Measuring Coastline - Numberphile

[@Zaskar1978]

Bwahaha, "The B. stands for Benoit B. Mandelbrodt", the first mathematical joke i instantly got!!

[@IRONMANAustralia]

I think the solution to this problem is simply to get the British population out there with a bunch of shovels to fill in all the little gaps and make the British Isle a square. Fix'd.

[@IRONMANAustralia]

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.

[@IRONMANAustralia]

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.

[@Hirvee5]

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

[@Supremebubble]

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

[@IRONMANAustralia]

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.

[@jeffreybernath6627]

"The 'B' stands for Benoit B. Mandelbrot." I have not laughed this hard in YEARS! Thank you Brady and Steve!

[@tastethejace]

HahahahahahahHahahahaAHAHAHHAHAHAHAHAHAH THANK YOU STEBE AND BRADY!!!° HAHAHAHAHA

[@FoxDren]

Surely you measure it in Planck lengths.

[@pineapplepie4929]

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

[@anandsuralkar582]

Of

[@killianobrien2007]

No physicists, thank you!

[@Gamer-uf1kl]

Ew physics

[@crowemagnum1337]

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.

[@gabiotta]

How do you measure a coastline? Sack the mathematician and employ an engineer.

[@Nia-zq5jl]

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

[@wesosdequeso8360]

That means tolls all over the coastline.

[@as8221]

pat a dont use a ruler

[@SpaghettiEnterprises]

lol I just commented the same thing!

[@xanderalaniz2298]

What do you think engineers are?

[@MarkusHobelsberger]

Benoit B. Mandelbrot made my day :D

[@firefish111]

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

[@WhyneedanAlias]

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

[@firefish111]

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

[@kaivalya931]

Mine too 😂

[@gabrieleporru4443]

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

[@kwinvdv]

Even if the coastline would be smooth at some level, how would you define it? Because due to waves, tides (and other phenomena) there will be fluctuations in where the sea stops and the mainland starts. And what about a river flowing into the sea, where does the coastline of the sea transit into the coastline of the river. If you would somehow take the average of the fluctuation of coastline it should become smooth at some level.

[@Wanderlust1972]

this reminds me of how archimedes found the circumference of a circle by inscribing it in polygons with more and more sides as if he was using smaller and smaller units or rulers

[@sopwithcamel5519]

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.

[@breakingglass27]

@@sopwithcamel5519 💯💯💯

[@markchadwick7972]

I ran into this same issue when attempting to calculate total elevation gain for a bicycle route. Different sources reported vastly different values.

[@MrAidanFrancis]

6:17 is, hands down, my absolute favorite math joke.

[@numberphile]

Get the paper from this video (and others) bit.ly/brownpapers

[@Adamantium9001]

The usefulness of the coastline measurement is in the knowledge of how much time it would take to travel it (or some equivalent formulation of the same). Therefore, I advocate a 1m ruler, as a nice round number that is close to the length of a human stride.

[@Layard1]

Dat Mandelbrodt joke

[@tadashimori]

When I want to know the length of a coastline, I'm asking how much I have to run to get to the other side of the coastline.

[@niaschim]

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

[@alvincay100]

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

[@AthAthanasius]

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? ".

[@Urstupidumbass]

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

[@osotanuki3359]

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

[@LimeGreenTeknii]

I'd say the ideal resolution would be the length of the average person's stride, or maybe the average stride of the shortest 5% of people or something, because if you're walking the coast, that's the highest resolution you'll really need, and I can't think of any need for a smaller resolution.

[@brodaclop]

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.

[@AaronSherman]

In practice, what we do is to measure a smooth curve between the high and low tide marks. This eliminates any complexity that cannot be practically relied upon.

[@energysage9774]

"The 'B' stands for Benoit B. Mandelbrot." Laughed way too hard at this.

[@altruistical]

I studied Fractals for my senior thesis and loved them! Thanks for this trip down memory lane!

[@Tupster]

Why is it that every time this channel talks about infinity there are countless responses trying to rationalize infinity back down to something more like what people learn in primary school. Infinity follows a set of rules that are not that complicated and make working with it a lot simpler than trying to shove it into a naive mathematical intuition developed when one was 7.

