
Coastline Paradox - Hooke
https://en.wikipedia.org/wiki/Coastline_paradox
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crowf
I once heard, from a potentially unreliable source, an interesting story about
this. When countries border each other and also share a coast there is a issue
of how to decide who owns the water. The simplest way is to find the point
where the two countries and the coast meet. Then from there draw a line along
the longitude. However, that is problematic, because there are countries like
Libya and Egypt where their boarder is along the latitude.

During those times the British were very colonial, and one of their methods of
keeping power was to cause the colonised people to fight amongst themselves
(as can still be seen with the Indians and Pakistanis, Israelis and
Palestinians). Well, they invented a method to split the coast by first
finding the angle of tangent on the point of the boarder between the countries
and the sea. Take 90 degrees of the tangent, and that is the sea boarder.

While this looks good upon first glance, due to the coastline paradox, you can
always find a different tangent line. That then could be the cause of conflict
between countries as they try to determine the sea boarder.

If anyone knows if there is any truth to the story, I would be interested in
hearing.

~~~
magicsmoke
Based on some reading online, maritime delimination of boundaries beyond a 3
mile "cannonshot" distance didn't become common until after the 1950s when
treaties were drawn up that would become the precursor to UNCLOS. In those
treaties, the starting point for border delimitation was the equidistant point
between two countries' coastline. When two countries have adjacent coasts,
this ends up roughly perpendicular, but it still works even if they have
irregular coastlines and doesn't involve any angle measurements, just
distance. Considering that ocean borders only became common after the age of
British colonialism and that perpendicular baselines were never specified as
the standard in UNCLOS, I'm inclined to say that this is a myth.

~~~
crowf
Thanks for fact checking. I guess I'll have to preface this story by saying
that it's a myth. Oh well

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RandallBrown
I recently went on a hiking trip and this issue caused a bit of grief from my
friends.

At the end of the first day, a few of us tracking our progress with GPS said
we had gone about 18 miles. The mapping software we had used (Gaia) had shown
we had only made about 12 miles of progress on our 82 mile track.

I'm used to GPS overestimating distances since it tends to jump around on the
trail a bit (and it still jumps around when you're paused) but the differences
throughout the trip were much larger than I would expect.

Our 82 mile trip ended up being 104 miles, which my friends weren't really
prepared for.

Examining the map closer, some of the trails we were on were pretty roughly
approximated on the map (way fewer switchbacks, long straight lines, etc.)

I'm not allowed to plan our hiking trips anymore...

~~~
curiousgal
How long was the trip?

~~~
RandallBrown
5 days

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baron_harkonnen
This is also a good intuition for why Brownian motion (frequently used to
model stock prices) does not have a derivative! If you imagine zooming in on a
stock price, with each successive zoom closer it remains self similar, looking
perpetually like a noisy stock chart. One of the basic assumptions of
derivatives is that as we zoom in on a smooth curve the more that curve
resembles a straight line. This doesn't hold true for fractal structures like
the coast of Britain and stock prices.

~~~
manfredo
Stock prices have discrete changes, though, in the form of individual
transactions. Zoom in far enough that your graph is entirely occupied by the
time between two transactions and you have a straight line.

~~~
jlawson
But you don't really have a straight line. You have an instantaneous step
change.

Of course, when you're zoomed in that much the concept of 'price' becomes a
lot more complex. You need to distinguish between bid, ask, and last
transaction price. So the whole mental model kind of falls apart.

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Waterluvian
I had a bit of a brain burst the other day when I recalled from undergrad
(GIS) that this was a real problem, but that area was a value that converged
on a number the more precisely you measured, so presumably you don’t have to
worry about massively different area values the more precisely you measure.
But then realized that the coastline paradox must also exist for 3D surfaces
(TINs or DEMs or whatnot) so asking “what is the area of a surface on earth”
can be a very complex question if dealing with elevation.

That got me wondering if it’s generally true for any number of dimensions.

~~~
Animats
Yes. In 2D, area enclosed by a noisy perimeter converges with more perimeter
points, but length only increases. Adding more points to the perimeter can
only increase length, because the shortest distance between two points is a
straight line. But adding a point can increase or decrease measured volume.

In 3D, volume converges with more perimeter points, but surface area does not.

This line of thinking leads to sampling theory. Given a noisy analog signal,
digitizing it at higher and higher resolution and sample rate can yield an
arbitrarily large number of data points, as you track the noise. But the area
of the envelope around the waveform converges.

~~~
Waterluvian
Thank you, especially for that last paragraph that gives me a lead on what to
begin digging a bit into.

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albemuth
Expected this to be a reference to Neal Stephenson’s “Fall; or, Dodge in
Hell”. Midway through and it’s fantastic.

~~~
scubbo
It's been a year or so since I read it - how is the Coastline Paradox related
to the book?

(Glad you're enjoying it! I absolutely loved some of the ideas - the treatment
of social media, in particular)

~~~
albemuth
In the book he attributes the fractal insight to measuring the crack in a bar
that is part of the border of the fictional enclave in NL/Belgium.

~~~
scubbo
Oh yeah! Thanks :D

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moron4hire
At some point, doesn't your measurement division hit the Planck length?

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mhh__
Not to a mathematician.

Or a physicist for that matter - we don't have a useful theory of quantum
gravity, space may or may not be quantized.

