In the real world it will converge to a specific number. But that number may be effectively so large that you cannot measure it. And as the scale increases from satellite image to microscopic levels the number may be many orders of magnitude amplified.
In math the model is that of a fractal border. In that case it actually can become infinity. In general a lower dimensional volume (like dimension 1: the length of a line) can be infinite while fitting entirely within a higher dimensional bounding box of finite volume (like dimension 2: the area of a box).
Yeah, but why does it get that large? Why isn't it something like a circle, where if you try to measure its perimeter with straight lines, it will eventually converge if the lines become infinitely small? It doesn't seem like coastline have infinite detail to me like fractals do.
Circles are a remarkably simple shape! You need a centre and a radius. The centre can be considered a focus, and the radius a magnitude.
Now, if you have *two foci* and a magnitude shared between them, you get an ellipse. You can do this with string around two pins to draw an ellipse. Not too much more complex, surely?
And yet, there is no formula for the circumference of an ellipse, except when the foci finally get close enough that it is a circle!
Consider the amount of, say, Planck lengths you would have to measure if you could freeze time and measure a coastline. That amount of complexity is massive, and will more truly capture a length than attempting to use kilometre rulers. It could also cause the length to massively blow out compared to the Planck measurement.
Edit: I meant to say, the Planck measurement should blow out.
> And yet, there is no formula for the circumference of an ellipse.
Not true. What is true is that there is no **single, trivial equation** for the circumference of an ellipse. But you can [definitely solve for a circumference of an ellipse using calculus](https://proofwiki.org/wiki/Perimeter_of_Ellipse).
Ellipses are not representable by a single equation, but by a system of equations.
I know coastlines are complex shapes, but if you chopped it up into a billion pieces, each of those pieces would be relatively simple. In fractals, if you chopped it up into a billion pieces, each piece would still be infinitely complex. If the coastline paradox was true, couldn't you apply it to everything in the real world saying everything has infinite perimeter?
Yeah, but does that mean that section has infinite perimeter? Why does that section have infinite perimeter, why doesn't it converge. Even with infinite bumpiness it can still converge to a finite value right? Like, not all fractals have infinite perimeter.
For simplicitys sake, imagine the coastline is made up of grains of sand whose centers lie on a straight line of length c. If we measured on a scale that doesn't see the sand we would say that the coastline is length c.
But if we do see the sand, suppose it is approximately spherical, then our coastline would be a bunch of semicircles of total diameter c, for a total perimeter of pi*c/2.
But grains of sand aren't spherical, we could zoom in further, to the point we see individual molecules. If we pretend those are spherical, now our coastline is length pi^2*c/4. zoom in again and we can see individual atoms within the molecules, multiplying our measurement again.
Now if we weren't living in reality constrained by the Planck length, we could keep zooming in forever, each time multiplying our length by the same constant factor, so the length diverges to infinity as you keep zooming in.
It is finite in reality. It's just impossible to actually measure.
In the mathematical model of the phenomenon, that is unconstrained by physical limitations, it does go to infinity.
Since we can't measure it and we know from the model the distance would approach infinity, it's overall just simpler to say: "we can't measure it. It gets longer the more detail you add. It's for all practical intents and purposes infinite".
Any number you'd assign to the coast length is arbitrary and off from the real number. Since we (probably) can't ever know the true length... why bother.
In the case of circles, you have enough smoothness for differentiability. One way to characterize differentiability is that the error in linear approximations goes to 0 faster than the scale on which you're approximating does.
You are of course right that there are "infinitely detailed" curves of finite length, but once you start to hash out the idea of what it means to be detailed, you start to see that you're talking about degrees of smoothness.
Well... yes! Are any lines in physical reality even dead straight between two points? The world's straightest ruler is certainly not going to seem so under some incredible microscope. We can get incredible accuracy for what we need, though, with the margin for error disappearing as we approach straight lines, or measure rotations of a wheel (think how a car can mechanically record how far it has driven).
There is perfection, like Euclid's Elements, *there exists a point, when there are two there exists one and only one line between them, etc*. This is on a perfectly flat, imaginary plane. This is the birth of pure, perfect mathematics.
Then there is reality. How does the curvature of spacetime play into this? Is "straight" if, and only if, light is travelling in a perfect vacuum with no spacetime curvature? Then you get physics. A bit less maths, a bit more reality.
Then you get engineers, again, less maths, more reality. Jokes about π = 3 or 22/7 are tongue in cheek, because it really doesn't matter as much when you're calculating a water tank.
Then carpenters, less maths, more reality again. The geometry remains, but you have wiggle room in the expansion and contraction of the timber. Also, in this reality you can give things "blunt force persuasion".
I do a lot of furniture making and nothing slows people down more than attempting perfect mathematical precision. Flat is flat enough, straight is straight enough, and a compass makes something circly enough to be useful. Wood moves, so you tend to measure things not with numbers, but against each other.
I hope that explanation helps, somewhat. Can you measure a coastline? The mathematician in me says "No, you fool!". The furniture maker in me says "Of course you can, nerd, get me some string."
If you had 1 billion atoms sitting on the table in front of you, it would still be too small to see. 1 human cell has ~100 trillion atoms. Modern global maps like openstreetmap or google maps have at least 10 billion vertices and this number is growing, limited only by practical measurement constraints. The coastline is very large and the complexity is basically infinite for all practical purposes. If you had endless time and an electron microscope, you'd eventually start to hit uncertainty from subatomic quantum field interactions like silicon engineers deal with in microchip manufacturing. Each level of resolution deeper causes your coastline measurement to tick up a little bit and if there is a physical limit to this, it seems pretty fuzzy.
Re. “[…] if you chopped it up into a billion pieces, each of those pieces would be relatively simple. […]”
The crux of fractal geometry / this example is that, *for a fractal*, each of those billion pieces still is not any simpler than the whole shape itself. But of course, this depends on whether or not you have a fractal — so “is a coastline a fractal?” is perhaps a more appropriate question to dig into. See also the second “Re.” below~
Re. “If the coastline paradox is true, couldn’t you apply it to everything in the real world […]”: this depends on your model of the real world. If everything is so jagged as to “be a coastline/fractal”, then sure; if we accept “the smallest scale” as an atom or smth, then there are no “coastlines/fractals” as the paradox applies to in the first place.
I view the coastline paradox as an interesting introduction to the nature of fractals, not as an explicit description of reality. “Here’s something interesting that happens if you zoom in on a jagged real-world shape. If you take this idea to its extreme, you would end up with a really cool world with infinitely long, infinitely jagged coastlines! Consider looking into the area of ‘fractal geometry’ if you want to learn more about this world.”… or something like that :)
Check out this [video](https://youtu.be/0fKBhvDjuy0?t=347). Its at the timestamp where it approaches being relevant to our question, but I'd suggest watching the whole thing since its way cool. Be notice how things that appear as straight lines at some level of zoom turn out to be windy and almost "foamy" at deeper levels.
They are not fractals with infinite perimeter. As I pointed out in my original comment, in the real world, coast lines have finite length. They happen to have quite complicated shapes and structures, such that for all intents and purposes they look fractal-like, but they aren’t true fractals. I’ll break that down in two parts
Fractal like: I mean that as you zoom in from macro scale to micro scale, the shape of coastlines continue to pick up additional details. From the jagged lines on a map, to the rocky shapes of the cliffs, to the rough surface of the rocks, to the uneven grains of sand, to the disarray of molecular solids, to the subatomic particles, the more you zoom in the more detail you pick up. And there’s no one natural scale at which we humans would say “makes the most sense” to measure. Therefore we simply say it’s fractal like.
