Gabriels Horn (extra) - Numberphile
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- čas přidán 18. 02. 2021
- With Tom Crawford... Following from main video at: • Gabriel's Horn Paradox...
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The paradox is resolved by assuming even a drop of paint can cover an infinite area, it's just going to be infinitely thin.
As area tends infinity thickness tends to zero ... Not infinitesimally thin ... Zero thin
@@derekcouzens9483 infinitesimally thin?
I don’t think that can be quite right, if you don’t mind me saying: The paradox is that at the same time as painting the inner surface, the horn is filled with paint - and the volume to fill it to an infinite length is pi cubic units. I think the paint would need to be infinitely ‘runny’ but clearly not infinitely thinly applied to the inner surface. (Infinitesimally thinly I mean) - having just written that I now think I know what you’re saying and that you’re right. That a finite volume can be spread over an infinite area if you assume zero thickness! Epiphany!!! Is that right. The painting of the infinite surface area requires no paint at all, because a surface has zero thickness. A surface has no volume is all the paradox is really saying.
@@gildedbear5355 cheers ... corrected
I agree: you could just pour out all the paint, leaving the inside covered with paint. Outside area is same as inside area (horn with zero thickness), so could be painted as well. Math paradox, but real possibility.
Presumably real (intuitive) paint would need to have a finite thickness to paint a surface. So at some point it really would become impossible to "paint". As well as the fact that the horn would rapidly become too narrow for a paint molecule to fit into. Paradox paint might work though...
I rather like the idea of making a (truncated) version and trying to convince Matt P to measure pi using it.
Matt wouldn't need too much convincing, I think. We just need an infinitely long 3D printer...
But what if you were painting the outside only?
@@BarbarianGod I think it still has to do with paint being a 3d thing, you can take any volume of paint and start spreading it out over the surface, the more you spread it out, the thinner it gets. So mathematically you can use any volume of paint to cover an infinite area as long as you allow the paint to be infinitely thin.
yeah if the paint truly fills the horn it must be infinitely sub-dividable. which means that it's not necessarily unpaintable on the outside, it would just take an infinite amount of time to paint it with a layer of zero thickness
I was going to say the same. It's still fascinating though that there are mathematical shapes with infinite surface area and finite volume. I enjoyed learning stuff like that in calculus.
He likes pokémon and is a formal mathematician.... I have hope
He also has a PS2 and Wii. Upright and vertical.
The paradox arises because you're comparing an area to a volume. Now, if you want to consider the volume required to paint the surface, you need to multiply the surface area by the thickness of the paint required. When you fill the horn, you realize that the thickness of the paint must get infinitely small (the thickness cannot be thicker than the radius). Therefore you get a finite volume. Another way you can see this is if you consider painting the horn with a coating of paint that is 1/x units thick, ie the coating of paint gets thinner as you go along. Now when you perform the integral, it will converge.
This was so helpful, thanks
That's what I thought and you explained it well. I think this is much more convincing than the "explanation" given in the video.
I think that is the real key to the "paradox". And very well explained.
Wow now suddenly I get it. Very well explained!
I had some thoughts bumping around my head about how there'd have to be a certain point in the horn where the paint could not flow past if it could coat the horn, but I couldn't figure out how to put it as nicely into more mathematical terms as you did. Thanks.
I can even paint a infinitely large area with infinitely small amount of paint - as long as the layer of paint can be infinitely thin.
this really reminds me of fractals. Coastline lengths are infinite, but they hold a finite surface area. Extend the coast line a bit below, you can never paint the coastal surface, but you could fill the ditch with finite paint...?
That maybe the best real-world equivalent.
You might be able to write change or coordinate so that you get oscillating functions which finite area but infinite length.
@@sadface7457 Koch's snowflake or Peano's curve. And their 3D equivalent, Menger's sponge.
Coastline lengths are finite. They are constructed by finite particles. They are pseudo-fractals. So you can indeed paint a coastline.
@@fatsquirrel75 You can paint a coastline up to the scale of a paint 'particle' (whatever that is; paint is usually not homogeneous at molecular scales and some 'paint-like' properties only emerge at larger scales...), but the minimal scale of the coastline may be smaller than that - though you are perfectly correct that a real coastline has a finite - if somewhat indeterminate - length. (Á corsaire, corsaire et demi, mon ami 😉)
I can paint an infinite floor with a single drop of paint.
I can make an infinitely thin cookie from a finite quantity of cookie dough.
Man I want an infinite cookie.
