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Coastline paradox
61–70 of 111 posts
Re: Coastline paradox
#62The 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?
You're taking the term "Coastline" literally. It's an abstract idea, and a coastline is a good way to conceptualize it as a real thing.
Re: Coastline paradox
#63I’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…
what if you want to take a hike along the beach and you want to estimate how long it will take to walk?
This depends on how big you are on a scale of “quark” to “ship”.
Re: Coastline paradox
#64Earlier quoted context omitted.
Calculus deals with things that are sufficiently smooth on a small scale, and thus converges. For example, approximating a circle with smoother and smoother polygons. The length of those polygons will converge to the circumference of the circle. The coastline paradox is the classic example of a fractal; the dimension of a fractal is actually greater than that of the space it lives within!
Unless when it doesn’t, see the problem of “squaring the circle”. Somewhat related to the coastline paradox as well.
I think what you're referring to is the false proof suggesting that pi=4. That is not what squaring the circle is, and does not in fact become smoother and smoother but rougher and rougher
Re: Coastline paradox
#65Earlier quoted context omitted.
> it doesn’t really seem like a fractal to me. That's because you don't look close enough.
The Wikipedia article states it's not really fractal.
It actually bottoms out pretty low though. The apparent fractal dimension changes over scales, but is sufficient that ruler length really does make a huge distance in coastline length
Re: Coastline paradox
#66Earlier quoted context omitted.
This unfortunately doesn't quite work, because while there's no well defined upper bound, there is a lower bound.
true, but you could just argue that any increase was just a result of over-estimating the last time you measured
The lower bound is generally very easy to measure precisely. So a change in that would be immediately noticed.
For example, how long is the North Carolina coastline? No idea. But we can measure the precise distance between the point where that state's border with South Carolina meets the ocean, and the point where the border with Virginia meets the ocean. The coastline must be at least this long.
And if you measure your waistline, you can draw a cord around your midsection with some known amount of force. Your true waistline may be larger, but it is at least that large. And if that number increases...well, we're in a period of relatively high inflation already, what's one more thing?
Re: Coastline paradox
#67I understand why the measurement of a coastline changes depending on how you measure it. I do not understand why it is important to measure coastlines. ¯\_(ツ)_/¯
It isn't
Re: Coastline paradox
#68It also imples that any border between countries that is defined by a natural border, eg a river is infinitely long, right?
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.
But the middle of the river is just a function of both shifting coasts.
Re: Coastline paradox
#69Another fairly counter-intuitive map-related thing is that Edinburgh is further west than Bristol, and roughly on the same longitude as Cardiff. There's no real advanced math or optical illusion going on here, it's pretty obvious when you look at the map. It's just that our minds just make a simplistic "it's on the east coast, and therefore further east than things on the west coast"-shortcut. It works most of the ti…
Reno, Nevada is further west than Los Angeles, California. Same effect, US perspective :)
Heck, Spokane, Washington is nearly the same longitude as Los Angeles.
Re: Coastline paradox
#70Earlier quoted context omitted.
true, but you could just argue that any increase was just a result of over-estimating the last time you measured
The lower bound is generally very easy to measure precisely. So a change in that would be immediately noticed. For example, how long is the North Carolina coastline? No idea. But we can measure the precise distance between the point where that state's border with South Carolina meets the ocean, and the point where the border with Virginia meets the ocean. The coastline must be at least this long. And if you measure y…
Much more complcated problem than you're implying as NC has no unbroken coast. It's all shoals and barrier islands - and when you're inside that, swampland.