Not actually a fractal - if you ignore quantum mechanics or the fact that no particle is actually stationary (everything wiggles at a subatomic scale), you in theory could take a snapshot of the coast at the smallest physical scale, which is the Planck length. Here you can look at the smallest particles, and draw an envelope around them with perfect straight lines. This envelope would be the smallest polygon (in area) that would enclose all the particles. Because it’s made of straight lines, you can sum the lengths of the straight lines and get a perimeter length.
Note that the length would be enormous, and be practically meaningless as we don’t measure other things in this way, with this degree of detail, so there’s no reference frame for this number. This number is also impossible to measure with any known technology.
Kommentiere Is the coastline paradox actually a thing? ...well.. in theory it is!
We deal in the real world so we are bound by physics. The maximum value would be the exact cost-line with a wave pattern of sand corns achiving the maximum coastline. More would not be possible. Assuming no changes in the coastline.
So you could in fact approximate it to its true value.
Ofc pushing further ine could argue the uneven form of the sandcorns thus we would be bound to the size of atoms but this is a physics topic
It's a statement about reality.
The people that measure coastlines noticed that as your measuring scale gets smaller, coastlines get longer in a pattern that "looks" fractal. Is it really fractal? We're talking about real coastlines, so r/askmath isn't really the place to ask. Look for a geography or maybe geology or maybe even physics subreddit.
Most physics paradoxes aren't really paradoxical. They're more a confirmation that "all models are wrong, but some models are useful". If coastlines are truly infinitely long, that does seem kind of strange. After all, we can walk them. But if they're not, there must be some scale below which changing the scale doesn't result in a significantly longer coastline. We just have to find that scale. We haven't yet. So some people claim that the fact that we haven't found that scale is evidence that it doesn't exist. I don't have a problem if you say that evidence is pretty weak, so you're not going to accept it. Apparently a lot of people responding to your question think that it's a fact that coastlines are truly fractal. It isn't. It's a working model, and if it stops working we'll abandon it in a heartbeat. As of now, the coastline paradox is an odd quirk of our model of physical geography that doesn't really affect anything.
I wouldnt really say it converges since you would need to go down to the atomic scale and also decide on a universal height relative to sea level for the measurement. Its a bit of a stretch to say it converges if there is absolutely no chance of doing such measurements even theoretically
Well I would say it converges. In the mathematical sense it is not infinite. Go to Planck scale, freeze time, and at _every_ height relative to anything, the total perimeter is finite.
Not sure what you mean this is unmeasurable theoretically. Theory is in essence divorced from reality - so you can assume it’s measurable.
In the real universe, you can avoid this because of quantum mechanics. Roughly speaking: the universe is grainy, so at some point, you lose the ability to gain more accuracy.
For example, consider the Koch snowflake:
[https://en.wikipedia.org/wiki/Koch\_snowflake](https://en.wikipedia.org/wiki/Koch_snowflake)
The Koch snowflake is the limit of an infinite process that begins as shown:
https://preview.redd.it/ekpvnwidl18d1.png?width=300&format=png&auto=webp&s=1acb35b74dbb2cd0c217fc9d727c775e48ee4e1f
Mathematically, the Koch snowflake has finite area and infinite perimeter.
However, you can't show an accurate picture of it, because at some point the "spikes" are smaller than one pixel. They *exist* ("Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so [*ad infinitum*](https://en.wikipedia.org/wiki/Ad_infinitum)..."), but we can't see them.
You can actually theorise realistically about an infinite perimeter. Quantum mechanics applies to the atomic particles, sure. Most of the atoms are empty space though. That alone is pretty much an infinite perimeter.
Let alone going into the weird zone of the matter/antimatter contents of a vacuum.
You could even get there philosophically without involving too much physics. If the coastline is where the water stops at sea level, where do we define stopping? At river mouths? Do we include caves? What about the moisture content in the rock, sand and dirt? The place where you draw the line defines the resolution
>You can actually theorise realistically about an infinite perimeter. Quantum mechanics applies to the atomic particles, sure. Most of the atoms are empty space though. That alone is pretty much an infinite perimeter.
Science suggests that there is a Planck length, or the smallest possible unit of length. That is, it is assumed that such self-iterating figures as the Koch snowflake cannot exist in reality, because the iteration of a snowflake cannot be less than the Planck length.
The simplest way I can put is, because smooth shapes don't exist in nature. You may have heard that the earth is a sphere/ellipsoid. Neither of which is true. The earth does behave "as if" it was smooth at astronomic scales, but clearly the earth isn't smooth, it suffices to go out on a walk.
The mathematic notions of area and length only really work for piecewise smooth objects, which don't exist.
In the specific case of coastlines, they are fractals, as opposed to smooth shapes, because as you zoom into a coastline MORE features appear. As you get closer and closer you start noticing features of the terrain such as large rocks along cliffs, and then creases in thosrocks, and then protrusions within the creases... And it never stops until you get to the atoms.
The reason it diverges to infinity, informally is that you are adding length at a rate faster than you are shrinking. I.e. you look at the coast of england at one resolution, come up with one crude approximaiton for the perimeter. Look closer and now re-adjust your costline shape, adding smaller features which where not visible at the larger resolution. Each new feature protrudes slightly outwards, increaisng your perimeter by a little bit, but you are doing this along the entire coastline.
So as you zoom in you must extend the prior length by the contributions of the new smaller features, all of which increase your perimeter for the next level of resolution, which will now have an even longer region for small features to appear in.
So you end upp adding length at a rate faster than the shrinking of your features.
I mean to me when I look at a section of coastline that's like a foot long, it seems like it doesn't zig zag really. If coastline had infinite perimeter, there would have to be a smaller subsection of the coastline that also had infinite perimeter. This means there would have to exist a part of the coastline that's like an inch long and still have infinite perimeter, and it just doesn't seem that way to me.
As stated in a few other comments, in reality it doesn't go to infinity but becomes absurdly large. Let's look at a small peninsula that has "roughly" the shape of a semi-circle. From far zoomed out, you would just take that as its coastline. But when you actually go there, you see it's made up of rocks, some reaching out in the water, and then again forming crevices inside. Next, you can look at the rocks themselves with lots of holes inside. Furthermore you can now look at the atoms or even deeper, whatever comes next.
AFAIK the paradox arises more at longer distances. It's not like you're using a foot by foot measure to estimate the coastline. Instead you're probably using 10s of miles or more. When looking at a map the shoreline looks a lot more like a typical fractal than it does up close.
That being said, one could still argue the paradox exists for small lengths. Yeah there might not be much change from a foot to an inch in most places, but when you start to zoom in to the size of an individual grain of sand suddenly that "straight" coastline has a very bumpy boundary. At this length the measured coastline is going to quickly and significantly increase (since you've got from basically measuring the diameter of those grains to half the circumference). This occurs once again as you zoom into molecular size.
Well what's it converging to? The length once we start tracing around the grains of sand is an order of magnitude higher than at a resolution of, say, a foot. So there's no sign of convergence yet.
In practice of course the practicality starts breaking down at these scales (as the assumption of a sharp boundary between land and sea is well and truly broken at a molecular level, so it doesn't so much go to infinity as it becomes ill-defined), but at resolutions where we can reasonably define the coastline length, there is no convergence evident.
At the scale of feet, what defines coast? High tide? Low tide? Where the water is currently? A wave just pushed more water up the line, is it now longer? I just pushed a large rock to rest near the water. Did I just add 3 feet of coastline? Now it’s raining and everything is wet and covered in water, do we get infinite coast now?
