[Youtube Review][Veritasium] What Is The Coastline Paradox?

(Recommended)Popular Videos : [Veritasium] What Is The Coastline Paradox?

 

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(Recommended)Popular Videos : [Veritasium] What Is The Coastline Paradox?

https://www.youtube.com/watch?v=I_rw-AJqpCM

 

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Summary Comments : [Veritasium] What Is The Coastline Paradox?

ur*****:

I was hoping he might introduce the concept of non-integer "fractal" or "Hausdorff" dimensions. So, the coastline of Australia stretched out as a single line (normally 1D) is of a dimension between 1 and 2 e.g. 1.25, the surface of your brain has a dimension of about 2.66, while the surface of your lungs has been measured as 2.97!

Al******:

I didn't believe this, so I actually did the math and ended up proving him right!
Here are the numbers :

Perimeter : Sets of added triangles:
3 0
4 1 set
5.3333 2 sets
7.1111 3 sets
9.48148 4 sets
12.642 5 sets
53.2732 10 sets
224.494 15 sets
946.011 20 sets
3986.48 25 sets

p.s. I created a computer program to do these numbers for me, but after about 550 repetitions, it crashed because the numbers were to large to compute!

Er********:

It is "getting larger" though, as 3.141 is still greater than 3.14. But yes, we use the variable for pi that gives it a static value. Just the thought there always being an extra decimal place to make it larger before was a comparison to the paradox in the video.

Es***************:

Let's say you are walking on a line, each step you take is 2 times smaller than previous step. If your first step is 1 unit, 2nd will be 0.5 unit, so you'll be in the 1.5 units away from your starting point. This will continue like this:
3rd step: 1.75
4th step: 1.875
5th step: 1.9375
...
Infinith step: 2

 

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Playtime Comments : [Veritasium] What Is The Coastline Paradox?

Ni**:

1:31 LOL.
I'm pretty sure it's pronounced like "coke" but as if the "o" had an umlaut...

 

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Top Comments : [Veritasium] What Is The Coastline Paradox?

Gr*******:

I've been there, and it was a wonderful time!  Would love to go back.

bl****************:

An actual physical coastline cannot be infinite. Because the laws of physics say that the shortest possible measuring stick is a Planck length. Very small, but finite. An it takes a finite number of Planck length steps to walk around - say - Australia once.
So it stands to reason, that if you have a finite stick you approach a limes.
The Koch snowflake - as a mathematical object - has no such limit. You can make the step length ever smaller and thus the circumference becomes of course infinite.

Al*******:

it isnt infinite at all. if you continued to zoom in you would reach a point where you cannot go any further. if you measured the coast in planck lengths you still wouldnt get a number approaching infinity

Ro****:

he is wrong, both perimeter and area are finite, even if you add infinitly more triangles to it.

Lu***:

How would the length be infinite if finite distance values exist between atoms and fundamental particles?

Bo******:

Excuse me, but wouldn't a finite distance divide into a finite set of segments, no matter how large or small either is? I guess I'm putting forward that even if you measured the coastline of Australia using Planck distance (so that there is no smaller distance that we can measure), we would get a very large number, true. The result of the measurement would be a finite length, however. 

I could give two theses, the one above: that there is a fundamental constraint to the distance our measurement is made in. Also, that there is a fundamental constraint to the distance we are measuring: .i.e. the actual length of Australia's coastline (at time t, to be pedantic). This constancy means that the length of the coastline is not fractal, as suggested. 

Regarding the paradox, it seems that smaller our unit of measurement is, the more precise our measurement of the length of the coastline. Since there is a constraint on the length of the unit of measurement, there should likewise be a constraint as to the length of Australia's coastline. In a fractal, we would with each successive zoom get more detail, and more distance to add to a sum of perimeter lengths, without a limit to the number of zooms. In real life, there is such a limit. Thereby the distance of Australia's coastline is finite. 

Ch************:

But with the Koch snowflake you are also adding infinite area as well. For every new step you also add more triangles with more area.

21**********:

I'm going to get high and watch fractals.

Ad**********:

Isn't this string theory?

Ma******:

The coast line is made of atoms. This puts a limit on how fine the fractal can get.

Ne****:

It may be immeasurable, but it is not infinite. 

Fl********************:

No paradox here.  There will be a finite number of water moleculs.

ch********:

this would explain why google maps told me a drive down the coast of California would take 6 hours but was 11 in reality.

Sg************:

the coastline is only a POTENTIAL infinite, not an ACTUAL infinite.  This is a very important distinction. With a potential infinite, you never actually reach infinity. Whereas with an actual infinite there literally exists an infinite number of physical things.  Big , big difference. 

Lu**************:

We have reached the point where science is no longer intriguing but is more confusing than the question you set out to answer. Wonderful.

el*******:

I don't understand ... If you were to get a hypothetical string, wrap it around Australia, cut it at the point where the end meets the start, straighten the string out and then measure it, surely that would give you the exact length of Australia's coastline?

Em*********:

I have a suggestion for an alternate coastline measurement scheme: Just measure the shortest distance required to circumnavigate the entire country (plus islands). That would count bays and fjords and deltas as being internal bodies of water, but it'd make a lot more sense. Give you a sense of how far you'd have to go to get from one side to the country to the other, or around it if it's an island.

du*****:

i'm having trouble with the notion: "a finite area can have an infinite perimeter". isn't this only true if the "fractalization" of the area's boundary would also be infinite, which is impossible because then the term "finite area" would not be applicable? therefor: an infinte perimeter implies an infinte area. or not?

