I think I've heard this in reference to the difficulty of software estimation. The project details and potential problems are unknowns until you get to that specific part of the project. Even though theoretically you can see the "shape" and rough size of the project when zoomed out.
Yes, there's a classic blig post (sorry, no bookmark handy) about an analogous trip from SF to LA.... Also, (maybe counterintuitively), in software estimation the more fine-grained the estimates, the more likely they are to _underestimate_ project scope. For areas that are better-understood, the estimate will be (properly) limited in scope (reducing the overall buffer) -- which leaves less margin for error in the are…
Coastline paradox
71–80 of 111 posts
Re: Coastline paradox
#72Next time you go to measure your waistline, just remember: have you really put on weight, or is it just the coastline paradox?
Interestingly there is no paradox for convex shapes, like measuring tape forms.
But isn’t the problem that the ‘object’ formed by the measuring tape/rulers isn’t representative of the (most likely) concave object you’re trying to measure.
Re: Coastline paradox
#73I’ve never taken calculus (the last math class I took was an algebra class in high school in the late 1980s). But I read a description of calculus once that used the example of trying to figure out the length of a coastline and described how one could use smaller and smaller sections to estimate the length and this is what calculus is all about. But, this article doesn’t mention calculus at all, so now I’m wondering…
Your impression is exactly right. The difference lies in something called smoothness. Loosely speaking, a smooth curve is one where if you zoom in far enough at any point, the curve starts to look like a straight line. You might have to zoom in a very long way, but as long as you eventually end up with something that looks straight, it counts as smooth, even if the overall shape is very wiggly. Loosely speaking, smooth curves behave well when you use calculus on them.
The coastline paradox is a situation where that doesn't happen. No matter how far you zoom in on a part of the coastline, you always find more detail -- bays and peninsulas, cliffs and inlets, pits and protrusions, tiny bumps, and so on. And the more of that detail you try to include in your measurement, the more your measured length will increase. In mathematics, we say that the length diverges. If you were measuring a smooth curve, the length would converge on a single value -- incorporating more detail would change the length, but only by tinier and tinier amounts.
Now of course a real coastline is a very complicated thing, and there are lots of practical obstacles to measuring one very precisely. So at some point we stop talking about real coastlines and start talking about a mathematical ideal, a kind of perfect roughness that's the opposite of the perfect smoothness we use for calculus. Such a perfectly rough curve is called a fractal.
There are also other things that can break calculus -- pointy curves, discontinuities, and infinities can all cause trouble.
Re: Coastline paradox
#74Earlier quoted context omitted.
Do this with Gabriel's horn and you'll observe that it has an area of pi, which does not reflect the fact that its perimeter is infinite.
Yeah, I'm not sure how I concluded you could go from volume to area.
It does seem intuitive that you'd be able to, I can certainly understand how you'd think that. But no!
Re: Coastline paradox
#75Earlier quoted context omitted.
as the crow flies at the speed of a bullet
Do bullets fly for minutes? (Edit: to be clear, that's a genuine question. I would imagine they drop to the ground before the "three minutes from the beach" have passed no matter the firing angle, but I don't know.)
Bullets take the same amount of time to fall as if you dropped it!
Re: Coastline paradox
#76Earlier quoted context omitted.
Do bullets fly for minutes? (Edit: to be clear, that's a genuine question. I would imagine they drop to the ground before the "three minutes from the beach" have passed no matter the firing angle, but I don't know.)
Bullets take the same amount of time to fall as if you dropped it!
Not exactly:
https://en.wikipedia.org/wiki/Magnus_effect#In_external_ball...
can make ULR bullets drop faster or slower depending on crosswind direction and strength.
Re: Coastline paradox
#77Earlier quoted context omitted.
The Wikipedia article states it's not really fractal.
It's not really a fractal in the sense that a fractal is a mathematical object, and a coastline lives in the real world. As your ruler gets smaller, it also becomes impossible to define what even is the coastline anymore - is it the wetted sandline? On a sufficiently small scale, where is the interface between land and sea? Thus at sufficiently small scale it does break down, but the fractal nature over a very broad…
On a sufficiently small scale, I guess you would have to start worrying about quantum effects. So, you would still have trouble measuring the coastline maybe not a paradox anymore but still quite a pain.
Re: Coastline paradox
#78The coastline paradox relies on the assumption that the coast is equally “jagged” at all levels. Is this actually true? I mean, if I go to the beach and look at the line where the water meets the land it doesn’t really seem like a fractal to me. The outer boundary of land masses gets smoothed out by erosion from the water, right?
If you look close enough the sand is pretty jagged I guess.
Re: Coastline paradox
#79I’ve always found this argument very cool (and obviously true in a mathematical sense) but not really ”practically” relevant. Like, the only reason you would be interested to know the length of a coastline is because you want to know how far a ship would need to travel, and that is pretty well-defined. Or, like, if you wanted to do a naval blockade, you would need to know how many ships you’d need to cover the entire…
The interesting thing is that while there is a 'practical' length, it varies depending on the application. So if you want to put a wind turbine every 1km along the coast, and a lamp post every 100m along the coast, the number of lamp posts you need won't be 10 times the number of wind turbines. It will be more.
Re: Coastline paradox
#80Earlier quoted context omitted.
No since water boundaries like rivers are defined as the middle of the river, not as the shifting coast of either side of the river.
genuine question: how is the middle defined when both shores are fractals?
Lawyers don't really think about these things in the same way that mathematicians do:
https://en.wikipedia.org/wiki/Thalweg#Thalweg_principle
> The Treaty of Versailles, for example, specifies that "In the case of boundaries which are defined by a navigable waterway" the boundary is to follow "the median line of the principal channel of navigation."