Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Coastline paradox

en.wikipedia.org

Post reply on HN
Print view View original

41–50 of 111 posts

Re: Coastline paradox

#41

Post by em-bee »

OskarS wrote: post #15

I’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?

Re: Coastline paradox

#42

Post by [deleted] »

infradig wrote: post #34
Remnant44 wrote: post #29

Earlier 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!

The Wikipedia article states it's not really fractal.

[deleted]

Re: Coastline paradox

#43

Post by macintux »

alistairSH wrote: post #24
tukajo wrote: post #16

Earlier 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.

Not always. The boundary between VA and MD (states in the USA) is the Virginia shoreline of the Potomac River. Which has caused oddities over the years… At one time, gambling and liquor were illegal in VA, so casinos set up boats at the low tide line. Customers parked in VA and walked onto the casino barges from the Va side, but they were technically in MD. You need to follow MD fishing regulations even from VA shore…

My uncle once argued a case in the U.S. Supreme Court regarding Indiana & Kentucky's border, which was initially established in the middle of the river.

The Ohio River has gradually shifted south since the border was established, so some parts of Kentucky are now on the "wrong" side of the river.

Re: Coastline paradox

#44

Post by yuriks »

Solvency wrote: post #30
Remnant44 wrote: post #29

Earlier 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!

The word converge always confused me here when learning English. It's not like the edges of a circular polygon "converge" even in a hypothetically perfect circle. They more....blend together, or lose their edge. Either way maybe I've never properly understood the term.

What converges is the resulting value of the calculation. As you add more lines the approximate result (of the circumference) gets closer and closer to the true answer. It's not necessarily referring to the geometric shape of the polygon. (Though you can also see it as, if you take any given point on the circumference of the polygon, as you add more segments, that point will get closer and closer to the true edge of a circle.)

Re: Coastline paradox

#45

Post by fellerts »

Remnant44 wrote: post #29
irrational wrote: post #27

I’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…

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.

Re: Coastline paradox

#46

Post by Remnant44 »

infradig wrote: post #34
Remnant44 wrote: post #29

Earlier 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!

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 range of scales is clear; I feel like the Wikipedia article is tripped into pedantry at that point :)

Re: Coastline paradox

#47

Post by naikrovek »

I 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)_/¯

Re: Coastline paradox

#48

Post by Aachen »

em-bee wrote: post #39
thih9 wrote: post #13

Earlier quoted context omitted.

Listings already have these and worse things. E.g.: properties are routinely described as “X minutes away from beach”, where in reality it’s impossible without teleportation.

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.)

Re: Coastline paradox

#49

Post by rootusrootus »

Aachen wrote: post #48
em-bee wrote: post #39

Earlier 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.)

> Do bullets fly for minutes?

Effectively no. MythBusters said a 30-06 shot straight up could hit 10K feet and would take just under 1 minute to hit the ground. I'd be comfortable asserting that a 30-06 is about the largest 'normal' cartridge. Bigger ones definitely exist, but they're a niche.

Handgun rounds are generally a lot slower and lighter than a 30-06, so they'll all be even quicker to reach the ground. Plus, to hit the beach, you'd shoot at about a 45 degree angle instead of straight up, which would reduce the height reached.

Re: Coastline paradox

#50

Post by [deleted] »

rootusrootus wrote: post #49
Aachen wrote: post #48

Earlier 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.)

> Do bullets fly for minutes? Effectively no. MythBusters said a 30-06 shot straight up could hit 10K feet and would take just under 1 minute to hit the ground. I'd be comfortable asserting that a 30-06 is about the largest 'normal' cartridge. Bigger ones definitely exist, but they're a niche. Handgun rounds are generally a lot slower and lighter than a 30-06, so they'll all be even quicker to reach the ground. Plus,…

[deleted]
Post reply on HN