How do you parse it?
Does that mean, “Would fill the universe,” to you? Why?
Does, “Impossible to measure,” mean, “Would fill the universe”?
Does an infinite one dimensional object necessarily fill the universe?
Nah…this is your question to answer. Don’t turn it back on me.
The sticking point is that no one has said that the actual real world coastline is infinite, yet he continues to act as though that is something that is on those in this discussion to prove.
As can be seen by the Mandelbrot Set. The boundary of it is infinite, yet fits quite comfortably in finite space.
That doesn’t really mean what you meant it to. An infinite extent one dimensional thing in 3D Cartesian space is a straight line of infinite length, and it cannot fit inside the bounds of a finite 3D part of the same space.
But overall the distinctions make no sense. Lines have no area or volume. None. It doesn’t make any sense to talk about lines filling anything that has volume or area, the idea has intrinsically no meaning. An infinite line still has neither area or volume.
You may as well argue about angels on a pin.
But in the real world you cannot have infinitely small things which you get to with fractals. Coastlines, being a real thing in the physical world, cannot be measured on an infinitely small scale (indeed, they do not have infinitely small parts). There are limits.
An infinite straight line is one possible thing of infinite length. But it’s not the only one, and there are plenty of others that do fit just fine inside (let’s say) a rectangle of small, finite dimensions.
A one dimensional object does not have to be a line. It just has to have no width or height. A curve, as plotted by x=y^2, is a one dimensional object.
It may curve in two dimensions, but it in itself is still one dimensional.
A line is defined as a one dimensional object that is straight, but not all one dimensional objects are lines.
Edit: Ninja’s be Chronos.
All of that only works on paper.
You can’t do any of that in physical space.
I think this comes down to what we are using dimension to mean. I would not consider an arbitrary curve in 3D space to only occupy one dimension no matter what choice of basis we used. But we are likely crossing up definitions, so I’m happy to let the criticism stand.
I would argue that a Knopp–Osgood curve, e.g., does precisely that.
If you do not define some notion of “dimension”, then of course there is not much to say about it.
Speaking of physical space, is it really known what it “really” looks like at small scales? There are certain mathematical models in use, of course.
My understanding is the limit is the Planck length and up-thread it was said that measuring below the size of an atom is not even doable (and the Planck length is way below that). Either way, there are limits in the real world. It is only turtles part-way down.
I think this is the statement of yours that people keep pushing back against. You seem to be insisting that it is impossible for a finite area (“Norway, a place of finite size”) to have an infinitely long border (“an infinite coast”). Mathematically, this is possible, as has been demonstrated in this thread. (But you don’t even need fractals to see how “infinite” depends on what dimension you’re talking about. Mathematically, a finite area, or a finite line segment, contains infinitely many points.)
Then you insist that you’re talking about the real Norway in the real world, not a mathematical abstraction, and that the real Norway can’t have an infinite coast. Maybe not, but the reason it can’t is not that it occupies a finite area.
The OP asked about “physical coastlines”. A real thing. The departure into abstract notions is the problem.
Tell that to this guy.
I think @Francis_Vaughan and I are on the same page on this point.
You can do it in physical space, just not with physical objects.
At least, probably. There’s some speculation that space itself might be quantized, but there’s no real evidence for that beyond some physicists saying “Hey, wouldn’t it be cool?”.
As for a curve being “one-dimensional”, the usual definition for that is that you can uniquely specify a point on the curve by specifying a single number (such as, how far it is from the end).
Huh?
You can do it but you can’t? 
If you are not using physical objects then you are not doing it in the real world.
I have no doubt you can write out an equation that describes it but that is not the same as actually doing it.
Now I am seeing why experimental physicists have a problem with theoretical physicists.
Right, with equations, not objects. But those equations can describe the real physical space we live in, rather than just an abstract X and Y axis, if you’d like.