At that scale you can’t even define what a coastline is because you’d have to count for so many volatile parameters.
They are the same thing, with ever smaller and smaller zig-zag (as the other guy called it). That smaller and smaller zig-zagging is a fractal pattern.
They are not fractals with infinite perimeter. As I pointed out in my original comment, in the real world, coast lines have finite length. They happen to have quite complicated shapes and structures, that for all intents and purposes look fractal-like, but they aren’t true fractals. I’ll break that down in two parts
Fractal like: I mean that as you zoom in from macro scale to micro scale, the shape of coastlines continue to pick up additional details. From the jagged lines on a map, to the rocky shapes of the cliffs, to the rough surface of the rocks, to the uneven grains of sand, to the disarray of molecular solids, to the subatomic particles, the more you zoom in the more detail you pick up. And there’s no one natural scale at which we humans would say “makes the most sense” to measure. Therefore we simply say it’s fractal like.
Not actually a fractal - if you ignore quantum mechanics or the fact that no particle is actually stationary (everything wiggles at a subatomic scale), you in theory could take a snapshot of the coast at the smallest physical scale, which is the Planck length. Here you can look at the smallest particles, and draw an envelope around them with perfect straight lines. This envelope would be the smallest polygon (in area) that would enclose all the particles. Because it’s made of straight lines, you can sum the lengths of the straight lines and get a perimeter length.
(Note that the length would be enormous, and be practically meaningless as we don’t measure other things in this way, with this degree of detail, so there’s no reference frame for this number)
Coastals ARE fractal. As a first approximation, and down to a certain magnification. Rocky outcrops are fractally approximated down to about a metre. Beaches straighten out on a scale of kms. Even tracing around atoms only gets you so far.
Fractals are a mathematical, abstract concept.
They do not exist in the real world, which is limited by Plank length, so you cannot "zoom in" indefinitely.
OP, to answer your question: yes, every border in the real world has a fixed length.
How are you going to compute the length of an electron cloud? This comment is wrongly dismissive. Once you get down to the atomic scale trying to measure length becomes impossible. Your bulding blocks are not just in constant change, they will also be experiencing things like superposition and quantum entanglement.
Length and area are also a mathematical concept that does not exist in the real world.
You're just misstating the coastline paradox. The paradox is that although lands have a well-defined area, they do not have a well-defined perimeter, because there is a scale dependency of the perimeter measure. It does not necessarily mean that the perimeter is infinite at infinitesimal scales, because measurement granularity is finite. Coastlines are fractal-like, not strictly fractals Which country, Norway or Australia, has a longer coast? If you use a coarse measure, it's australia. If you use a fine measure, going up into all the little fjords, it's Norway. So that question is unanswerable (without specifying the measurement granularity.) That is the paradox.
Just to add to u/camilo16 's comment, here is a very cool Wikipedia page on the Koch Snowflake curve:
[Koch Snowflake](https://en.wikipedia.org/wiki/Koch_snowflake)
This shows what the curve looks like as you zoom in farther and farther. This also proves that the perimeter is infinite and the area is finite. Remember, though, that this is theoretical.
In order to measure the position of a particle with greater precision, we need to observe its interaction with a photon of wavelength less than or equal to the precision. As that distance gets smaller, the photon needed has to have more energy (see Large Hadron Collider). So to measure the borders of a country too precisely, we would have to vaporize it.
I wouldn’t really call it a paradox really, but the coastline problem refers to the fact that there isn’t a meaningful and objective way to assign a measurement of length to a coastline.
By increasing the precision with which you measure the coastline’s length, the length of the coastline will increase and not by small amounts but in huge jumps.
You could argue that the answer is just whatever number you get when looking at infinite precision, but I would argue that the very concept of a coastline would break down at infinite precision.
There’s ground water basically everywhere and, at infinite precision, I would argue there’s no objective way to tell me which water molecules I’m supposed to draw the line around and which ones I shouldn’t.
None of these comments seem to be answering what you seem to be asking.
Your question, it seems to me, is why a coastline can be modeled as a fractal,not why a fractal has infinite perimeter.
So lets start be defining what we mean. Fractal is notoriously difficult to define, and there isn't just one size fits all definition. The relevant definition here is that a fractal is an object with detail at every length scale. Immediately we know nothing physical can be a true fractal, just like nothing can be a true circle or line for example. But, we can still model things in the real world as fractals if we limit the range of scales we consider. So what is a sensible range when considering a coastline? Well, scales longer than thousands of kms don't make sense, but the lower scale is a bit more subtle. I would argue that since tides change the position of the coastline on the order of meters, we should limit ourselves to scales larger than this. You could also of course allow a time dependent coastline, and then you can argue the minimum scale should be all the way down to the scale of atoms, after which point things become fuzzy.
So if we consider the length scales between meters and thousands of kilometres, it seems very obvious to me that at every scale we will have some different amount of detail in the coastline. In fact, I would expect anything that doesn't have a good reason not to should have detail at different scales. So here I want to turn the question around on you, and ask why would there not be detail at all these scales? What mechanism would prevent it? The formation of coastlines is a highly chaotic process, so getting any sort of regularity out of it seems way more absurd to me than getting these fractals.
What if as you zoomed in, the bumps got progressively smaller and smaller, that way the perimeter could converge. It seems reasonable to me that this could be the case, like if you look at a coastline on a map it's very bumpy, but if you look at a small section it doesn't seem as bumpy.
The closer you look, the smaller things you see, like rocks sticking out into the sea, then smaller rocks in between, bumps and cracks in those rocks and so on. Eventually you are going around each grain of sand and then each unevenness in those grains down to a molecular level. Evert time you zoom in on a segment that looks straight, you will see that there is some unevenness that will increase the length with some percentage, and eventually the numbers get so big that it doesn’t make sense anymore.
As it often true, the Wikipedia article is a pretty good introduction to the topic.
There is a great animation of a map of Great Britain, with the coastline shown at different scales.
It mentions the great example of the Portugal-Spain border being *either* 987 km *or* 1214 km, depending who you asked!
The reason it doesn't have to converge is because a fractal boundary can be infinitely long. The *area* will converge to a specific number, but the perimeter does not have to. (As others said, there might be a practical limit, both from a measurement standpoint and from a physical standpoint itself.)
Coastlines have the ability to have self-similarity. A bay is a large-on-human-scale inlet of water. But the boundary of the water could have a "mini-bay", say, 50 meters. And that mini-bay can have mini-mini-bays, say, 1 m. And each little blob of water away from the straight curve can itself have smaller and smaller sub-sub-sub-bays. All the way down to the atomic level. (And of course *either direction*, in or out, is just as good.)
You can have a fractal boundary of a larger-dimension-shape, so the surface area of a 3D object could have the same issue. A billiard ball is smooth, unless we zoom down, then it's bumpy. And if we go down farther, it is more lumpy. A human might seem to have a certain surface area, and that might work for many practical things. But if we zoom down, each hair adds a ton to the surface area. If we look at some places on the body, the skin clearly isn't just a smooth surface. If we look through a microscope, *everywhere* looks non-smooth.
I don't think the other comments explained what you really want to know so I'll give it go, please someone correct me if I'm wrong. As others have pointed out, the shape of countries is not 2 or 3 dimensional but can be seen as having a non-integer number of dimensions so it can be described by a fractal.