[@wdyt2121]

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

[@frisianmouve]

The real answer is that britain has no coastline because there's no land touching water, when you get down to the molecular level

[@bakedpotato3734]

WOAH

[@iamthinking2252_]

Of course, that means we never touch the ground either

[@carvman217]

VSAUCE!

[@eds2570]

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

[@proloycodes]

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

[@josie421]

The filming locations of Numberphile vids greatly amuse me.

[@WMfin]

..and this is why you don't hire mathematician to measure length of coastlines!

[@ThomasBomb45]

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

[@WMfin]

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

[@texannationalist5887]

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

[@Beertraps]

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

[@maythesciencebewithyou]

Pi is exactly 3

[@Arm4g3dd0nX]

I guess the main difficulty here is you can calculate the average fractalness of a coastline, but not the average bumpiness. If you knew the average bumpiness, you could use a sinusoidal circle or something of equal bumpiness and total volume was equal and calculate the circumference from there.

[@Macgki]

6:19 lol, nice one :D

[@SLaYeRDutch]

Whis that was min name XD

[@iamthinking2252_]

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

[@Gamer-uf1kl]

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

[@DavidSmyth666]

If I remember right, functions similar to the coastline are almost always continuous but nowhere differentiable, which is interesting.

[@TheHalalPolice]

I wasted good 7 minutes waiting for a method to do so, having calculus in mind, and then you tell me it can't be done?!

[@rehabfarid172]

lol

[@russellcowling3631]

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!

[@TheHalalPolice]

There must be a way

[@xolotltolox7626]

@@TheHalalPolice there isn't

[@ManintheArmor]

I'd use a string, measure the string, then curve the string as much as possible until I got the true length of the coastline.

[@pardeepgarg2640]

Huh ready to measure thousands of kilometres with just string

[@KainYusanagi]

THANK YOU for including the actual coastline length at the end. It would have been infuriating otherwise.@_@

[@Zanpaa]

The one real-life applications of fractals!

[@oO_ox_O]

The one application shows how things are not applicable.

[@IMortage]

Not even close.

[@Momohhhhhh]

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!

[@ThePreston159]

* scans QR code * "Zanpa" :3

[@StuartWoodwardJP]

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

[@Savvy07]

4:52 Wait I think we assumed something wrong here. It's definitely 4 times when we think of perimeter but MF(magnification factor) is 3 times. So it is possible to measure coastline if we keep scale and MF constant.

[@dermaniac5205]

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.

[@SuperAngryPacman]

I love Steve videos :)

[@kanva4]

Woah!! This is the first time I'm seeing Steve over here. I have subbed to his private channel and little did I know that he is in many Numberphile videos!

[@PatGilliland]

Great stuff! The first computer I ever saw was a PDP8 used in part to calculated lake surface areas in Canada by tracing from maps. I -think- they traced inner and outer circles to define upper and lower limits then traced the lake edges. to get the "real" number

[@putumban96]

I love that their videos touch on more complex topics now than just basic number properties!

[@alpistein]

"It's lumpy and bumpy all the way down" - That's my new motto

[@JohanCarrion]

What about measuring the distance by putting a theoretical rubber band around the coast, then taking the inner distance in a similar way (backwards rubber band) and then take an average?

[@DoctorKarul]

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

[@nicosmind3]

A practical number is how a farmer would measure it. A number which allows you to get the right area for the area of your fields. Or to be really exact. The farmer asks him/herself "how many blades of grass can I get in this field". But reality is they're going to be more concerned with metres square. And since it's a practical number related to agriculture. Then it's interest for government

[@scrambledmandible]

Steve Mould and Numberphile! In the same video! This is happiness

[@Tfin]

Well, a coastline isn't made of things smaller than molecules, and the _length_ of two molecules is a straight line, since other molecules can't get in there without there being three of them. There's your shortest ruler, as any additional length cannot be experienced. Atoms and sub-atomics will not follow any path of greater resolution _if_ they are traversing the coastline. So determine the makeup of the coast, and you've got an answer. For practical purposes, any resolution finer than the potential change in the coastline over the course of a reasonable period of time, due to tides, erosion, deposits, or whatever, is meaningless.