As area tends to infinity then thickness tends to zero ... Can you do either?
You can't really do either of those, because the paint or cookie can only get so thin before you have to start splitting atoms, at which point they're no longer paint or cookies.
At that point they become explosive...
I can make a gray sky blue.
I can make it rain whenever I want it to.
I can build a castle from a single grain of sand.
I can make a ship sail on dry land.
@@PhilBagels And I can ride my bike with no handlebars ;)
Forget painting the whole surface area of it, you can't even paint one stroke down the whole length of it.
forget painting a stroke down the length of it, you cant even pick a stroke with the right thickness
The paradox is resolved when you think about what we actually mean when we say "you can fill it", and "you can't paint it". TL;DR: the word "can" in "you can fill it" doesn't mean the same thing as the word "can" in "you can't paint it".
When we refer to this being a surface you "can't paint", that means it would take an infinitely long time to paint assuming a finite rate of paint application. In other words, someone going at it with a paintbrush would take an infinitely long time. So surely, when we say "you can fill it", that means that it can be filled in a finite time, right?
Not so fast. Remember that this container is infinitely deep, and fluids cannot move at infinite speeds. In order to fill a container, fluid has to reach the bottom of the container. But even a perfect fluid, with zero viscosity, that is completely continuous at any magnification, would not be able to reach the bottom in finite time. So we really can't fill it, if our definitions are consistent - even though it takes a finite volume to fill, we'd be stuck holding the bucket forever.
That said, there exist surfaces with a finite volume, finite surface area, and finite depth - think about a bowl with a fractal bottom, for example. If you were to fill that bowl with a perfect fluid, completely free-flowing and completely continuous at any magnification, then you would have no trouble filling it - any point on the bottom is some finite distance from the top, and so any packet of fluid moving at some finite speed can reach the bottom in some finite time, despite the fact that painting the bottom at a finite rate of paint application would take infinitely long. So how do we resolve the paradox in this case?
The resolution turns out to be the existence of atoms, or, more precisely, the fact that completely continuous fluids at any magnification simply do not exist in the real world. A bowl with a fractal bottom will, by necessity, contain nooks and crannies that are smaller than the atoms of any real fluid, so while the bowl may look full to the human eye, closer inspection will reveal that it, in fact, cannot be filled. There will be tiny, tiny spaces that remain empty. (Also, the bowl couldn't be made of any real material, but by that point we've already resolved the paradox.)
In other words, it's not too surprising that a fluid that can be divided into infinitesimally small packets can assume an infinitely complex shape in a finite time. For any fluid without this property, there isn't any paradox - you can neither completely paint nor completely fill the bowl.
Edit: the second line of reasoning also applies to the original horn - there will eventually come a point where the diameter of the hole is smaller than the smallest particle of whatever it is you're filling the horn with, so physically, you can never completely fill it. There will always be a tiny empty portion near the bottom.
Soundbite of the week: "Don't mess with infinity, man"
Followed by an aggressive vuvuzela sound!
To be brief, you can paint infinite surface with a finite volume of mathematical paint, so long as the "thickness" of the paint changes fast enough.
It's like the joke of the infinite mathematicians who walk into a bar. The first one says, "I'll have a beer." The second one says, "Half a beer please." The third says, "I'll have a quarter beer." The bartender says, "Stop, I know this one," and pours two beers. Why? Because a half, plus a quarter, plus an eighth and so on forever... converges to one beer.
As long as the paint's volume "divides" or "spreads out" or "thins" fast enough, it can be stretched out to cover infinite area. Similarly, so long as the bartender's mathematical beer is divided up fast enough, we can put some portion of beer in infinite glasses. There is nothing inconsistent about this. :3
Supertask
We dont care about imaginary universes where you can use a finite amount of paint to paint an infinite surface
That's just adding another non intuitive idea onto of the existing one
You might as well explain Santa by claiming he can travel close to the speed of light, and then if he STILL doesnt have enough time posit time traveling or faster than light travel.
@@nosuchthing8 "We dont care about imaginary universes where you can use a finite amount of paint to paint an infinite surface"
Then why do you care about imaginary universes where you can build an infinitely long horn which gets arbitrarily thin?
to paint the outside in finite time you simply place gabriel's horn inside a scale model of itself and fill that instead
Right. That works because the coat thickness decreases as you move towards the narrow end of the horn. The reason there is an apparent paradox for painting the inside of the horn is because we are incorrectly assuming that the coat thickness is constant, which is impossible. Of course for the outside of the horn, if we do choose to have a constant coat thickness, then we would use an infinite amount of paint. This is remedied by decreasing the coat thickness as we move along the horn, which would be accomplished by your solution.