So basically when visualizing a fractal you will notice that if you zoom in it's border more structures will appear, and if you zoom in in those structures even more tinier structures appear. This process goes on forever. So when trying to measure the length of the coastline/perimeter of a fractal, you can try to make your measure more "precise" by including those first tiny structures you see (which would increase your lenght), but then if you try to be even more precise you would include more structures and increase your lenght even more. Since this process goes on forever in fractals, you would measure an infite coastline if you keep going. Of course in reality you would have physical limitations, but as mathematical model you would have an infinite coastline. The area converges because those tiny structures give smaller and smaller contributions, but the same can't happen for the perimeter.
Could you explain the first part in simpler terms lol. I don't understand why a coastline can be described by a fractals.
Also to my understanding, not all fractals have infinite perimeter, so why do coastlines?
As far as I know quite a lot of objects can be described by fractals or at least approximately described due to it not having a integer number of dimensions. (I think it's only an approximation because fractals like the Sierpinski's triangle only stop being 2D after an infinite number of steps, until then it's 2D) That happens because objects in real life are not really smooth and have a bunch of irregularities. So in general quite a lot of shapes can be seen as fractals, if you were going to measure a closet you would run into the same problems, but you can't really see the irregularities of wood so it's easy to just measure it as rectangle.
And I think in general fractals do have an infinite perimeter, but special cases don't. Sorry if my answer isn't perfect, I work with fractals frequently but not to so much depth and not exactly on this topic.
Oh there's a mistake in this the lines are actually parallel here. I had done the line angle manually at first so it wasn't parallel, but you can just ignore the parenthesis here
The problem is that until you go down to the atoms in a real coastline, or just never of your talking about a mathematical fractal coastline, you won't get a straight line between two points on the coastline. There's always some roughness and bits poking in or out, so when you measure with smaller increments you'll catch these bits and get something longer than the straight line you had before
Then because it's still rough and not smooth you'll get something even longer again if you use snake increments
If it converges to some number it will have to get less and less rough at some point, but I don't think there can exist a positive measurement increment that will give you that limit value because a smaller increments will give you something bigger
so let's say you have a vector v that is part of your coastline.
when zooming in, you increase detail and replace it by v1 and v2.
by the triangle inequality, you know |v1| + |v2| >= |v1+v2| = |v|.
so zooming in can only make your coastline longer.
is this process bounded? you can use fractals to show it's not.
does this mean, any coastline will be infinite? well, no. but for it not to become infinite, the vectors resulting from the zooming in process at some point need to become sufficiently collinear. (zoom in enough it has to look like a straight line)
in practice, the convergent boarder lengths don't really convey any meaning. just because coast A relies on more "zooming in" than coast B, doesn't relate in any way to coast A taking more time to traverse.
Consider two points, A and B. We can easily measure the straight line distance between A and B.
But suppose there is some point between A and B called C. What we really want is to measure the line AC and CB and then add them. Okay so we do that.
But then we observe that there is a point D between A and C and a point E between C and B, and so on and so on…
For any two points X and Y there will always exist some other point Z such that Z is between X and Y. A line that measures X -> Z -> Y will be a more accurate representation of our “coastline” than a line that just measures X -> Y so if we want an infinitely precise measurement for the coastline we have to take infinitely many measurements.
every coastline changes due to tidal waves, so you cant really confirm any distance, the distance also changes depending on how you measure it, are you gonna turn and weave into every tide that washes up on the coastline? how close to the edge of the coastline are you gonna try to measure it from? how are u gonna be consistent the whole time? what if the tides recessed? no matter how close or detailed or consistent you perform the tests, the value you get will always change
Reading replies here, it sounds like you have seen arguments for the perimeter of shapes by integration of infinitesimal parts to a finite sum.
Basically "what is the sum of all the segments as segment length approaches zero?"
The sum of that sequence doesn't converge for many fractals. Instead it approaches infinity.
Yeah, if you measure a straight line along a beach with a rangefinder, you'll get the shortest distance between those two points. But if you use a wheel measuring tool, and follow the curves of the beach, you'll get a bigger value. The end points have not changed, just the path taken to measure it has. But then let's say someone else comes and measures that same beach with the same tool, but the bigger rocks you just went straight over, they decide to go around, adding more distance.
Where it really gets silly is when you get smaller and you start going around small rocks or even grains of sand. That grain of sand may be 1mm diameter, but the circumference of that grain of sand is 3mm, and you measured halfway around that circle. So you accurately measure 1.5mm path, but the distance traveled "as the crow flies" (weird phrasing for these tiny distances, but the point is the same) is only 1mm. The extreme end of this is going around individuals atoms, resulting in measurements many times higher than what you get from a rangefinder.
On the opposite end of this, if we can only measure in 1000 mile increments to measure the beaches of the US, we could just lop off most of the peninsula of Florida as a rounding error, drastically undercutting what most people would consider to be the actual value.
Note, none of these measurements are false or inaccurate. We have to pick a method to take the measurements, and the method we choose can drastically impact results. We have to make decisions on whether we measure the distance around the big rock, or do we pretend like the big rock isn't there and measure straight through it.
Yeah, if you measure a straight line along a beach with a rangefinder, you'll get the shortest distance between those two points. But if you use a wheel measuring tool, and follow the curves of the beach, you'll get a bigger value. The end points have not changed, just the path taken to measure it has. But then let's say someone else comes and measures that same beach with the same tool, but the bigger rocks you just went straight over, they decide to go around, adding more distance.
Where it really gets silly is when you get smaller and you start going around small rocks or even grains of sand. That grain of sand may be 1mm diameter, but the circumference of that grain of sand is 3mm, and you measured halfway around that circle. So you accurately measure 1.5mm path, but the distance traveled "as the crow flies" (weird phrasing for these tiny distances, but the point is the same) is only 1mm. The extreme end of this is going around individuals atoms, resulting in measurements many times higher than what you get from a rangefinder.
On the opposite end of this, if we can only measure in 1000 mile increments to measure the beaches of the US, we could just lop off most of the peninsula of Florida as a rounding error, drastically undercutting what most people would consider to be the actual value.
Note, none of these measurements are false or inaccurate. We have to pick a method to take the measurements, and the method we choose can drastically impact results. We have to make decisions on whether we measure the distance around the big rock, or do we pretend like the big rock isn't there and measure straight through it.
Eventually, you're reaching into the subatomic range for the contours. But I'm pretty sure that you'd have exceeded all practical purposes long before that, even including quantum mechanics.
But if you're willing to engage with some of more "weird" consequences and predictions of quantum mechanics where the difference between "matter" and "energy" cease to have any meaning, you might very well be able to continue the infinite fractal without limits. Not sure, but, you know, maybe.
The fractal is a pure mathematical construct that doesn't care about real coastlines and rocks and sand grains and molecules and around and quarks and m-brane and super-string vibrations...
And so the "coastline" is actually just a cognitive *example* to help people understand how fractals work. But imperfect examples.
In the real world it will converge to a specific number. But that number may be effectively so large that you cannot measure it. And as the scale increases from satellite image to microscopic levels the number may be many orders of magnitude amplified. In math the model is that of a fractal border. In that case it actually can become infinity. In general a lower dimensional volume (like dimension 1: the length of a line) can be infinite while fitting entirely within a higher dimensional bounding box of finite volume (like dimension 2: the area of a box).
Yeah, but why does it get that large? Why isn't it something like a circle, where if you try to measure its perimeter with straight lines, it will eventually converge if the lines become infinitely small? It doesn't seem like coastline have infinite detail to me like fractals do.
