http://blogs.discovermagazine.com/d-brief/2018/04/17/measure-us-coastline-unknown/#.WtpICHXwbRY

How long is the U.S. coastline? It’s a straightforward question, and one that’s important for scientists and government agencies alike. The U.S. Geological Survey could give you an answer, too, but I’m going to tell you right now that it’s wrong.

In fact, no one could give you the right answer, and if you look around, you’ll find a number of estimations that differ by seemingly improbable amounts. One government report lists the number as 12,383 miles. The same report admits that a different government agency says the figure is actually 88,612 miles. That’s an almost eight-fold disparity for a fact that seems simple to obtain. We all know how to use a ruler, right?

http://faculty.washington.edu/cet6/pub/Temp/CFR521e/Mandelbrot_1967.pdf

 

I always thought that was a problem of fractals. That dates back a very long time. Is it news?

 

It's not news, it's just cool information.

 

http://blogs.discovermagazine.com/d-brief/2018/04/17/measure-us-coastline-unknown/#.WtpICHXwbRY
http://faculty.washington.edu/cet6/pub/Temp/CFR521e/Mandelbrot_1967.pdf

They don't agree on the fractal dimension of a coastline. I wonder if it is because they have different goals for their measurement.

 

How high is the tide when measurements made ?

Full moon or new moon ?

Wind and wave activity eroding coastline or calm ?

Sunny or cloudy, rainy or dry, freezing or hot....

There is no one answer and any answer subject to Heisenberg Uncertainty Principle
impact on measurement data and precision.

Regards, Dana.

 

I don't think fractals have anything to do with it. The variable T, time mucks it up.

If you snapshot the whole coast, then the length can be found for that snap. The coast length is in constant change.

 

The coast length is in constant change.

That may be true, but it doesn't affect the basic problem. Even if the coastline were literally frozen, the length you measure still depends on the size of your stick.

 

Hi,

Fractals have been brought into the problem because the coastline acts like a fractal ... the closer you look, the more detail you see. With more and more detail, the actual length (circumference) changes. It gets longer and longer.

Consdier looking at a part of the coast in a very highly detailed photo. At one distance it may look straight, and the shortest distance between two points is a straight line. Now zoom in a little more, and we might see a triangular coast, where instead of a straight line (shortest distance) between those same two points we see it angle in then angle out again. It must be longer because it's no longer a straight line.

Yes, this problem has been around for a long time now.

 

Changing your stick length does not change the coast length. There is no limit to long sticks. But there is a limit on short sticks. A snap of absolute coast length can be made once the gradient of shore is defined.

 

Changing your stick length does not change the coast length. There is no limit to long sticks. But there is a limit on short sticks. A snap of absolute coast length can be made once the gradient of shore is defined.

Nope. At higher resolution, the coastline just gets longer. Suppose you had to add the length across each atom that juts out and back at the interface. That measure would give a very different answer than using, say, a one-mile stick that misses any feature less than its length.

 

Then use 1/10 of an atoms diameter for your stick. We can measure the hills and valleys of atoms with AFM......We can separate the horizontal component from the depth component too.....for a horizontal length....or not, for a true absolute length.

It all depends on how you define the shore gradient and how you want to define a line for length....for this measurement.

Does your shoreline occupy 2D or 3D? One angle or two. Sounds like a qubit job.

 

One has to set limits. It's like how big is a state. From what angle? Straight down?

What about the surface area that isn't horizontal? How big would Colorado be....if we flattened all the surface area?

And old school problem said you could put every family on earth and give them five acres. Granted that was 4-5 decades ago.

 

Then use 1/10 of an atoms diameter for your stick. We can measure the hills and valleys of atoms with AFM......We can separate the horizontal component from the depth component too.....for a horizontal length....or not, for a true absolute length.

Besides the fundamental uncertainty issues, you have the practical problem of trying to differentiate which atoms belong to the coastline and which to the water. At that scale, it'll look like a chaotic maelstrom of molecular motion.

 

Then use 1/10 of an atoms diameter for your stick. We can measure the hills and valleys of atoms with AFM......We can separate the horizontal component from the depth component too.....for a horizontal length....or not, for a true absolute length.

It all depends on how you define the shore gradient and how you want to define a line for length....for this measurement.

Does your shoreline occupy 2D or 3D? One angle or two. Sounds like a qubit job.