But if the horn has an infinite length, it would take an infinite amount of time to slide it inside a scale model of itself?
@@FostersLab Yes, that in itself would be a supertask because you'd never be able to reach the terminal end to stick it inside. Of course, people also forget that paint is *discrete*, and eventually the horn gets too narrow for even one paint molecule to go lower, meaning that you can't paint it completely from the inside.
@@MisterUnlikely if the paint is discrete then so is the material the horn is made out of, and therefore couldn't continue to get arbitrarily smaller. I think its fair to assume that we aren't working with discrete materials.
@@jetison333 Regardless, in order to nest horns, you'd still be doing a supertask because you can't physically move the shape far enough.
It's an infinite 2d surface area vs a finite volume. If you were to hypothetically pick a thickness for painting the surface, at a certain distance along the paint inside is filling it by being thinner than the thickness of your paint on the outside, since it gets arbitrarily thin inside.
No
You know... that makes me think. What if you flipped things around? Once you paint the outside of the horn with an arbitrarily (but constantly) thick layer of paint, _now_ the horn has _infinite_ volume, because at some point the horn itself has shrunk to negligible volume, and you thus have an infinitely long cylinder of paint of radius equal to its thickness.
@@barefootalien Except that you could take a large container of paint that held somewhat more than PI units of paint and dip the horn into it, coiling up the thinnest end and as you lowered it into the container. The coiled horn's infinite outer surface would gradually become painted without a "thickness" of paint having to be considered yet the volume of paint displaced would tend towards a maximum volume of PI. What's interesting is that time doesn't appear anywhere in the maths and yet a process of dipping/filling and painting over time is implied in conceptualising the problem.
@@peteneville698 Hmm.... that's a very interesting concept, but I don't think it works. If you really do coat the horn evenly with paint as you lower it in and curl it up, and assume you can't squish it all out from between the curling segments, then it has infinite volume, and you could never fit the whole thing into the container.
If you _can_ compress it to the point that you displace the paint, then I would argue that you haven't _really_ painted the horn. You've created a new object using what used to be the horn, say, an infinite cornucopia, and painted _that,_ but at that point you've redefined the question.
The way I think of it is this: You cannot paint the entire surface with any finite thickness of paint, but you can paint it with an infinitesimal thickness. By filling the horn, you are effectively painting the entire interior surface with an infinitesimal thickness of paint; but for any finite thickness of paint you choose, the majority of the horn will have a smaller interior diameter than that, and therefore will not be painted to that specification.
The real answer to the paradox is simply that a film of paint has a finite thickness, and as you get towards the end it's much thicker than the actual horn.
The other simple thing to realize is that as objects get larger, their volume to surface area ratio increases, and conversely, as they get smaller, the volume to surface area ratio decreases. That means, that as you approach the end, you're adding much more area than volume.
No. Imagine a normal cylindrical coffee cup. You can fill the cup and coffee touches all the surface.
Now imagine an infinitely long cylindrical coffee cup. It takes an infinite amount of coffee to fill it, and to coat the inside. As expected.
Gabriel's horn is odd because it's a mix of the two situations. Volume is finite, surface area infinite
I would say that you're close to resolving the paradox, but you've got things flipped.
In the physical world, you're right. There is a sort of "minimal" thickness paint could have. But in mathematics, we can have arbitrarily small thickness to the paint. Any nonzero amount of paint can cover an infinite surface area if we just have the thickness get small enough at a fast enough rate.
So the fact that Gabriel's horn has an infinite surface area doesn't really mean that we need an infinite amount of paint to paint it.
honestly, part of the paradox comes from the fact that paint, which to us intuitively has a volume but can also cover a surface, can't exist mathematically.
Ooh, now this, I like! Very clever!
I don't think this is as much of a paradox as one may think. The surface area of a given volume is not a constant but is dependent on the shape. A sphere has the smallest SA/A ratio of any shape. A cube has a larger ratio and a pyramid is even bigger. If a volume is in a prism or cylinder that is stretched so that it has an infinitely small thicknesss or diameter (but not 0) the length can be stretched until it is infinitely long giving an infinite surface area.
How I resolve the paradox is by understanding that mathematically with any fixed amount of paint I can paint infinite amount of surface by lim thickness of the paint to 0.