[@2Cerealbox]

Ive thought about this same thing before, but I assumed there was some simple way since Ive heard people talk about how many miles of coastline a country has before. Weird that I never pursued that thought beyond just a passing observation before. Fascinating stuff.

[@ravenlord4]

The plank length is 10^-35 meters. Any "ruler" smaller than that is meaningless. However for all practical purposes, you can use the size of a water molecule, since "coast" indicates the interface between land and water. Anything smaller than 3 angstroms (0.3 nm) would be meaningless since water can't touch it. So, the real world answer would be to use rulers of 0.3 nm at a coastline at mean tide. Again, engineers always have an answer for when mathematicians lose their way ;)

[@Bradstuffer]

Spot on.

[@Reydriel]

Angstroms? Never heard of that before :D

[@RascalityBass]

Hardly usefull, the answer would be massive

[@IceMetalPunk]

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

[@damienamabel]

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.

[@SpaghettiEnterprises]

Now that I've thought about it, since coastlines change constantly (erosion, etc.) the only way to get an "accurate" measurement would be to record the entirety of the coastline in a single instant. Therefore, the resolution of the "ruler" would be the highest possible resolution(at the time of the measurement) of the camera used to photograph the entirety of the coast, taken from a perspective in which the image's longest traversal is the longest axis of the coast in question.

[@LannisterFromDaRock]

Put a thread on it... Not rulers... :D

[@PinkChucky15]

I had heard before that measuring the coastline of a country couldn't be done but this video was definitely more helpful in explaining why.

[@antiHUMANDesigns]

1:38 "Steveland", epic lol! :D

[@IllusionBlade]

This guy's thumbnails are the best

[@Kaitain]

I'd have thought the only meaningful measurement was the length of the all coastal footpaths (and on that resolution where there isn't one). Because why else would you need to know the length? You could never drive along it, driving round it in a boat you can choose your distance, but you can walk along it and only really on the paths. So you might need to know that length.

[@DPortain]

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

[@seigeengine]

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

[@ryleexiii1252]

As soon as you drew that damn triangle I knew where this was going.

[@lladerat]

Fractals. Fractals everywhere. If you think about it everything can be measured as infinite because there is nothing in nature that is perfectly straight.

[@lladerat]

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?

[@jwc3o2]

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

[@lladerat]

@@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?

[@jwc3o2]

@@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?

[@Beer_Dad1975]

Love how all of these draw together. I can't wait until my oldest son is a bit older so I can share these videos with him. He loves maths already, but he's only 6, so he's not quite up to this level yet.

[@donniebryant9038]

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.

[@ashwith]

You've been quite interested in infinity lately ;-)

[@maxchristopherson9909]

I like the bubble wrap method (a name I just made up for it). just wrap it in the tightest way possible that never requires the border to curve outwards. It would be a little biased toward's certain island shapes, but at least it's a number.

[@johngrey5806]

Why does Brady ask about which side the sea is on? What does it matter?

[@numberphile]

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

[@johngrey5806]

Hahaha.

[@VedanthB9]

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.

[@JWQweqOPDH]

You simply need to know what the use is then you can calculate. Is is for building coastline houses? Run a virtual circular marker tip down the coastline with the radius of how far away a house sits from the beach, and measure length of the edge of the line the virtual marker draws.

[@tabularasa0606]

We cannot measure beyond the planck length, so how many planck lengths is it ?

[@nachoijp]

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

[@LamdaComplex]

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.

[@pandoradoggle]

Eric Hebert What about erosion and tides?

[@Gytax0]

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?

[@citrusz0rz212]

Guys. Come on, the answer is 0.

[@NAATHAAN]

Every Mathamatician ever: You can't measure the true shore/border of a country USA: *S T R A I G H T L I N E S*

[@nevesjh]

I don't think it is totally correct. If you consider what this video is saying you could not actually measure anything at all, as everything is lumpy and bumpy if we keep zooming in. You can solve that by having a standard when measuring things.

[@Asharon]

Just a question - if you have a country that is a bumpy square, then the size of the landmass would be bumpy-A*bumpy-B and since A and B would be infinite, it would be an infinite big country?

