Coastline paradox

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invisifly2
Posts: 9
Joined: Thu Aug 18, 2011 10:23 pm UTC

Coastline paradox

Postby invisifly2 » Sat Sep 24, 2011 10:47 pm UTC

Forgive me if I'm incorrect, but apparently it is a paradox that pops up when measuring a coastline that causes the total length of the coastline to increase each time you measure it with a smaller unit of measurement, due to the extra features that can be measured. For example, official measurements of the U.S. coastline range anywhere from (in miles) 12,380 to 95,471. That is a huge difference.

So, if I kept measuring a coastline using smaller and smaller units of measurement, would the total length go on toward infinity? And how excatly does this work?

IIAOPSW
Posts: 131
Joined: Sat Dec 27, 2008 1:52 am UTC

Re: Coastline paradox

Postby IIAOPSW » Sat Sep 24, 2011 10:57 pm UTC

in theory, the total length does go to infinity. In practice, do you really want to measure the length of coastline of each individual atom. Even if you could, your measurement would get totally messed up by the tide.

Basically, coast line is a type of fractal in that its statistical properties are scale invariant.
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gorcee
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Re: Coastline paradox

Postby gorcee » Sat Sep 24, 2011 11:38 pm UTC

Also, if you're going to make a thread for every item in the recent cracked article, just make a single thread. Or, like, read the Cracked comments.

invisifly2
Posts: 9
Joined: Thu Aug 18, 2011 10:23 pm UTC

Re: Coastline paradox

Postby invisifly2 » Mon Sep 26, 2011 7:54 pm UTC

I don't read cracked, actually.

Turtlewing
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Re: Coastline paradox

Postby Turtlewing » Wed Sep 28, 2011 8:19 pm UTC

It's been a while since I studied this but, what you're descibing is the "fractal dimension" http://en.wikipedia.org/wiki/Fractal_dimension of an object.

I usually think of it as a "crinkly-ness factor". But really it's a measure of to what extent it makes sense to talk about length vs area vs volume etc. Coastlines have a dimension between 1 and 2 meaning that depending on the complexity they may act more like 1 dimensional objects (a line) vs 2 dimensional objects (a square) with regards to what measurements make sense.

A real coastline will not have an infinite length as it's not actually infinitely complex, but they're one of the better real world examples of the mathematical principles involved.

gorcee
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Joined: Sun Jul 13, 2008 3:14 am UTC

Re: Coastline paradox

Postby gorcee » Wed Sep 28, 2011 9:09 pm UTC

Turtlewing wrote:It's been a while since I studied this but, what you're descibing is the "fractal dimension" http://en.wikipedia.org/wiki/Fractal_dimension of an object.

I usually think of it as a "crinkly-ness factor". But really it's a measure of to what extent it makes sense to talk about length vs area vs volume etc. Coastlines have a dimension between 1 and 2 meaning that depending on the complexity they may act more like 1 dimensional objects (a line) vs 2 dimensional objects (a square) with regards to what measurements make sense.

A real coastline will not have an infinite length as it's not actually infinitely complex, but they're one of the better real world examples of the mathematical principles involved.


That's pretty much it. Length is a strictly 1 dimensional measurement. Area is a strictly two dimensional measurement. But you can resolve a coastline to increasingly small detail, to the point the length of the curve that you draw along the coast will go to infinity.

The coastline is actually a 1 + x dimensional object, where 0<x<1. So you can't really measure it in terms of length. Doing so would basically be like asking "what's the length of the surface of a square?" Well, draw a line along the left edge, and that has length s. Move that line over by an infinitesimal amount, add that to the previous length, now you have 2s... and so on to infinity. It's not that the surface of a square is of infinite length, it's that asking "what's the length of a 2-d object?" is a nonsensical question. If you partitioned that area in a series of fixed, finite width segments, and asked what the length of those segments would be when put end to end, that's a question that makes sense. Obviously, the wider the segments, the less resolution of the area you have, and the shorter that length gets.

Same thing with a coastline.


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