In real world it is indeed paradox, but that's why we can imagine math world with different "limits".
Agreed! The paradox comes from thinking about "mathematical paint" when considering the volume, but thinking about "physical paint" when thinking about the surface area.
What I mean here is that mathematical paint can become arbitrarily thin, whereas physical paint has a smallest possible thickness. If we're dealing with physical paint, we definitely cannot paint Gabriel's horn with a finite amount. But neither can we fill Gabriel's horn - the thinness of the tube will eventually become too small for the paint to fit inside, so we will never be able to actually "fill it up". However, with "mathematical paint", we absolutely can paint an infinite area with any positive volume of paint, since we can make it arbitrarily thin.
The fact that any improper integral can converge (or similarly, that any infinite series can converge) is basically what this supposed paradox boils down to.
I totally agree that the natural way to come to terms with the paradox is consider that infinite objects cannot exist in the real world (the world of 'concrete things') but only in the ideal world of abstract things.
Even in the ideal world of abstract things, there's no paradox. The "coating" of paint on the inner surface that occurs when you fill it is thinner than any finite thickness you choose for the majority of its infinite length. You can paint an infinite area with a finite amount of paint as long as you don't mind that the thickness of the paint reduces towards zero.
@@Eagle0600 Your point is well made. However, it is paradoxical enough if you need an infinite surface for a finite volume.
I understand the 2 results like this: The inner volume of the world longest horn, assuming it has the same size and shape of this "unit" Gabriel's horn, is bounded by pi; as you make it longer, it contains more volume, but it never contains more volume than pi units cubed. While the surface area, for any reasonable definition of surface area for curved surfaces, is unbounded; you can ask for a finite surface area as large as you want, and you can make the horn longer (still finite) until its surface area is bigger than what you asked.
It requires π amount of volume of paint... that's true, but because of it's infinite nature, the amount of time required for paint to reach the bottom will be infinite..it means you can fill it but to fill you will have to wait for ever...which is kind of infinite but not infinite.....😫 Paradoxical!!!
I've never think about it, wow
Hmmmmm. Continuity says that any amount you pour in the top must enter the horn, so it must be that you can fill it in a finite period of time. Assuming these are “real fluids” the speed of the paint would increase to infinity as it traveled down the tube to satisfy the continuity equation, so you can still fill it in a finite amount of time
@@spencertracks2720 Why would the speed of paint increase as it travels 'down'? What does 'down' even mean in an infinitely long structure?
That aside, continuity is a simplifying assumption to deal with fluid dynamics on a macroscopic scale, and is an assumption that doesn't even hold observationally in everyday conditions.
@@dlevi67 this is a very idealized scenario. Assuming it is a pure structure of geometry, there is no friction, and the fluid must be made of infinitely small "particles," if you want to consider it that. If you hold the horn so that the force of gravity would pull the fluid down the length of it, then it would in fact accelerate faster than gravity because of the weight of the fluid above it and the work being done by the walls. Assuming no viscosity, some of the fluid would have to accelerate and approach an infinite velocity, otherwise, as the original comment suggested, you wouldn't be able to fill it at all.
But it can be filled. Because P = density * gravity * height, the pressure at the lowest point of the fluid increases proportionally with the depth. take one "unit" of fluid to be an amount of fluid occupying the full diameter of the horn with a height of 1, so that its volume is essentially the area of the pipe it is in, and we treat this unit as a point-mass of fluid, so the pressure is even throughout it. Well, as Pressure is Force/Area, so Force = Pressure * Area = Mass * acceleration, because the surface area and the volume of our unit of fluid will be changing at the same rate (1/x^2), the pressure will be proportional to the acceleration, and therefore the acceleration is proportional to height. So, the further the fluid goes, the greater its acceleration and you have an exponentially increasing velocity to infinity.
And yes, Continuity doesn't always hold in the real-life because it's real life, but this is extremely idealized, no part of this is reflective of real-life except assuming a constant density and the basics of 3-dimensional geometry.
And assume that gravity is at the very least consistent, so either a planet with an infinite radius or instead just have the horn accelerating at a constant rate through space
Assuming the paint isn’t made of finite molecules, any amount of paint could cover the surface of the horn, it would just be an infinitely thin layer.
If we take an ideal fluid with a lot of adhesion definitely lol.