[@zelda12346]

You can just have an international standard of measurement for coastline, and everyone just uses the same ruler. Additionally, we can use some nice maths to calculate the square-acreage of Britain since the infinite coastline is an analytic boundary. I don't think perimeter is that important (area is), but that's just me, and I don't work for the CIA who spends hours everyday obsessing about qualities of all the countries in case of...something.

[@IDoNotFeelCreative]

First I was like "nice, but why am I watching this" and then the Koch curve came up and I was like yay ^^ and then it all went to "nope can't do it" and I was like awww xD

[@napoleon_bonaparte2462]

You're substituting a theoretical answer for what is originally posed to be a practical question. The question we typically have on our minds, is more like: roughly how many paces would it take for me to walk around the coastline? Or how many kilometers would it take for my car to drive around it? Which could be determined by a piece of string and a scale map. I think this actually reveals a lot about human nature as well as nature itself and the mathematics behind it. When we ask a question, we intuitively interpret a level of detail associated with the answer. If we were all the size of atoms, the immediate answer we'd come to would be the final answer you have concluded - that it is near infinite. I wonder if there is such a thing as information relativity?

[@superfluidity]

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.

[@adamgray9212]

The British way to measure the coastline is in terms of how many cups of tea the Queen could drink in the time it takes to travel around the coastline waving at people 😂

[@HalvorRaknes]

The one unambiguous scale that to me seems to stand out is to measure where you can walk if the vertical terrain isn't too challenging.

[@Ghost00117]

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.

[@ScarletAssasin]

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

[@TheSerpent753]

wouldn't this apply to rivers as well? and if it does -- what about that video you did on the length/curvature of rivers and their relationship to Pi?

[@ChibiRuah]

Can we get more videos about Fractals? I have seen them around and stuff, but my knowledge is very limited about them. PS thanks for the cool video. math never stops impressing me

[@kye4840]

Somewhat related question. Say you have a square, and transform it into a half-square (diagonally, so it looks like a quadrant with axes) and you join them with a stair pattern. You would still have the perimeter. If you keep adding more detailed stairs, how does that stair pattern end up straight? (I.e. P = 4s =/ 2s+s*sqrt2)

[@cwjalexx]

the Mandelbrot joke made me laugh

[@anthonykunda815]

Would it be possible to measure the fractal dimensions of abstract artworks with similar kinds of edges using fractal dimensions or is it only limited to the coastlines

[@marclink0]

6:22 That's where I started to laugh XD

[@QuasarRiceMints]

So, if the "B." in "Benoit B. Mandelbrot" [and not Mandelbrodt] stands for "Benoit B. Mandelbrot", there is no way of saying the extended version of Mandelbrot's name. You would keep saying "Benoit Benoit Benoit Benoit Benoit (...)" ;)

[@kingemocut]

i had to luck up fractal for my A-level just yesterday.... coincidence? i think not!.

[@AndyChesterton]

If you are walking the coastline the relevant ruler size could be considered to be the length of your stride.

[@pixelater4943]

Trace the land with a string, stretch out the string, measure the string, and boom.

[@felipebrandel5436]

That would require a string of infinite length.

[@jasondashney]

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?

[@yeahuh4128]

every Numberphile thumbnail: A person, and a background image, 1 more detail

[@daadadada]

Britain's coastline cannot be infinitely long. It's not a theoretical, mathematical problem like your triangle coastline. While you can infinitely do another iteration of a fractal coastline, you can only go down to atomic level, or subatomic, or even Plank's length level, but cannot continue to iterate infinitely. The result would be an absurdly large number, but still not infinite.

[@RFC3514]

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.

[@daadadada]

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.

[@RFC3514]

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.

[@guineapig55555]

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

[@daadadada]

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.

[@uberchops]

But for practical purposes, if we measure a coastline using, say, meters and then measure another coastline also using meters, can we draw any meaningful comparisons between the two coastlines? If Coastline A is twice the size of coastline B with meters, will it still be twice as big measured with Kilometers?

[@Flargenyargen]

"Mandelbrot's in heaven..."