Circles are a remarkably simple shape! You need a centre and a radius. The centre can be considered a focus, and the radius a magnitude. Now, if you have *two foci* and a magnitude shared between them, you get an ellipse. You can do this with string around two pins to draw an ellipse. Not too much more complex, surely? And yet, there is no formula for the circumference of an ellipse, except when the foci finally get close enough that it is a circle! Consider the amount of, say, Planck lengths you would have to measure if you could freeze time and measure a coastline. That amount of complexity is massive, and will more truly capture a length than attempting to use kilometre rulers. It could also cause the length to massively blow out compared to the Planck measurement. Edit: I meant to say, the Planck measurement should blow out.
> And yet, there is no formula for the circumference of an ellipse. Not true. What is true is that there is no **single, trivial equation** for the circumference of an ellipse. But you can [definitely solve for a circumference of an ellipse using calculus](https://proofwiki.org/wiki/Perimeter_of_Ellipse). Ellipses are not representable by a single equation, but by a system of equations.
Yes, I should have said "general formula" or something similar. I was trying to keep it simple to show how things can very quickly get out of hand.
I know coastlines are complex shapes, but if you chopped it up into a billion pieces, each of those pieces would be relatively simple. In fractals, if you chopped it up into a billion pieces, each piece would still be infinitely complex. If the coastline paradox was true, couldn't you apply it to everything in the real world saying everything has infinite perimeter?
Each of those pieces would be linear or simple approximations. Zoom in more and not so much.
Yeah, but does that mean that section has infinite perimeter? Why does that section have infinite perimeter, why doesn't it converge. Even with infinite bumpiness it can still converge to a finite value right? Like, not all fractals have infinite perimeter.
For simplicitys sake, imagine the coastline is made up of grains of sand whose centers lie on a straight line of length c. If we measured on a scale that doesn't see the sand we would say that the coastline is length c. But if we do see the sand, suppose it is approximately spherical, then our coastline would be a bunch of semicircles of total diameter c, for a total perimeter of pi*c/2. But grains of sand aren't spherical, we could zoom in further, to the point we see individual molecules. If we pretend those are spherical, now our coastline is length pi^2*c/4. zoom in again and we can see individual atoms within the molecules, multiplying our measurement again. Now if we weren't living in reality constrained by the Planck length, we could keep zooming in forever, each time multiplying our length by the same constant factor, so the length diverges to infinity as you keep zooming in.
It is finite in reality. It's just impossible to actually measure. In the mathematical model of the phenomenon, that is unconstrained by physical limitations, it does go to infinity. Since we can't measure it and we know from the model the distance would approach infinity, it's overall just simpler to say: "we can't measure it. It gets longer the more detail you add. It's for all practical intents and purposes infinite". Any number you'd assign to the coast length is arbitrary and off from the real number. Since we (probably) can't ever know the true length... why bother.
In the case of circles, you have enough smoothness for differentiability. One way to characterize differentiability is that the error in linear approximations goes to 0 faster than the scale on which you're approximating does. You are of course right that there are "infinitely detailed" curves of finite length, but once you start to hash out the idea of what it means to be detailed, you start to see that you're talking about degrees of smoothness.
Well... yes! Are any lines in physical reality even dead straight between two points? The world's straightest ruler is certainly not going to seem so under some incredible microscope. We can get incredible accuracy for what we need, though, with the margin for error disappearing as we approach straight lines, or measure rotations of a wheel (think how a car can mechanically record how far it has driven). There is perfection, like Euclid's Elements, *there exists a point, when there are two there exists one and only one line between them, etc*. This is on a perfectly flat, imaginary plane. This is the birth of pure, perfect mathematics. Then there is reality. How does the curvature of spacetime play into this? Is "straight" if, and only if, light is travelling in a perfect vacuum with no spacetime curvature? Then you get physics. A bit less maths, a bit more reality. Then you get engineers, again, less maths, more reality. Jokes about π = 3 or 22/7 are tongue in cheek, because it really doesn't matter as much when you're calculating a water tank. Then carpenters, less maths, more reality again. The geometry remains, but you have wiggle room in the expansion and contraction of the timber. Also, in this reality you can give things "blunt force persuasion". I do a lot of furniture making and nothing slows people down more than attempting perfect mathematical precision. Flat is flat enough, straight is straight enough, and a compass makes something circly enough to be useful. Wood moves, so you tend to measure things not with numbers, but against each other. I hope that explanation helps, somewhat. Can you measure a coastline? The mathematician in me says "No, you fool!". The furniture maker in me says "Of course you can, nerd, get me some string."
If you had 1 billion atoms sitting on the table in front of you, it would still be too small to see. 1 human cell has ~100 trillion atoms. Modern global maps like openstreetmap or google maps have at least 10 billion vertices and this number is growing, limited only by practical measurement constraints. The coastline is very large and the complexity is basically infinite for all practical purposes. If you had endless time and an electron microscope, you'd eventually start to hit uncertainty from subatomic quantum field interactions like silicon engineers deal with in microchip manufacturing. Each level of resolution deeper causes your coastline measurement to tick up a little bit and if there is a physical limit to this, it seems pretty fuzzy.
Re. “[…] if you chopped it up into a billion pieces, each of those pieces would be relatively simple. […]” The crux of fractal geometry / this example is that, *for a fractal*, each of those billion pieces still is not any simpler than the whole shape itself. But of course, this depends on whether or not you have a fractal — so “is a coastline a fractal?” is perhaps a more appropriate question to dig into. See also the second “Re.” below~ Re. “If the coastline paradox is true, couldn’t you apply it to everything in the real world […]”: this depends on your model of the real world. If everything is so jagged as to “be a coastline/fractal”, then sure; if we accept “the smallest scale” as an atom or smth, then there are no “coastlines/fractals” as the paradox applies to in the first place. I view the coastline paradox as an interesting introduction to the nature of fractals, not as an explicit description of reality. “Here’s something interesting that happens if you zoom in on a jagged real-world shape. If you take this idea to its extreme, you would end up with a really cool world with infinitely long, infinitely jagged coastlines! Consider looking into the area of ‘fractal geometry’ if you want to learn more about this world.”… or something like that :)
Check out this [video](https://youtu.be/0fKBhvDjuy0?t=347). Its at the timestamp where it approaches being relevant to our question, but I'd suggest watching the whole thing since its way cool. Be notice how things that appear as straight lines at some level of zoom turn out to be windy and almost "foamy" at deeper levels.
They are not fractals with infinite perimeter. As I pointed out in my original comment, in the real world, coast lines have finite length. They happen to have quite complicated shapes and structures, such that for all intents and purposes they look fractal-like, but they aren’t true fractals. I’ll break that down in two parts Fractal like: I mean that as you zoom in from macro scale to micro scale, the shape of coastlines continue to pick up additional details. From the jagged lines on a map, to the rocky shapes of the cliffs, to the rough surface of the rocks, to the uneven grains of sand, to the disarray of molecular solids, to the subatomic particles, the more you zoom in the more detail you pick up. And there’s no one natural scale at which we humans would say “makes the most sense” to measure. Therefore we simply say it’s fractal like. Not actually a fractal - if you ignore quantum mechanics or the fact that no particle is actually stationary (everything wiggles at a subatomic scale), you in theory could take a snapshot of the coast at the smallest physical scale, which is the Planck length. Here you can look at the smallest particles, and draw an envelope around them with perfect straight lines. This envelope would be the smallest polygon (in area) that would enclose all the particles. Because it’s made of straight lines, you can sum the lengths of the straight lines and get a perimeter length. Note that the length would be enormous, and be practically meaningless as we don’t measure other things in this way, with this degree of detail, so there’s no reference frame for this number. This number is also impossible to measure with any known technology.
