But then it’s basically a fractal because you measure between atoms

Of

No physicists, thank you!

Ew physics

That's how I see it. A coastline might be fractal-like, but it exists in the real world and cannot be an actual fractal... Because the plank length is the smallest meaningful measurement you can make in the real world.

Man you Poms are worse at reading than you are at cricket, (check my name again). Also I'd also like to point out that even New Zealand has a bigger coastline than you guys because it has two main islands, so that means it has a coastline of two times infinitiy, while you only have your one lousy little infinity. So suck on that ya Limey.

Good point. Don't anyone tell Obama, because he'd probably like the sound of that and attempt to spend infinity dollars, (0.6 infinity in Pounds Sterling), creating eternal shovel-ready public sector jobs to remake the US into a trapezoid or something.

IRONMANAustralia Two times infinity is not bigger than one times infinity. That is where the whole video is based on.

IRONMANAustralia Two times infinity is equal to infinity. Even infinity times infinity is equal to infinity. So both coastlines do have the same length.^^

Nope. Infinity is more like an arbitrary set. So if I get a line and divide it into an infinite number of points, I have a set of infinity points. If I then make the line twice as long I can also divide that into infinity points. Yet that second set is obviously bigger than, (and contains), the first. Pretty simple really.

HahahahahahahHahahahaAHAHAHHAHAHAHAHAHAH THANK YOU STEBE AND BRADY!!!° HAHAHAHAHA

Gabiotta still, what resolution or how long should the ruler be you are measuring with?

That means tolls all over the coastline.

pat a dont use a ruler

lol I just commented the same thing!

What do you think engineers are?

His real name is Benoit Benoit Benoit Benoit Benoit Benoit Et Cetera

@@firefish111 Now the question becomes: Did he really have a last name?

@@WhyneedanAlias Well..... yes but actually no

Mine too 😂

Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit ((Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit (Benoit

Yeah I bet it's different though, because when your choosing smaller and smaller rulers to measure a circle, you're getting closer and closer to a definite answer. But with the coastline problem, when you choose smaller and smaller rulers I think it just keeps getting bigger and bigger forever.

@@sopwithcamel5519 💯💯💯

Tadashi Mori, depends on the tide and how fast you run, and whether your toes are wet or not

Are you planning to traverse the molecular canyons of every rock on the beach?

Then you want the length as measured by your typical running stride. In general the answer is "what's the total length as measured by a step length that's useful for your particular use case? ".

Tadashi Mori that’s just as arbitrary as picking a 10m ruler or a 10m ruler

Ian Schimnoski how far, not how long they’re out running

countries have decided upon a "mean tide coastline" and you can use that as reference.

Whis that was min name XD

RIP is RIP In Peace, which stands for RIP In Peace In Peace, or RIP In Peace In Peace In Peace, I.e RIP In Peace In Peace In Peace

@@iamthinking2252_ brb = bad recursion brb = bad recursion bad recursion brb = ...

There is no ideal resolution. There is an ideal resolution for a specific purpose and there are good compromise resolutions that are sort of okay for most things and are easy to produce. The size of a pixel on a satellite's image for example, or one millimetre on the OS map.

I agree. This video isn't intended to explain real-world solution of measuring a coastline. It is only the recreational one.

lol

thats maths for you.. sometimes i regret starting this as a degree.. now i'm in my third year! fyi.. fractals are cool, but horrible to work with!

There must be a way

@@TheHalalPolice there isn't

Because they try to be consistent and honest in their answers? Just imagine a lawyer trying to measure a coastline...

***** Ok, that's even worse! :D Bear with me, I study engineering and therefore I like practical solutions

and that's why engineers are better than mathematicians, they find simple and easy solutions, where mathematicians simply wish to create problems

That is not true. Each field has its purposes. Most of engineering is build upon things that physicists and mathematicians found out.

Pi is exactly 3

Huh ready to measure thousands of kilometres with just string

The one application shows how things are not applicable.

Not even close.

I'm doing some research this summer with using fractal patterns in architectural acoustics to get sound waves to travel in ways desirable for auditoriums or classrooms. I know your comment was probably tongue-in-cheek, but it's surprising how many applications there really are for some of the math out there!

* scans QR code * "Zanpa" :3

c.f. The Misbehavior of Markets: A Fractal View of Financial Turbulence Paperback - by Benoit Mandelbrot for another real-life application of fractals.

