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Coastline paradox

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Re: Coastline paradox

#81

Post by pjerem »

paulddraper wrote: post #55
School-Cotton wrote: post #9

The 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 the coast is a sandy beach, then it is usually pretty smooth. But in other cases ... https://www.nps.gov/articles/images/acad-ship-harbor-rocky-c...

> If the coast is a sandy beach, then it is usually pretty smooth

But it underlines another issue in this problem. What is defining the coastline ? If it’s water touching the land, it’s always moving, if it’s land at sea level, it’s eight always moving or approximated depending on your definition. And billions of billions of billions of billions of … of grains of sand seems like very near to fractal that could arguably add billions of coastline kilometers depending on your max definition.

Re: Coastline paradox

#82

Post by [deleted] »

justinl33 wrote: post #72
remram wrote: post #57

Earlier quoted context omitted.

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.

[deleted]

Re: Coastline paradox

#83

Post by denzil »

justinl33 wrote: post #5

Next time you go to measure your waistline, just remember: have you really put on weight, or is it just the coastline paradox?

Of course, if you are using tape to measure your weight, you are using the wrong tool for the job.

Re: Coastline paradox

#84

Post by bmacho »

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…

> 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.

According to https://en.wikipedia.org/wiki/Outline_of_calculus:

> Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

The coastline paradox is definitely an example of something that interests calculus people, and to tackle the problem they most likely will use a bunch of concepts and theorems of calculus.

However calculus, as a field is not this paradox.

Re: Coastline paradox

#85

Post by Asraelite »

n_plus_1_acc wrote: post #26
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.

genuine question: how is the middle defined when both shores are fractals?

I've thought about this before and came to the non-rigorous conclusion that if you define the middle as always being equidistant between the nearest points on the shoreline, then its length must be finite. The same is true if you define it as the shortest path through the river.

Basically this is because an infinite-length line must "loop back" on itself, so that at some point it curves by at least 90° from the main direction of movement. This can never happen at a small scale for the middle of a river.

I would really love a mathematician to weigh in on this though. The one main problem I can see is that it might lead to something akin to the Weierstrass function https://en.wikipedia.org/wiki/Weierstrass_function which itself has infinite length, although I don't quite understand why this is or if it's applicable to the case of rivers.

Re: Coastline paradox

#86

Post by thih9 »

naikrovek wrote: post #47

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)_/¯

Note that it can apply to borders too.

> the Portuguese reported their measured border with Spain to be 987 km, but the Spanish reported it as 1214 km.

Re: Coastline paradox

#87

Post by 082349872349872 »

ilaksh wrote: post #18

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.

Schröder's Fractals, Chaos, and Power Laws had a digression about a family of distributions where the longer you've gone without observing something, the longer the expected mean time to first sighting becomes.

I read that and immediately thought well, this explains software schedule slips.

"The first 90% took us 90% of the scheduled time; the last 10% will take the other 90%..." (editor's note: 9% of that last 10%, anyway!)

Re: Coastline paradox

#88

Post by Animats »

A simple way to think about this is this: draw circles along the coastline, all of radius R, such that the entire coastline is within some circle, with the circles centers are spaced R apart. For a given R, there is some minimum number of circles N which will cover the coastline. Smaller values of R require more circles.

The length of the coastline for a value of R is RN. Plot RN vs. R.

Trying to use a ruler on an irregular edge is ill-defined. If you formulate the problem as a chain of circles, there's a well defined result.

Re: Coastline paradox

#89

Post by OscarCunningham »

Animats wrote: post #88

A simple way to think about this is this: draw circles along the coastline, all of radius R, such that the entire coastline is within some circle, with the circles centers are spaced R apart. For a given R, there is some minimum number of circles N which will cover the coastline. Smaller values of R require more circles. The length of the coastline for a value of R is R N. Plot R N vs. R. Trying to use a ruler on an…

What you will find is that in the plot the length doesn't level off as R gets smaller. It will just get larger without bound.

Re: Coastline paradox

#90

Post by honzabe »

ilaksh wrote: post #18

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.

> I think I've heard this in reference to the difficulty of software estimation.

I don't like linking to Quora but I have to link the legendary "coastline trip" story posted by Michael Wolfe as a response to the question "Why are software development task estimations regularly off by a factor of 2-3?"

https://www.quora.com/Why-are-software-development-task-esti...

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