Infinitely thin = 0. It wouldn’t be painted at all
@@Xboxiscrunchy Not true. If each pain "particle" were infinitely small, then there would be infinite particles, and you could therefore cover an entire surface with it. If you start with a layer thickness of, say, 1 inch (a lot of paint) for the first inch down the horn. Then, spread half of that pain down so that it covers the next portion with a layer 0.5 inches thick. Then use half of that paint to cover the next area with a layer 0.25 thick. Continue down the tube, and because the surface area decreases exponentially, the amount each volume of paint covers increases exponentially, and so it will be able to cover an infinite volume with the started amount of paint. The paint, although near the end will be infinitely thin, is still a layer of paint. Just like how any volume can be split into an infinite layer of infinitely thin slices
@@spencertracks2720 haha reminds me of super tasks..
@@spencertracks2720think of it like a limit. Thickness = volume/area. For any finite volume As area approaches infinity thickness approaches zero. So At infinite area the thickness must be zero.
The calculations are pretty straightforward and very cool, but I ended up feeling like there was a lack of discussion on the idea of why the inside surface wouldnt be painted if the horn was filled with paint. The explanation that "well, you can't actually build this" was unsatisfying.
The point is, you can't actually fill the horn with paint. Although the volume has a finite lower bound [because you are chronically _under_estimating it in your calculus; the curvature is such that you are always finding something that is less than or equal to the actual volume], the length is infinite, so the first drop of paint will never reach the end of the horn.
Because, if you watched the video, the volume is finite and the surface area is infinite
1:28 yes okay this may resolve the case of Gabriel's horn, but there are objects with an infinite boundary but a finite interior, which fit in a finite region of space - they're called fractals. The Mandelbrot set, the Koch snowflake etc, have infinite perimeter bit finite area.
The way I consolidate this paradox is considering what happens as you continue along the horn toward the pointy end. The space inside the horn is decreasing as you continue along the positive x-axis. Eventually, it will be smaller than the width of any molecules that make up the paint. So technically if you filled it with paint, you'd have slightly less than Pi units of paint inside the horn, and you will have also painted as much of the inside surface of the horn as our physical world allows, which is definitely finite in this scenario. Trying to paint the outside, you would still need infinite paint, as there are no similar physical constraints on the volume outside the horn, assuming the universe is of course infinite in size. Otherwise you do run into a physical limit.
Of course there are similar physical constraints on the outside of the horn. The size of a molecule of paint remains the same, and at some point the circumference of the horn will be too small to accommodate the size of the molecule; you can't paint 'around' too small a tube just as much as you can't paint 'inside' too small a tube. But that is not the paradox.
That only works if the universe is granular like ours. It doesnt work in non granular universes.
The paradox consists in comparing an area to a volume. Since paint is sold in buckets, clearly it has volume. If we are allowed to spread the paint in an infinitesimally thin layer, one drop of paint could cover an arbitrarily large area, even an infinite area.
As another example, a Koch snowflake has an infinite perimeter but a finite area.
The sound of vuvuzelas will never not be annoying
If we can imagine an infinitely long horn, surely we can imagine it being filled with paint. We don't have to restrict ourselves to conform to physical behaviour of objects here. I think where Tom is wrong is in insisting that infinite surface can't be painted with finite volume of paint. We are talking idealised paint with zero thickness and there is no limit to how much area a finite volume of it can cover. That's the resolution for me. If we can imagine it being filled then we must concede it can be painted.
Yes!
Your point about a finite volume of paint doesn't completely resolve the paradox, IMO. Consider the following: you can fill the horn with a finite amount of paint, but you can't cover it with a finite amount of wrapping paper. The thickness of the wrapping paper is actually irrelevant here. The "resource" that is being consumed when covering something is area, not volume.
@@ruroruro I see what you are getting at. I believe that conceptually speaking the difference between the wrapping paper and the paint is that paper cannot be stretched while the paint can get thinner and thinner and the area it covers can grow without limit.
@@bumlic I think that they just left out the phrase "a layer of uniform thickness" when discussing painting it. No matter what that thickness is, as long as its non-zero, the volume of it will still be infinite because the volume of a layer r units thick will always be greater than an infinitely long cylinder of radius 2r, which is of course infinite, and not in a way that converges to zero as r gets smaller. (If you start with r=0 then the result is either undefined (0*∞ is not a well-defined operation) or zero (as it's just a line which has zero volume by definition) but if you approach it as a limit it's infinity, and that's what counts.)
