6 ms·

Coastline Paradox

63 points by Hooke 6y ago

Waterluvian 6y ago
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 6y ago
  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 6y ago
    Thank you, especially for that last paragraph that gives me a lead on what to begin digging a bit into.
- sudosysgen 6y ago
  Well, there is a good way of measuring the area; instead of measuring actual area, measure the area of the surface on a perfectly smooth spheroid with an identical perimeter.
- swilliamsio 6y ago
  >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. StandUpMaths did a great video on this recently, asking whether national land area includes elevation and which nations would gain/lose the most land area if it were/weren't: https://www.youtube.com/watch?v=PtKhbbcc1Rc https://www.youtube.com/watch?v=PtKhbbcc1Rc
- growlist 6y ago
  Another GIS-er here. Your comment made me think about the modifiable areal unit problem: https://en.wikipedia.org/wiki/Modifiable_areal_unit_problem https://en.wikipedia.org/wiki/Modifiable_areal_unit_problem The coastline paradox and MAU effectively seem to be of the same class, i.e., change the rules for your measurement and you get different results. This hadn't occurred to me before. I wonder if there's other similar?
baron_harkonnen 6y ago
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.
- nordsieck 6y ago
  > 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. Unless some exchange has allowed traders to infinitely divide currency, that's doesn't seem like it could be true.
  - baron_harkonnen 6y ago
    Likewise you could argue that the coastline problem is limited to at most the Planck length. Most models of stock movement are modeled mathematically with Brownian motion which is by definition a continuous time process. You can even more trivially show that stocks are non-differentiable if you want to argue that they are fundamentally discrete time processes, in which, by definition, they are not continuous, differentiable functions. But the argument of about "what does continuous even mean when we talk about physical systems" is another and older argument altogether.
    - pdonis 6y ago
      > you could argue that the coastline problem is limited to at most the Planck length Only if the Planck length were really a minimum possible length. But it might not be; even in a theory of quantum gravity, it might not be, since what is quantized in quantum gravity does not have to be lengths and times.
- manfredo 6y ago
  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.
albemuth 6y ago
Expected this to be a reference to Neal Stephenson’s “Fall; or, Dodge in Hell”. Midway through and it’s fantastic.
- scubbo 6y ago
  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 6y ago
    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 6y ago
      Oh yeah! Thanks :D
crowf 6y ago
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 6y ago
  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 6y ago
    Thanks for fact checking. I guess I'll have to preface this story by saying that it's a myth. Oh well
- 082349872349872 6y ago
  "Let's you and him fight." In recent news, also greeks and turks, irish and ulster scots, ... Arguably the same strategy exists between those who take The Sun and those who take The Economist.
moron4hire 6y ago
At some point, doesn't your measurement division hit the Planck length?
- mhh__ 6y ago
  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.
RandallBrown 6y ago
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...
- namibj 6y ago
  GPS is quite a bit better on determining velocity though. That said, using GPS for this task without adding an IMU for sensor fusion seems prone to measurement errors.
- curiousgal 6y ago
  How long was the trip?
  - RandallBrown 6y ago
    5 days

63 points by Hooke 6y ago

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 6y ago
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 6y ago
  Thank you, especially for that last paragraph that gives me a lead on what to begin digging a bit into.
sudosysgen 6y ago
Well, there is a good way of measuring the area; instead of measuring actual area, measure the area of the surface on a perfectly smooth spheroid with an identical perimeter.
swilliamsio 6y ago
>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. StandUpMaths did a great video on this recently, asking whether national land area includes elevation and which nations would gain/lose the most land area if it were/weren't: https://www.youtube.com/watch?v=PtKhbbcc1Rc https://www.youtube.com/watch?v=PtKhbbcc1Rc
growlist 6y ago
Another GIS-er here. Your comment made me think about the modifiable areal unit problem: https://en.wikipedia.org/wiki/Modifiable_areal_unit_problem https://en.wikipedia.org/wiki/Modifiable_areal_unit_problem The coastline paradox and MAU effectively seem to be of the same class, i.e., change the rules for your measurement and you get different results. This hadn't occurred to me before. I wonder if there's other similar?

baron_harkonnen 6y ago

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.

nordsieck 6y ago
> 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. Unless some exchange has allowed traders to infinitely divide currency, that's doesn't seem like it could be true.
- baron_harkonnen 6y ago
  Likewise you could argue that the coastline problem is limited to at most the Planck length. Most models of stock movement are modeled mathematically with Brownian motion which is by definition a continuous time process. You can even more trivially show that stocks are non-differentiable if you want to argue that they are fundamentally discrete time processes, in which, by definition, they are not continuous, differentiable functions. But the argument of about "what does continuous even mean when we talk about physical systems" is another and older argument altogether.
  - pdonis 6y ago
    > you could argue that the coastline problem is limited to at most the Planck length Only if the Planck length were really a minimum possible length. But it might not be; even in a theory of quantum gravity, it might not be, since what is quantized in quantum gravity does not have to be lengths and times.
manfredo 6y ago
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.

albemuth 6y ago

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

scubbo 6y ago
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 6y ago
  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 6y ago
    Oh yeah! Thanks :D

crowf 6y ago

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 6y ago
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 6y ago
  Thanks for fact checking. I guess I'll have to preface this story by saying that it's a myth. Oh well
082349872349872 6y ago
"Let's you and him fight." In recent news, also greeks and turks, irish and ulster scots, ... Arguably the same strategy exists between those who take The Sun and those who take The Economist.

moron4hire 6y ago

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

mhh__ 6y ago
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.

RandallBrown 6y ago

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...

namibj 6y ago
GPS is quite a bit better on determining velocity though. That said, using GPS for this task without adding an IMU for sensor fusion seems prone to measurement errors.
curiousgal 6y ago
How long was the trip?
- RandallBrown 6y ago
  5 days