size has an infinite coast which, by definition, would fill the universe.”
What definition are you referring to?
The dictionary:
- limitless or endless in space, extent, or size; impossible to measure or calculate.
That definition doesn’t say anything about filling the universe.
Huh?
How do you parse:
“limitless or endless in space”?