@@killerbee.13 This is some very nice food for thought! I certainly agree that a layer of finite thickness would have infinite volume. However, my original point was exactly that the thickness of idealised paint is zero. The cylinder comparison is interesting. The limit is inifinity because the cylinder is infinite in length to start with. However if I choose to vary the length also, say I define it as 1/r, and then send r to 0 the limit will be zero, if I am not mistaken. Sorry, I am a bit rusty on technicalities of calculus ;)
Fill the horn with paint, then take a second horn and slide it inside the first horn. BAM, painted.
Wouldnt you be sliding one onto the other forever though? They're infinitely long. 🤣
@@PokerPlayerJames First you've got to get to the "open" end of the horn for the insertion to begin, which could take a little time in and of itself... 😉
Nope. Even the inside of the first horn is not coated with paint. How would that coat the second?
@@nosuchthing8 Um - ignoring for a moment some insignificant physical constraints like a way of building and then filling an infinitely long, infinitely narrow pipe, but why would a horn filled with (mathematically ideal) paint not be coated by the same?
@@PokerPlayerJames Just slide them infinitely fast... but you need to use an infinitely viscous ink too... the only trouble I see with this brilliant idea is that you need to put an infinitely long horn in front of another infinitely long horn and make them perfectly aligned :)
If the horn existed in our universe, and we assume our universe is infinite, the horn would fit within the universe (perhaps even multiple times), so we could say the area of the horn is a smaller infinite than the volume of the universe. So if the universe was filled with paint there would be enough paint to cover the horn's surface.
If the horn is infinite thr surface area is infinite. So, no
It's another proof of an infinite process that cannot complete a finite task. It's related to Xeno's paradox, but it's the opposite, in which an infinite process paradoxically seems like it couldn't finish a finite task. But in reality, it does.
That's only the volume.
The real paradox is that although you can completely fill the horn, there is not enough paint to coat the inside surface.
It doesn't matter if its not infinite, since the surface goes to infinity, there's a point at which it would take more paint to cover its surface than to fill it!
like a cube cube with its top missing (so it is fillable) of which each side (edge) is 1.5 cm has the same "property"
The slight of hand comes from comparing unrelated units. Volume and area are distinct separate things, apples and oranges. Any finite volume can be spread over an infinite area.
Something not as paradox-y that this illustrates is that there is no upper bound to the surface area of an object with a given volume. One can always construct a new object with the same volume but with a larger surface area. I find this concept much more intuitive.
I said this in the last video and I'll say it again here: there are alternate ways of finding the volume of the horn which return an infinite result. For example, you could consider the horn as the function 1/x rotated about the x-axis. This makes its volume 2*pi*area under 1/x. But the integral of 1/x is ln(x) so that is an infinite area. Infinite area time 2*pi is infinite volume.
I think you're forgetting that the curve is truncated at x=1
@@killerbee.13 Sorry I don't understand. Could you elaborate please?
@@WhosBean 1/x approaches infinity as x approaches 0, so most of the area under it's curve will be in the first unit of its length. But they start at x=1 instead, changing the integral and making it finite.
That math does not work.
@@killerbee.13 The definite integral of 1/x from 1 to infinity is ln(infinity) - ln(1), ln(1) is 0 so basically the are under 1/x is ln(infinity) which is obviously undefined but lim as x goes to infinity of ln(x) is infinite so we can say that ln(infinity) diverges and so the area under 1/x is infinite, even considering only the part from x = 1 onwards.
The volume of a solid is a double integral. In the method I'm describing we take 1/x integrated first with respect to x with 1 and infinity as limits, then w.r.t t where t is the angle of rotation around the x-axis and has limits 0 and 2*pi (measuring angles in radians). So the double integral is SS(1/x)dxdt. Integrating w.r.t t first gives us 2*pi*S(1/x)dx but as we saw above S(1/x)dx with infinity as an upper limit diverges, so the volume is in fact infinite.
Brady's question about the potential slight-of-hand was an excellent question and is worth pointing out as a critical part of mathematics. This thingie makes no sense yet seems to work on paper. Where is the disconnect between the idealized form and the real-world form? The question is the impetus to understand. Wonderful.
There's no paradox painting the inside. As long as you don't incorrectly assume a constant coat thickness, the volume to paint the inside is finite, because the coat thickness cannot exceed the radius of the horn for any given slice.
The paradox is solved by being consistent about the properties of the paint. Here's the mathematical explanation (to which various intuitive takes correspond).