Da**:

But eventually you reach a point that the stick cannot get smaller, right? You could go as far as the planck length and then its over, so actually it isnt infinite but finite after all.

na************:

Why Do You Capitalize Every Word In The Title?

Al*******:

This is a phenomenon that happens with measuring the length of any curve, not just a coastline. Since the shortest distance between two points is a straight line, taking a particular curve segment and approximating with a line will give you the minimum distance estimate on that segment. Pick a point on the segment and now use two lines to approximate the curve. By triangle inequality, this is at least as big as the single line (it's only equal if the original segment is a line already). So, as you continue to increase the number of points in your approximation, you get a non-decreasing sequence which converges to the true length of the curve. For relatively tame curves (all derivatives are sufficiently small), this convergence happens quickly and so increasing the amount of lines yields a smaller and smaller increase in the measured length. However, for something like a coastline, we have large derivatives and so the convergence is very slow. This gives the appearance of no upper bound.

In reality, it's very difficult to calculate any lengths very accurately at all, since even a seemingly straight edge like the end of a ruler can have thousands of tiny dents, plus the texture of the material, and then the structure of the molecules and atoms. So, in order to measure any length (or surface area), we have to decide what shape it most closely represents and calculate that. It just goes to show that something that seems so fixed as length is actually quite arbitrary and is simply based on how we approximate the world around us.

Ph***************:

+greg77389 Actually this is not math at all. In reality there's no such thing as coastline. There are atoms, and you can't even know their exact positions in space, even if you somehow tackle with a fact that atoms don't make a really solid object to draw line on. Math is about fractals and other debately useless models of the real world.

Ga******:

You should make a longer vid on fractals. They're too interesting for 2 minutes.

IO*****:

So where is the paradox? The length is always fixed, no matter how long the measuring stick you use. Even with molecule-length measurement stick, it's a fixed number (at least in a fixed moment in time). By the way, in your example with triangles, with each round of adding smaller triangles you actually add more length to the perimeter. Ofcourse if you infinately add more length, the length is "infinite". But even that is not true. The perimeter has a fixed measurable perimeter with each round of adding triangles.

BG***********:

The perimeter is not infinite because australia is not a fractal shape.  The limit of the perimeter as the length of the measuring stick approaches zero is certainly not infinity, and if it was, you'd have to prove it a lot better than that.

Su*********:

Just use a tape and wrap it around the coast

ia*******:

 Even measuring at the atomic scale in Planck lengths, you'd eventually get a number.  Just because the number is huge doesn't mean it's infinite. 

Lets not get lazy here.

ar*******:

I remember one geography test from my high shool where one question was: How long is the border of your country. If I knew Veritasium back then...

To*****:

at 0.35 australia looks like a very happy woman with huge hair in heavy wind. D:

Ho*********:

I might propose that you use a fixed number of significant digits to measure any coastline. The CIA World Factbook number of 25,700 km has 3 significant digits. Suppose that this means that you should use 1/10 of the least digit, or in this case, 10 km, to measure the coastline. This would leave three accurate significant digits. A smaller island example would be Oahu, whose coastline is 365 km. By this system, Oahu would be measured with a 100 m rod.

This limitation will not diverge-- if too small a rod is initially chosen, the coastline length would read quite long. This would call for a remeasurement with a longer rod, leading to a shorter measurement. The measurement and rod length would converge.

Sh********:

So what was the length used for the other measuremnt you gave?  Is there a worldwide standard increment for measuring coastlines?
Love the channel - I just started with your oldest entry and I'm slowly catching up...
-Shawn

sp********:

So, the length of your stick IS important after all?

Os*********:

What makes the triangle's perimeter infinite is that you keep adding other triangles to it. You keep adding triangles, you keep adding surface, so you can't say it has a finite surface. The moment you stop adding triangles to it, you will get a finite surface, and with it a finite perimeter. It's the same with Australia: at some point, you will have a measure that is small enough to account for all the turns in the coastline, and then you would find a finite perimeter. Sure, you could always take a smaller measurement, but it would get the same result, because you've already accounted for all the turns.

le*******:

I was aware of that paradox related to the way fractal work. When I heard about it years ago, it blew my mind and it still does. It's weird but it makes a lot of sense at the same time. Thinking that somehow finite borders measurement can head towards infinity (finite at the Planck length, that is) is just amazing.
I'm just disappointed a bit that the video only talks about this very specific aspect of fractals and doesn't explain the fractals as a whole. These things, while not always sexy to look at, have very strange other properties that are worth a video. Especially if you bring the chaos theory to the topic.
Fractals don't seem to have anything entertaining/interesting to tell but they really do.

Ge**************:

The coastline paradox was really interesting to learn about! I didn't know that changing the length of the measuring stick you use could impact the total length of the Australian coastline so much. One question I have is that CIA World Fact Book measured the length of the Australian coastline as 25,700 kilometers by measuring with shorter measuring sticks than the 500 km ones, but has anyone measured the length of the Australian coast with a shorter measuring stick and gotten an even more precise length? Also, I didn't understand how the length of Australia's coastline would be infinite using drops of water to measure it, wouldn't it just be really hard to measure? I did learn a lot from watching this video such as that a fractal is a shape with a finite area and an infinite perimeter, and many coastlines are fractals.

 

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[Veritasium] We gathered comments about popular videos and looked at them in summary, including play time, and order of popularity.

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