[@SendyTheEndless]

6:24 was that a reference to Godel Escher Bach? :)

[@JarodBenowitz]

Sure a mathematician would say it is infinite but a physicist wouldn't. Physicist's' arch nemesis is infinity, we do everything we can to get rid of em'. If we make the assumption that the proportion of the triangles' height added to every side is equivalent to a convergent series such that we don't have perfect self-similarity then the length too converges. It would literally be an infinite nested sums that all converge.

[@primeHeretic]

Confused, surely as you increase the resolution it doesn't tend to infinity, it tends towards an actual discrete(?) value, but with a higher degree of accuracy as the resolution increases? i.e. 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ..... etc.. It would never come out millions of times longer as claimed in the video?

[@IceDave33]

A quick counterexample is the Koch snowflake in the video - for each additional layer of bumps you consider, the calculated length doubles. In general, a finite area (say, the area of Britain), can be contained within a curve of infinite length. :)

[@primeHeretic]

IceDave33 ***** I'm not sure the second measurement can ever increase over the first by 100% or more. For example, talking about n1 + n2 + n3, where each iteration is measured using a ruler half the length. If n2 were equal to or greater than n1, then surely that mesurement (or part of it) would have already been included under the n1 measurement? If you see what I mean.

[@primeHeretic]

Or in other words - it's not possible to draw a triangle where the sum of the length of the two adjacent sides is double (or greater) than the length of the longest side. Hmm, actually unless were talking about a 3D "plane" but I have no idea what happens there! And nothing about curveed surfaces was mentioned in the video.

[@IceDave33]

Sorry, thanks for the correction, my comment was indeed wrong!! I meant the length increases by a factor of 4/3 at each iteration. But you still get the same result - if I start with any finite number and keep multiplying by 4/3 then the result tends to infinity! (Or said another way, for any L, I can find an N such that after N iterations, my length is larger than L.)

[@IceDave33]

To get the 4/3 figure, just consider that each line: _ _ _ is replaced by: _ /\ _ In other words, I replace 3 pieces of equal length by 4 pieces of the same length :)

[@174wolf]

Just another example how some pretty silly math can make life far more difficult. A physicist would just take a long rope, lay it along the coast, straighten it and measure it out.

[@AlchemistOfNirnroot]

That just made my day : P

[@TheWindWaker333]

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?

[@174wolf]

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.

[@ryanisme1989]

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.

[@robertwilsoniii2048]

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

[@MattPieti]

The problem with the example where the coastline is (allegedly) simultaneously three AND four times the length of one part is that they use different standards of measurement. When it is said that it is three times the length of one part, one of the 'thirds' is bigger than the other two, so it really isn't a third. When it is said that it is four times the length of one part, however, they are all measured equally as far as length is concerned. Also, this video reminded me a lot of Riemann sums :)

[@bloodyadaku]

Can't you use a limit...?

[@RPBiohazard]

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.

[@bloodyadaku]

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.

[@anurodhmishra5555]

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 :)

[@anurodhmishra5555]

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.

[@bloodyadaku]

This does! A lot :) Thank you!

[@aiswarya.v575]

Is richardsons effect is used to determine the length of coastline in fractal dimension

[@chamalinni]

As a mathematician, as soon as he starts talking about the increasing complexity I totally disregard the possible "real" meaning, real world is not ideal and hence not as elegant as the "abstract" world (we can argue for hours about this with physicists). Then I see people arguing about the purpose of the measurement and such. People, you are missing the point, the coastline problem is an analogy to introduce you to fractal dimension and how that may contradict our "common sense". Measuring the coastline was never important here.

[@Yupppi]

I didn't realize that this video of steve mould on my lost was from numberphile and I wrote "if this was a math video, it would lead to the discovery that the coast line is infinite. Something like the coast line is like a fractal." And alas, we came to the conclusion that the coast line of Britain is infinite. The engineer question would be "why do we zoom in so much and use smaller rulers, why not zoom out and try to come up with a more useful measure, we don't need to use every meter of sand in the coast line for anything."

[@dzigerica666]

1) So at what scale coasts ARE measured in modern geometry? 2) Explain how in general we get fractal dimension - for koch's curve its log4/log3

[@JWQweqOPDH]

Geography, not geometry

[@seigeengine]

1) At whatever scale we think sounds good. There really isn't any huge consensus, especially between countries.