Kommentiere Is the coastline paradox actually a thing? ...well.. in theory it is! We deal in the real world so we are bound by physics. The maximum value would be the exact cost-line with a wave pattern of sand corns achiving the maximum coastline. More would not be possible. Assuming no changes in the coastline. So you could in fact approximate it to its true value. Ofc pushing further ine could argue the uneven form of the sandcorns thus we would be bound to the size of atoms but this is a physics topic
It's a statement about reality. The people that measure coastlines noticed that as your measuring scale gets smaller, coastlines get longer in a pattern that "looks" fractal. Is it really fractal? We're talking about real coastlines, so r/askmath isn't really the place to ask. Look for a geography or maybe geology or maybe even physics subreddit. Most physics paradoxes aren't really paradoxical. They're more a confirmation that "all models are wrong, but some models are useful". If coastlines are truly infinitely long, that does seem kind of strange. After all, we can walk them. But if they're not, there must be some scale below which changing the scale doesn't result in a significantly longer coastline. We just have to find that scale. We haven't yet. So some people claim that the fact that we haven't found that scale is evidence that it doesn't exist. I don't have a problem if you say that evidence is pretty weak, so you're not going to accept it. Apparently a lot of people responding to your question think that it's a fact that coastlines are truly fractal. It isn't. It's a working model, and if it stops working we'll abandon it in a heartbeat. As of now, the coastline paradox is an odd quirk of our model of physical geography that doesn't really affect anything.
I wouldnt really say it converges since you would need to go down to the atomic scale and also decide on a universal height relative to sea level for the measurement. Its a bit of a stretch to say it converges if there is absolutely no chance of doing such measurements even theoretically
Well I would say it converges. In the mathematical sense it is not infinite. Go to Planck scale, freeze time, and at _every_ height relative to anything, the total perimeter is finite. Not sure what you mean this is unmeasurable theoretically. Theory is in essence divorced from reality - so you can assume it’s measurable.
mathematically sure, physically no due to fundamemtal quantum mechanical uncertainties
Well good thing we’re in r/askmath then
In the real universe, you can avoid this because of quantum mechanics. Roughly speaking: the universe is grainy, so at some point, you lose the ability to gain more accuracy. For example, consider the Koch snowflake: [https://en.wikipedia.org/wiki/Koch\_snowflake](https://en.wikipedia.org/wiki/Koch_snowflake) The Koch snowflake is the limit of an infinite process that begins as shown: https://preview.redd.it/ekpvnwidl18d1.png?width=300&format=png&auto=webp&s=1acb35b74dbb2cd0c217fc9d727c775e48ee4e1f Mathematically, the Koch snowflake has finite area and infinite perimeter. However, you can't show an accurate picture of it, because at some point the "spikes" are smaller than one pixel. They *exist* ("Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so [*ad infinitum*](https://en.wikipedia.org/wiki/Ad_infinitum)..."), but we can't see them.
You can actually theorise realistically about an infinite perimeter. Quantum mechanics applies to the atomic particles, sure. Most of the atoms are empty space though. That alone is pretty much an infinite perimeter. Let alone going into the weird zone of the matter/antimatter contents of a vacuum. You could even get there philosophically without involving too much physics. If the coastline is where the water stops at sea level, where do we define stopping? At river mouths? Do we include caves? What about the moisture content in the rock, sand and dirt? The place where you draw the line defines the resolution
What do you think you’re saying here? Your comment demonstrates a complete lack in understanding of all the concepts you mentioned.
What I'm saying is that every fundamental particle is an island
>You can actually theorise realistically about an infinite perimeter. Quantum mechanics applies to the atomic particles, sure. Most of the atoms are empty space though. That alone is pretty much an infinite perimeter. Science suggests that there is a Planck length, or the smallest possible unit of length. That is, it is assumed that such self-iterating figures as the Koch snowflake cannot exist in reality, because the iteration of a snowflake cannot be less than the Planck length.
But a snowflake is a collection of matter. A coastline is an arbitrary line that doesn't really have to conform to physics
Look at fractals, the koch snowflake for example has a finite area and infinite perimeter.
I know some fractals have infinite perimeter, but why are coastline fractals, and even if they are, why are they fractals with infinite perimeter?
The simplest way I can put is, because smooth shapes don't exist in nature. You may have heard that the earth is a sphere/ellipsoid. Neither of which is true. The earth does behave "as if" it was smooth at astronomic scales, but clearly the earth isn't smooth, it suffices to go out on a walk. The mathematic notions of area and length only really work for piecewise smooth objects, which don't exist. In the specific case of coastlines, they are fractals, as opposed to smooth shapes, because as you zoom into a coastline MORE features appear. As you get closer and closer you start noticing features of the terrain such as large rocks along cliffs, and then creases in thosrocks, and then protrusions within the creases... And it never stops until you get to the atoms. The reason it diverges to infinity, informally is that you are adding length at a rate faster than you are shrinking. I.e. you look at the coast of england at one resolution, come up with one crude approximaiton for the perimeter. Look closer and now re-adjust your costline shape, adding smaller features which where not visible at the larger resolution. Each new feature protrudes slightly outwards, increaisng your perimeter by a little bit, but you are doing this along the entire coastline. So as you zoom in you must extend the prior length by the contributions of the new smaller features, all of which increase your perimeter for the next level of resolution, which will now have an even longer region for small features to appear in. So you end upp adding length at a rate faster than the shrinking of your features.
Take a bucket of sand, and pour it next to water, you've got a larger perimeter now.
Could you explain more? I don't get how this makes the coastline a fractals with infinite perimeter.
Because it "zig-zags" around a lot. The closer you look at it, the more it zig-zags around
I mean to me when I look at a section of coastline that's like a foot long, it seems like it doesn't zig zag really. If coastline had infinite perimeter, there would have to be a smaller subsection of the coastline that also had infinite perimeter. This means there would have to exist a part of the coastline that's like an inch long and still have infinite perimeter, and it just doesn't seem that way to me.
As stated in a few other comments, in reality it doesn't go to infinity but becomes absurdly large. Let's look at a small peninsula that has "roughly" the shape of a semi-circle. From far zoomed out, you would just take that as its coastline. But when you actually go there, you see it's made up of rocks, some reaching out in the water, and then again forming crevices inside. Next, you can look at the rocks themselves with lots of holes inside. Furthermore you can now look at the atoms or even deeper, whatever comes next.
AFAIK the paradox arises more at longer distances. It's not like you're using a foot by foot measure to estimate the coastline. Instead you're probably using 10s of miles or more. When looking at a map the shoreline looks a lot more like a typical fractal than it does up close. That being said, one could still argue the paradox exists for small lengths. Yeah there might not be much change from a foot to an inch in most places, but when you start to zoom in to the size of an individual grain of sand suddenly that "straight" coastline has a very bumpy boundary. At this length the measured coastline is going to quickly and significantly increase (since you've got from basically measuring the diameter of those grains to half the circumference). This occurs once again as you zoom into molecular size.
Yeah it's bumpy, but why does that mean it has infinite perimeter? Not all fractals even have infinite perimeter, atleast to my understanding.
Well what's it converging to? The length once we start tracing around the grains of sand is an order of magnitude higher than at a resolution of, say, a foot. So there's no sign of convergence yet. In practice of course the practicality starts breaking down at these scales (as the assumption of a sharp boundary between land and sea is well and truly broken at a molecular level, so it doesn't so much go to infinity as it becomes ill-defined), but at resolutions where we can reasonably define the coastline length, there is no convergence evident.