But normally you'd expect the perimeter to scale equally with the magnification factor. For any shape that can be described as a finite number of measurable segments, that is true. The problem here is, that due to it's very construction, there are no measurable segments no matter how small you go. For the actual figure he drew, the perimeter of the large image, is obviously 4 times bigger than the perimeter of the small image, but that's just because he didn't "finish" drawing the image (which is impossible, seeing how it would become infinitely complex). The "big" image has been drawn to 4 steps of "lumpyness" while the smaller arms only have 3 steps of "lumpyness". The 4-step coastline is 4 times as big as the scaled down 3-step coastline, that's true. But if he had drawn the small arms with 4 steps of lumpyness as well, the perimeter of the small arms would actually be only a third of the perimeter of the full image.

That's called the convex hull, and it seems like a useful approximation. en.wikipedia.org/wiki/Convex_hull

WOAH

Of course, that means we never touch the ground either

VSAUCE!

are you sure about that ah ah you never go to school

@@eds2570 1. you didnt go to school 2. r/whoooosh

I doesn't matter which is why Steve was poking fun at me - I just like to know! :)

Hahaha.

Numberphile It actually does matter, because the shape, and subsequently the length, of the coastline keeps changing based on the changes in the sea level.

It's a crappy analogy for the difficulty of determining the properties of fractals.

Foot paths are about as arbitrary a measurement as any other.

But this answer tells you something important that you can use when you get your string and map out. The answer is scale-dependant, so you need to make sure that your map and string are at the right scale so can follow the coast with the string as precisely as you will follow it with your feet, otherwise you may get a wildly inaccurate answer. And you have to decide how exactly you're going to follow the coast with your feet - are you planning to keep your right foot in the sea and your left foot on dry land, stay fully on land but as close as is possible to walk comfortably to the sea, or follow the road closest to the coast? I think you'd get orders of magnitude differences in numbers of paces, especially on the west coast of Scotland.

Brandan09997 Well yes and no at the same time... because time is pretty abstract concept and can be measured only relative to changes that is around us (how we define a second, for example), it does not really exist by itself, does it?

isn't sunlight, as it beams between clouds, a part of "nature"? they look pretty straight to me...

@@jwc3o2 sunlight is not straight, it's just a collection of events and play of the lights and shadows which happens to look straight from distance, same as horizon so what? You do realize light is made out of photons?

@@lladerat yes, i realize light is made of photons; it's the path of light from source to wherever that certainly appears straight in a way in which the horizon does not. too, are there not crystal structures that occur with straight edges?

how do you define something being coastline instead of water or land at plank length? O.o

nachoijp I would guess you'd need to calculate the probability that any particular segment of planck length is coastline, water, or otherwise and use that to adjust the final result.

Eric Hebert What about erosion and tides?

Eric Hebert Planck length is way smaller than protons or electrons. If you look at an electron how do you know it is one of water or sand? Or maybe wet sand?

Guys. Come on, the answer is 0.

You'd be surprised. Just let him hang out and see the videos while you're watching them, you never know what's going to stick.

That's what I was thinking. Just walk along the coast with one of those wheel measuring things until you reach the end. Done deal.

Ghost00117 But it would still not satisfy the need for a universally accepted answer since creatures with smaller and smaller size/feet would find the coast longer and longer

Spot on.

Angstroms? Never heard of that before :D

Hardly usefull, the answer would be massive

***** Angstroms are usually used in chemistry.

according to my calculations, if the coastline of UK (supposedly 12,429 km [wikip.]) was measured using a 1m unit and followed the model of the Koch snowflake, then using a 0.3nm unit ruler then the coastline would be 3,919,322km long, which is roughly 10 times the average distance to the Moon. So not very useful.

Just a thought: wouldn't that apply to all distances measured that are not just the straightest line between two points? If I were to measure distance to work e.g. with a small enough ruler, it would become infinite, too, right? Way to increase travel allowance:)

Sebastian Börner However you would have to actually drive all those curves too. No shortcuts ;)

Sebastian Börner You'd have to be very drunk for your way to work to get infinitely long. ;) Since you manage to get to work in a finite time the length can't be infinite or else it would mean you're mving infinitely fast.

My bad - I've put an annotation and something in the description - but await the many comments! You were first! :)

That would require a string of infinite length.

Would you measure the "coastline" of a pair of scissors that were just a few degrees open? You'd be off by more than a third. Why should a coastline be different if it has a ton of inlets?

What he's shown there is that the limit is infinite - the sum of the lengths as the number of "lumps" approaches infinity diverges/has no solution/is infinite.

Well, sure, if that's the way you'd like to measure coastline. But that's what contour integrals are for. If you map the coastline with a series of parametric functions, you could find the contour integral of the coastline, thus finding the "arc length", i.e. the coastline.