[Option 1] If the paint is to be "mathematical" (or "continuous") that is, our finite volume of paint corresponds to a set in R³ with a Lebesgue measure of pi, then the set is uncountable, as it has a nonzero Lebesgue measure. Since this set is uncountable, it can surject onto (paint) the infinite surface area of the horn, because they have the same cardinality. In this case, the paint can fill the horn, and it can also paint the horn. There is no paradox here because there is nothing paradoxical about a surjective function f: P -> H, because there is nothing paradoxical about a set with finite Lebesgue measure surjecting onto another infinite set, IE the surface of the horn.
[Option 2] If you want the paint to be more physical in nature, meaning it is composed of a discrete, finite number of particles ("molecules"), the paint cannot paint the infinite surface area, but it cannot fill the entire finite volume either. The horn's opening becomes smaller without bound, so it eventually becomes smaller than the paint's discrete molecules, at which point the paint stops flowing through the horn. This leaves a portion of the paint's volume unfilled, and an infinite amount of surface area untouched by the paint.
The paradox is caused by an inconsistency. When you state that it _can_ fill the finite volume of the horn, you are giving it the properties of mathematical paint, but when you say it _cannot_ coat the infinite surface area, you are giving it the properties of the discrete physical paint. If you are consistent, the paint either does both things (if it's continuous) or does neither thing (if it is discrete). Also, when going through this problem, you should keep in mind that Epstein didn't kill himself.
The way to make sense of the paradox IMO is to think about the area increments vs the corresponding volume increments as x increases. Just like x^3 increases much faster than x^2 (think 10^2 vs 10^3), for values of x smaller than one, x^3 will decrease much faster than x^2 (0.1^2 vs 0.1^3). So, as x approaches infinity the area increments will get smaller and smaller - but smaller in such a way that you can always group a finite amount of "terms" to keep increasing the total sum (ie the harmonic series diverges). But the corresponding volume increments will vanish much quicker, meaning that you'll soon stop making progress (ie the sum of the inverses of squares converges)
***"A solution" for PARADOX: if you dump Pi ink volume units, it will take infinite time to fill the Pi volume, as that number used infinite limit on the x-axis...
No, you need to fill with paint the same way you built the horn in the first place: instantaneously (neither task can be performed progressively, for the reason you point out).
The actual reason behind this "paradox" is the confusion between 2d measure and 3d measure. When we paint an area we think of the paint having a small amount of thickness. We may make this thickness quite small, but we tacitly assume it to be the same everywhere. So we kind of equate the question "how much paint is needed?" with the 2d measure of the surface (if we have a plane surface). The actual amount of paint-volume needed then is c * surface-area with some small constant c. But the c is still important here! With Gabriels Horn this does not work, as we need to make c smaller and smaller to even fit inside the horn. In this way we avoid an infinite volume by letting one dimension go to zero while one other goes to infinite.
Note that the surface of Grabriels Horn has 3d-measure 0. So mathematically we would need zero volume of paint. That's true for any (reasonable) 2d-area. So the intuitive euqating of the amount of paint needed to paint a surface (a volume) with its surface-2d-measure is just wrong! And you really don't need an infinite example for this. How much paint is mathematically needed to paint the unit square? The answer is 0, not 1!
This comment is the real explanation of the paradox. Put another way, any real painting of “the surface” is always a volume-filling exercise - no matter how thin - hence finite in this case and actually unrelated even conceptually to the mathematical surface area which happens to be infinite.
On the issue of "it is infinite so you could never build it", I disagree with the coomentator's comment that you could never fill it. You could buy a pot of π(?) paint and on day one build the 1st metre length (temporarily seal the bottom) and pour in the paint. On day 2 build the next metre and pour in the paint etc. You never need to revisit the paint shop. If you were painting the outside you would need a standing order at the paint shop.
No because the surface area is infinite. You might as well claim you could walk an infinite distance.
After filling up the horn with paint we only see that the surface has finite volume, but that's trivial because we already know it has zero volume. Obviously the "paint" on the surface has finite volume, but we are interested in the surface area. The problem is that real paint isn't spread infinitely thin. With a bucket of π litres of paint you could paint Gabriel's horn with the paint having thickness, but this is basically the same as filling the horn with such paint, and the surface area will still be infinite because that's not how we measure the quantity of paint.
The paradox arises because real paint DOES have a volume and a thickness, paint is not equivalent to area.
Nope. The 'paradox' (which is not a paradox) arises because you have a finite volume encapsulated by an infinite surface. You can use 'ideal mathematical paint' with zero thickness, it will still take you an infinite amount of time to paint this if you paint at a finite m^2/s rate.