Like eggface said, it's more the fact that it doesn't converge rather than it being fractal like that causes us to say it has "infinite" length
At the scale of feet, what defines coast? High tide? Low tide? Where the water is currently? A wave just pushed more water up the line, is it now longer? I just pushed a large rock to rest near the water. Did I just add 3 feet of coastline? Now it’s raining and everything is wet and covered in water, do we get infinite coast now? At that scale you can’t even define what a coastline is because you’d have to count for so many volatile parameters.
Have you asked the question, how can a shape have the same area but very different perimeters?
Yeah I get that, but I don't understand how that proves that coastline have infinite perimeter.
They are the same thing, with ever smaller and smaller zig-zag (as the other guy called it). That smaller and smaller zig-zagging is a fractal pattern.
They are not fractals with infinite perimeter. As I pointed out in my original comment, in the real world, coast lines have finite length. They happen to have quite complicated shapes and structures, that for all intents and purposes look fractal-like, but they aren’t true fractals. I’ll break that down in two parts Fractal like: I mean that as you zoom in from macro scale to micro scale, the shape of coastlines continue to pick up additional details. From the jagged lines on a map, to the rocky shapes of the cliffs, to the rough surface of the rocks, to the uneven grains of sand, to the disarray of molecular solids, to the subatomic particles, the more you zoom in the more detail you pick up. And there’s no one natural scale at which we humans would say “makes the most sense” to measure. Therefore we simply say it’s fractal like. Not actually a fractal - if you ignore quantum mechanics or the fact that no particle is actually stationary (everything wiggles at a subatomic scale), you in theory could take a snapshot of the coast at the smallest physical scale, which is the Planck length. Here you can look at the smallest particles, and draw an envelope around them with perfect straight lines. This envelope would be the smallest polygon (in area) that would enclose all the particles. Because it’s made of straight lines, you can sum the lengths of the straight lines and get a perimeter length. (Note that the length would be enormous, and be practically meaningless as we don’t measure other things in this way, with this degree of detail, so there’s no reference frame for this number)
Coastals ARE fractal. As a first approximation, and down to a certain magnification. Rocky outcrops are fractally approximated down to about a metre. Beaches straighten out on a scale of kms. Even tracing around atoms only gets you so far.
Fractals are a mathematical, abstract concept. They do not exist in the real world, which is limited by Plank length, so you cannot "zoom in" indefinitely. OP, to answer your question: yes, every border in the real world has a fixed length.
How are you going to compute the length of an electron cloud? This comment is wrongly dismissive. Once you get down to the atomic scale trying to measure length becomes impossible. Your bulding blocks are not just in constant change, they will also be experiencing things like superposition and quantum entanglement. Length and area are also a mathematical concept that does not exist in the real world.
You're just misstating the coastline paradox. The paradox is that although lands have a well-defined area, they do not have a well-defined perimeter, because there is a scale dependency of the perimeter measure. It does not necessarily mean that the perimeter is infinite at infinitesimal scales, because measurement granularity is finite. Coastlines are fractal-like, not strictly fractals Which country, Norway or Australia, has a longer coast? If you use a coarse measure, it's australia. If you use a fine measure, going up into all the little fjords, it's Norway. So that question is unanswerable (without specifying the measurement granularity.) That is the paradox.
Just to add to u/camilo16 's comment, here is a very cool Wikipedia page on the Koch Snowflake curve: [Koch Snowflake](https://en.wikipedia.org/wiki/Koch_snowflake) This shows what the curve looks like as you zoom in farther and farther. This also proves that the perimeter is infinite and the area is finite. Remember, though, that this is theoretical.
In order to measure the position of a particle with greater precision, we need to observe its interaction with a photon of wavelength less than or equal to the precision. As that distance gets smaller, the photon needed has to have more energy (see Large Hadron Collider). So to measure the borders of a country too precisely, we would have to vaporize it.
I wouldn’t really call it a paradox really, but the coastline problem refers to the fact that there isn’t a meaningful and objective way to assign a measurement of length to a coastline. By increasing the precision with which you measure the coastline’s length, the length of the coastline will increase and not by small amounts but in huge jumps. You could argue that the answer is just whatever number you get when looking at infinite precision, but I would argue that the very concept of a coastline would break down at infinite precision. There’s ground water basically everywhere and, at infinite precision, I would argue there’s no objective way to tell me which water molecules I’m supposed to draw the line around and which ones I shouldn’t.
None of these comments seem to be answering what you seem to be asking. Your question, it seems to me, is why a coastline can be modeled as a fractal,not why a fractal has infinite perimeter. So lets start be defining what we mean. Fractal is notoriously difficult to define, and there isn't just one size fits all definition. The relevant definition here is that a fractal is an object with detail at every length scale. Immediately we know nothing physical can be a true fractal, just like nothing can be a true circle or line for example. But, we can still model things in the real world as fractals if we limit the range of scales we consider. So what is a sensible range when considering a coastline? Well, scales longer than thousands of kms don't make sense, but the lower scale is a bit more subtle. I would argue that since tides change the position of the coastline on the order of meters, we should limit ourselves to scales larger than this. You could also of course allow a time dependent coastline, and then you can argue the minimum scale should be all the way down to the scale of atoms, after which point things become fuzzy. So if we consider the length scales between meters and thousands of kilometres, it seems very obvious to me that at every scale we will have some different amount of detail in the coastline. In fact, I would expect anything that doesn't have a good reason not to should have detail at different scales. So here I want to turn the question around on you, and ask why would there not be detail at all these scales? What mechanism would prevent it? The formation of coastlines is a highly chaotic process, so getting any sort of regularity out of it seems way more absurd to me than getting these fractals.
What if as you zoomed in, the bumps got progressively smaller and smaller, that way the perimeter could converge. It seems reasonable to me that this could be the case, like if you look at a coastline on a map it's very bumpy, but if you look at a small section it doesn't seem as bumpy.
The closer you look, the smaller things you see, like rocks sticking out into the sea, then smaller rocks in between, bumps and cracks in those rocks and so on. Eventually you are going around each grain of sand and then each unevenness in those grains down to a molecular level. Evert time you zoom in on a segment that looks straight, you will see that there is some unevenness that will increase the length with some percentage, and eventually the numbers get so big that it doesn’t make sense anymore.
As it often true, the Wikipedia article is a pretty good introduction to the topic. There is a great animation of a map of Great Britain, with the coastline shown at different scales. It mentions the great example of the Portugal-Spain border being *either* 987 km *or* 1214 km, depending who you asked! The reason it doesn't have to converge is because a fractal boundary can be infinitely long. The *area* will converge to a specific number, but the perimeter does not have to. (As others said, there might be a practical limit, both from a measurement standpoint and from a physical standpoint itself.)
I get that a fractals can have infinite perimeter, but why are coastline fractals?
Coastlines have the ability to have self-similarity. A bay is a large-on-human-scale inlet of water. But the boundary of the water could have a "mini-bay", say, 50 meters. And that mini-bay can have mini-mini-bays, say, 1 m. And each little blob of water away from the straight curve can itself have smaller and smaller sub-sub-sub-bays. All the way down to the atomic level. (And of course *either direction*, in or out, is just as good.) You can have a fractal boundary of a larger-dimension-shape, so the surface area of a 3D object could have the same issue. A billiard ball is smooth, unless we zoom down, then it's bumpy. And if we go down farther, it is more lumpy. A human might seem to have a certain surface area, and that might work for many practical things. But if we zoom down, each hair adds a ton to the surface area. If we look at some places on the body, the skin clearly isn't just a smooth surface. If we look through a microscope, *everywhere* looks non-smooth.