[@MultivectorAnalysis]

For the log4/log3 dimension, watch Vi Hart's video on "dragon scales."

[@paradoxica424]

JWQweqOPDH No, he means Geometry. Or both, possibly.

[@dzigerica666]

No, i tought geography actualy

[@jackthagingerjesus]

4:20 smaller forever wouldn't it be just to the plank length because after that point it has no meaning within reality just trying to link concepts

[@Marckvdv]

Brady, Mandelbrodt is spelled as ... Mandelbrot

[@numberphile]

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

[@frmcf]

It seems to me that the resolution or 'step length' that you need to use depends on your purposes. If you were planning to walk around the coast, following beaches and cliffs as far as possible, then you'd want to know the distance with a step length of the order of your actual step. If you were sailing or flying around the coast, you would need a much longer step length, because you're not going to be turning the boat or plane every 50cm; rather, your turns might be of the order of every km or so.

[@StephenMortimer]

the "see of brady"??

[@badhbhchadh]

*sea

[@PhilHibbs]

The coast line is constantly moving due to the tides. So any length shorter than the high-to-low-tide distance is meaningless, so there's your ruler length, you just have to keep changing the length to match the local tidal gap, and measure along the mid-line.

[@RFC3514]

So if I keep running 10 metres to the left and to the right, it's "meaningless" to measure my _height_ with a resolution below 10 metres...? A bit of a problem with your logic, there. What matters isn't how far the boundary moves along its normal, it's what the turning circle of anything that's going to _follow_ the coastline is. In other words, if your car can't turn 359.9999 degrees in the time it takes to move 1 cm forward, then you don't need a resolution of 1 cm (for a road map).

[@airsoftIreland602]

Why not just use a flexible ruler?

[@IRONMANAustralia]

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

[@gerardhuegel5397]

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

[@JWQweqOPDH]

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.

[@atleandersson9723]

04:35 Am i just dumb or is it really not The same shapes. When he's minimizing The whole thing you could see that the smaller triangles are stickning outside the little shapes. Doesn't this matter? Please explain I'm just curious not trying to prove someone wrong

[@Vebinz]

"Can't be done" isn't really an answer since obviously there are methods used by officials and government agents, and whatever those methods are should've been mentioned.

[@Caseman99]

Wasn't expecting that answer (or no answer). Great video

[@MathHacker42]

That was a terrible way to explain why that fractal (koch curve) has an infinite length. You can just show that each iteration is 4/3 times the previous iteration, which means it will get exponential larger and tend toward infinity.

[@cr10001]

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

[@RobertSzasz]

This is one of those Paradoxes where the answer feels intuitive. A more interesting paradox would be finding something that measured smaller the smaller your unit of measurement and longer the longer the unit of measurement. Are there any classes of objects that act like this?

[@DrGerbils]

No. To measure length, you need a metric, i.e., a distance function. In any distance function d(a,c)

[@seigeengine]

My I-have-no-idea-what-I'm-talking-about answer is no. All objects get more detailed the smaller you look because of how they're composed. At best we can look at a perfectly straight line, but you'd always end up with the same answer. You'd be looking for an object that got less precise the more precisely you measured it.

[@paradoxica424]

Unfortunately, we live in a physical world, not a mathematical one, so the smallest unit of measurement is the planck length (though strictly speaking, we don't really use it for measuring stuff)

[@vasilivanich3842]

The triangle inequality says no :)

[@paradoxica424]

You can't create a triangle with each side one planck length because you'd never have anything in real life small enough. Nice try, but the planck length defeats any mathematical logic you attempt to apply to deny the fact that the universe is finite.

[@Jxckers]

Sea of Brady.

[@d.lawrencemiller5755]

Does a shape with an infinite perimeter have a finite area? If so, you could choose the length of the ruler to be such that the measured perimeter is as close as you'd like to the perimeter of a square with the same area. Or circle, or ellipse, or whatever. Just as long as we agree which shape. If many shapes measured this way come up with the same length of ruler, then we can use that as our ruler length universally.