Hello,

That's not really what this is about. It's not about having the resolution to be able to "see" all deviations, it's about not seeing deviations that were not even there before you zoomed in.

In other words, say you look at the coast and see 1 mile long. Then you zoom in and see it's really 2 miles long because of various features that dig in and out of the coast. Now you zoom in again, and one place where you used a straight line becomes an alcove that is actually a portal to a stream that leads to a lake, and the lake has it's own coast which you never even noticed yet. The stream opens up a whole area that was not even considered yet, and the coast of the stream and lake have to be considered part of the coast too because it's all binding the same water.
So the straight line and triangle example i gave is just the start of it. We might progress from a straight line to a very narrow stream that opens up into a large lake which when added to the original calculation adds a lot to the "coastline".

That's where fractals come into play, because they can mimic the coastline because there is no end to their resolution and complexity.

We know we have a certain area to consider, bound by the entire coast line. But try to put a maximum bound on the coast line based on the area seems to be impossible because the area is independent of the length of the coast line. It seems like we should be able to do that, but the coast line can be so complex that it just does not relate to the area itself. It's not like a simple curve like a circle, and it's not even as simple as 100 partial circles joined end to end using random arc lengths. The curve is not defined in the usual way or else we could calculate anything we wanted to calculate about that and the area as well.

If you look into this like on the web a little more you will get a better feel for what this problem is trying to tell us. It's not about connecting atoms, or any other tiny things. In fact it is just the opposite, it tells us that connecting tiny things to form straight lines doesnt work on this problem. But look into it a little more on the web and see what you find out.

 

As is always the case, an analogy can't be pushed too far. Yes, a fractal pattern such as the Mandelbrot set has an infinite perimeter, but that's only because the definition of the boundary goes down to an infinite resolution.

But the coast line is a physical thing with physical limits.

What it really comes down do is whether the contribution of a given particle, whether it be a molecule or a subatomic particle, to the perimeter is finite or infinite (at a particular moment in time so as to remove time variability).

If it is finite, then the length of the coastline if finite because there are a finite number of particles making finite contributions to it.

In order for that contribution to be nonfinite, it has to be sensible to talk about the contribution of each particle at an arbitrarily fine resolution, including infinitely beyond the scale, for instance, at which the resolution relative to the particle is greater than the size of the particle relative to the known universe. And you can't stop there. You can't ever stop.

But does that make sense? At some point you get to the scale where the resolution of the measurement has to be finer than the fundamental uncertainty in the measurement. Would such a measurement have any validity at all? Would anyone buy a claim that they measured the perimeter of an orange to be a mile by making the measurement with nanometer resolution using their child's school ruler?

 

Would anyone buy a claim that they measured the perimeter of an orange to be a mile by making the measurement with nanometer resolution using their child's school ruler?

FWIW, measuring a convoluted surface area, like an orange peel, is actually easier than one might think. There are some cool methods where you can measure, for instance, how much mercury can intrude into the pores of a surface. The result depends on the size of the mercury atoms and probably a few other parameters, but since it “measures” at the atomic scale, the result is good enough for many applications.

 

That may be true, but it doesn't affect the basic problem. Even if the coastline were literally frozen, the length you measure still depends on the size of your stick.

Yes but, the smaller the stick, the bigger summation of errors. In the Naval Academy, when I was a rookie (1st year cadet), I recall one of my classmates being "punished" by telling him to measure the length of a room (some 100 m long) using a match (wax). Had he used any garden variety metric tape, sure the error would have been smaller.

Not derogatory my question but just out of curiosity: what is the benefit of knowing the "actual" length of a coastline? Any really practical application?

 

Not derogatory my question but just out of curiosity: what is the benefit of knowing the "actual" length of a coastline? Any really practical application?

Overcharging for beachfront property.

 

For practical purposes and a sain and healthy life- stop worrying about it.
That said, get rich by smashing at the rocks and increasing the surface area of your country.
They say invest in land- its just one thing the're not making more of. Prove them wrong!

 

Not derogatory my question but just out of curiosity: what is the benefit of knowing the "actual" length of a coastline? Any really practical application?

I can see the value of knowing the length at the scale of a few meters, to know how long a shoreline trail might be or how long a near-shore kayak trip might be. About the only thing I can think of being useful at a smaller scale would be to know how hard a clean-up (eg. after an oil spill) might be.