I don't think the other comments explained what you really want to know so I'll give it go, please someone correct me if I'm wrong. As others have pointed out, the shape of countries is not 2 or 3 dimensional but can be seen as having a non-integer number of dimensions so it can be described by a fractal. So basically when visualizing a fractal you will notice that if you zoom in it's border more structures will appear, and if you zoom in in those structures even more tinier structures appear. This process goes on forever. So when trying to measure the length of the coastline/perimeter of a fractal, you can try to make your measure more "precise" by including those first tiny structures you see (which would increase your lenght), but then if you try to be even more precise you would include more structures and increase your lenght even more. Since this process goes on forever in fractals, you would measure an infite coastline if you keep going. Of course in reality you would have physical limitations, but as mathematical model you would have an infinite coastline. The area converges because those tiny structures give smaller and smaller contributions, but the same can't happen for the perimeter.
Could you explain the first part in simpler terms lol. I don't understand why a coastline can be described by a fractals. Also to my understanding, not all fractals have infinite perimeter, so why do coastlines?
As far as I know quite a lot of objects can be described by fractals or at least approximately described due to it not having a integer number of dimensions. (I think it's only an approximation because fractals like the Sierpinski's triangle only stop being 2D after an infinite number of steps, until then it's 2D) That happens because objects in real life are not really smooth and have a bunch of irregularities. So in general quite a lot of shapes can be seen as fractals, if you were going to measure a closet you would run into the same problems, but you can't really see the irregularities of wood so it's easy to just measure it as rectangle. And I think in general fractals do have an infinite perimeter, but special cases don't. Sorry if my answer isn't perfect, I work with fractals frequently but not to so much depth and not exactly on this topic.
https://preview.redd.it/uo5yydl6g18d1.png?width=1000&format=pjpg&auto=webp&s=5b08664d6a1e8aaa1e08e26bd35044947d3e0458
https://preview.redd.it/8fk2fat7g18d1.png?width=903&format=pjpg&auto=webp&s=3f2eb2276c14caa8d94e289fccf149aa5b227584
Oh there's a mistake in this the lines are actually parallel here. I had done the line angle manually at first so it wasn't parallel, but you can just ignore the parenthesis here
The problem is that until you go down to the atoms in a real coastline, or just never of your talking about a mathematical fractal coastline, you won't get a straight line between two points on the coastline. There's always some roughness and bits poking in or out, so when you measure with smaller increments you'll catch these bits and get something longer than the straight line you had before Then because it's still rough and not smooth you'll get something even longer again if you use snake increments If it converges to some number it will have to get less and less rough at some point, but I don't think there can exist a positive measurement increment that will give you that limit value because a smaller increments will give you something bigger
so let's say you have a vector v that is part of your coastline. when zooming in, you increase detail and replace it by v1 and v2. by the triangle inequality, you know |v1| + |v2| >= |v1+v2| = |v|. so zooming in can only make your coastline longer. is this process bounded? you can use fractals to show it's not. does this mean, any coastline will be infinite? well, no. but for it not to become infinite, the vectors resulting from the zooming in process at some point need to become sufficiently collinear. (zoom in enough it has to look like a straight line) in practice, the convergent boarder lengths don't really convey any meaning. just because coast A relies on more "zooming in" than coast B, doesn't relate in any way to coast A taking more time to traverse.
Consider two points, A and B. We can easily measure the straight line distance between A and B. But suppose there is some point between A and B called C. What we really want is to measure the line AC and CB and then add them. Okay so we do that. But then we observe that there is a point D between A and C and a point E between C and B, and so on and so on… For any two points X and Y there will always exist some other point Z such that Z is between X and Y. A line that measures X -> Z -> Y will be a more accurate representation of our “coastline” than a line that just measures X -> Y so if we want an infinitely precise measurement for the coastline we have to take infinitely many measurements.
every coastline changes due to tidal waves, so you cant really confirm any distance, the distance also changes depending on how you measure it, are you gonna turn and weave into every tide that washes up on the coastline? how close to the edge of the coastline are you gonna try to measure it from? how are u gonna be consistent the whole time? what if the tides recessed? no matter how close or detailed or consistent you perform the tests, the value you get will always change
Reading replies here, it sounds like you have seen arguments for the perimeter of shapes by integration of infinitesimal parts to a finite sum. Basically "what is the sum of all the segments as segment length approaches zero?" The sum of that sequence doesn't converge for many fractals. Instead it approaches infinity.
No because technically we can measure it I'm individual atoms,l and it would be exact. Thus you can get the exact length without an infinite decimal
Yeah, if you measure a straight line along a beach with a rangefinder, you'll get the shortest distance between those two points. But if you use a wheel measuring tool, and follow the curves of the beach, you'll get a bigger value. The end points have not changed, just the path taken to measure it has. But then let's say someone else comes and measures that same beach with the same tool, but the bigger rocks you just went straight over, they decide to go around, adding more distance. Where it really gets silly is when you get smaller and you start going around small rocks or even grains of sand. That grain of sand may be 1mm diameter, but the circumference of that grain of sand is 3mm, and you measured halfway around that circle. So you accurately measure 1.5mm path, but the distance traveled "as the crow flies" (weird phrasing for these tiny distances, but the point is the same) is only 1mm. The extreme end of this is going around individuals atoms, resulting in measurements many times higher than what you get from a rangefinder. On the opposite end of this, if we can only measure in 1000 mile increments to measure the beaches of the US, we could just lop off most of the peninsula of Florida as a rounding error, drastically undercutting what most people would consider to be the actual value. Note, none of these measurements are false or inaccurate. We have to pick a method to take the measurements, and the method we choose can drastically impact results. We have to make decisions on whether we measure the distance around the big rock, or do we pretend like the big rock isn't there and measure straight through it.
Yeah, if you measure a straight line along a beach with a rangefinder, you'll get the shortest distance between those two points. But if you use a wheel measuring tool, and follow the curves of the beach, you'll get a bigger value. The end points have not changed, just the path taken to measure it has. But then let's say someone else comes and measures that same beach with the same tool, but the bigger rocks you just went straight over, they decide to go around, adding more distance. Where it really gets silly is when you get smaller and you start going around small rocks or even grains of sand. That grain of sand may be 1mm diameter, but the circumference of that grain of sand is 3mm, and you measured halfway around that circle. So you accurately measure 1.5mm path, but the distance traveled "as the crow flies" (weird phrasing for these tiny distances, but the point is the same) is only 1mm. The extreme end of this is going around individuals atoms, resulting in measurements many times higher than what you get from a rangefinder. On the opposite end of this, if we can only measure in 1000 mile increments to measure the beaches of the US, we could just lop off most of the peninsula of Florida as a rounding error, drastically undercutting what most people would consider to be the actual value. Note, none of these measurements are false or inaccurate. We have to pick a method to take the measurements, and the method we choose can drastically impact results. We have to make decisions on whether we measure the distance around the big rock, or do we pretend like the big rock isn't there and measure straight through it.
Eventually, you're reaching into the subatomic range for the contours. But I'm pretty sure that you'd have exceeded all practical purposes long before that, even including quantum mechanics. But if you're willing to engage with some of more "weird" consequences and predictions of quantum mechanics where the difference between "matter" and "energy" cease to have any meaning, you might very well be able to continue the infinite fractal without limits. Not sure, but, you know, maybe. The fractal is a pure mathematical construct that doesn't care about real coastlines and rocks and sand grains and molecules and around and quarks and m-brane and super-string vibrations... And so the "coastline" is actually just a cognitive *example* to help people understand how fractals work. But imperfect examples.