No you can't...The point being you can put a limit to the area under the curve by using this method, but not the length. To put in a bigger picture, you can put a bigger parametric curve around England, you can put an upper limit on its area, but this cannot be done for a length because for whatever parametric curve you choose I can just go one order of magnitude lower in size, add a few random lines connecting the points and show to you that the length is longer that what you claim. Think for a while about it imagining and you will find that whatever arc you choose to measure the length, you can always make it larger if it is a fractal. Hope this helps :)

You are right from a practical standpoint. It's intuitively hard to say that you cannot measure the coastline because, for all intents and purposes, it seems doable. Now you pointed to one really important thing, although maybe not in the right context. Coastline indeed adopts the nature of a line, in the sense that it does not have a thickness. It is a mathematical construct. If I ask you how many meters of 'line' you can insert between a given width of space, say between two points 1 cm apart, your answer should be infinite. Like a winding pipe going up and down between the space, you can do this infinitely many times because these so called pipes have no width or thickness but only length. If you had the other dimension, apart from length, you could not build fractals. This is the reason why, in a seemingly small space, you can have an infinite length. Maybe this explanation helps.

This does! A lot :) Thank you!

*sea

Hmm, got to watch it...

Ding! Ding! Ding! Ding! What does she win Mr. Pardo?

How flexible? Flexibility would work the same way length does for straight rulers.

You'd still get an answer over 1000 times larger than the width of the country if you ran the ruler along all the bumps of the molecules. Besides, there's no difference between an infinite amount of connected but independently turning rulers and a flexible ruler.

That is actually a good metaphor for what happens at a sub-atomic level - the particles themselves don't have a well-defined boundary.

Unless you measure at the highest tide and lowest, and say it's length is x to y.

An infinite length but enclosing a finite (and quite small) area. Nice paradox.

Are you also going to say that pi is not irrational, because any real circle that you draw will be composed of a finite number of atoms, that you can count (and therefore express as a ratio of integers) ? That's kind of missing the point. In the real world, there is no hard "boundary" to follow. An atom (or even an electron, etc.) isn't a solid little sphere where one point in space is "inside" the particle and another (infinitely close) is "outside" the particle. Everything is on a probability gradient. You can only find real boundaries in abstract geometry and maths. The coastline in this example is just a practical example that people can relate to. The fact that the water moves back and forth is a good (large-scale) example of why the notion of a hard boundary doesn't make sense in physical terms.

Well the idea of a circle is a concept rather than a drawn or constructed figure. Therefore, it can be infinitely precise: it's only in the mind. Britain's coastline and the atoms of which it is composed aren't just a thought, they are real things. Even if atoms aren't little spheres, like you mentioned, each of them is separated from the other by large voids. What you could do is calculate a median or mean of each atom's electron cloud and take this for measurement, and then link each atom to the next with a straight line going through the void. The most precise measurement that could be made would be at the Plank length's scale, with means or medians and approximations. Unless you froze an image of the subatomic particles of the entire coastline of Britain and then used that image to calculate the length of the coastline.

Dave Tremblay The idea of a boundary is as abstract as a circle. All you can have in the physical world are approximations, that inevitably break down (becoming fuzzy or imprecise or meaningless) as you go to smaller and smaller scales. Your suggestion of "calculating a median" is, itself, an abstraction (even if it were possible, which it isn't due to the uncertain way matter behaves at those scales), that would return an imaginary line.

>implying the Planck length is a Law and not a theory

Well, no. You're comparing the 'concept' of a circle to a real-life shoreline. Of course, you can imagine a mathematically perfect circle where pi has an infinite amount of decimals, but it would be impossible to recreate in real life. Now, Britain's coastline is real, but, indeed, would suffer crazy amounts of variability and uncertainties down to the atomic level, so let's not even imagine the uncertainties on a Planck's length's scale. I agree that any hard boundary doesn't make sense in real life, but my original point was only to point out that it cannot be infinite. It must be finite.

That just made my day : P

You've missed the point of the entire video. What is the "resolution" of the "rope"? What holes and bumps matter and which ones don't? How much does a segment of the coast need to deviate from a straight line to no longer be considered straight? You could count how many footsteps it would take to walk a coast but is that the solution you want? Maybe the distance a car would drive but what would be your unit of measurement then?

TheWindWaker333 As I said, typical for mathematicians. The point of measuring something like a coastline is to have something to judge that coast by and compare it to other coasts. Therefore, we don't require an incredibly precise measurement, but a standardized one. Proposal: Use 1cmx1cmx100cm bars, and arrange them along the water line so that they touch each other, with each also touching the water line. Do this all around the world, assign it some unit and you're done.

Towe96 Ah, but countries will have different types of coastline. So two countries which have ostensibly the same coastline will be measured differently under your rope system, depending on whether they have smooth beaches or sharp rocks, for instance.

I quite like this physics approach. It seems to be the common sense way to do it.