@@dlevi67 It seems to me that you're saying the same thing, ideal paint = area != real paint.
@@GuzmanTierno No, I'm not saying the same thing at all. I'm saying that even if you have mathematical paint that can be thinned down to zero thickness, it will still require an infinite _time_ to paint it. Not an infinite amount of paint.
In fact with mathematical paint that goes to zero thickness you require exactly zero volume of paint; you'll have all the paint you poured in the horn at the start, when you are done painting, but you'll never be done painting.
(edit: typo 'pourent' --> 'poured')
@@dlevi67 Ok, thanks, now I get what you mean ... it takes finite time to pour the paint inside and it takes infinite time to paint the surface even with an ideal 0 thickness paint ... But this only happens because we are comparing two non-comparable concepts, volume and area, by using an analogy with the real world that doesn't work, paint ...
@@GuzmanTierno Well, actually, the interesting thing is that it takes an infinite time to pour the paint as well: assume that somehow there is a constant gravitational acceleration equal to Earth's all along the horn (which in and of itself is a much greater stretching of physical laws than the infinite horn already is...): the first infinitesimal drop of paint that you pour in moves down the horn, and at any moment t it is located at a point _x(t)_ = 1/2 at². Now, that first drop needs to get to the 'end' of the horn... which is at _x_ = ∞. That only happens at t = ∞, so you'd be waiting forever to have enough space to pour in your last drop of paint, even if the total amount of paint is finite.
I think the point is that there is no paradox - it's just that the (unphysical) object that is the Horn has the attribute of infinite size (length and area) but finite volume. It's not that the analogy of paint doesn't work; it's that the properties of such an object are counterintuitive - not paradoxical, which implies contradiction.
The apparent contradictions arise only when we don't think carefully enough about the implications: e.g. we say "it will take an infinite volume of paint to cover, but a finite volume to fill" - not true, if we allow that paint can have zero thickness (i.e. we use comparable units); we then go to a second level and say "ah, but it will take an infinite time to paint, but we can fill it very quickly", and then we realise that actually filling takes as long as painting - if we assume the same 'realistic' constraints on finite speed of a brush or finite acceleration of a falling drop.
I think they make sense, an object with finite volume but infinite surface area. But if you can find an object with finite surface area but infinite volume, that will be truly amazing.
I like Tom's videos because he often is describing an applied maths problem. As he says an infinitely long (and infinitesimally thin) horn is not a real physical object so you see strangeness like this with a finite volume and infinite surface area. I like to resolve the paradox by thinking of the thickness of the coat of paint. If the coat of paint gets thinner as you go down the horn (for example thickness is 1 at x=1, 1/2 at x=2, 1/3 at x=3, and so on) then you can paint the horn with a finite amount of paint.
Dr. Crawford probably knows of real world examples where taking an infinite sum or integral gives some useful result, so the point of an example like this is just to point out that even while theoretical math sometimes sees strange or counterintuitive, it doesn't mean that the same maths won't be useful to solve real world problems.
So the paradox comes from the idea that you could fill the horn with pi amount of paint and by doing so you could paint an infinite surface area with finite amount of paint. I think the paradox is resolved by the fact that this is actually a non-sequitur. Filling the horn with paint does not actually paint the surface area because at some point the horn gets thinner and smaller than even a single quanta of paint. Even if you tried to fill it with paint, the paint molecules would get stuck in the tube and there would be an infinite surface left that would not be covered with paint at all. The only way to cover the entire surface area with paint would be to allow the thickness of the paint to be zero, but in that case the volume of the paint required would also be zero. I think the same logic also applies to any other object which has infinite surface area but finite volume.
"The paradox is resolved by realizing that a finite amount of paint can in fact coat an infinite surface area - it simply needs to get thinner at a fast enough rate (much like the series 1/2^N gets smaller fast enough that its sum is finite)." - Wikipedia
Any amount of volume greater than zero, even the tiniest finest positive volume, can cover an infinite amount of area
I think the paradox arises because paint has a finite thickness whereas the inside of the horn becomes arbitrarily thin.
No
I was thinking of convergent series when watching the first video! A nice reminder that an integral is essentially an infinite series.
Some of the points already arose in the comments of the original video. The most important part is that there is a difference between 3D paint and 2D paint. And yeah, you can't build it in the first place, so it's like asking what happens if you travel FTL
I would still repeat from the main video
The last paradox that, you can fill in pi volume, and it will automatically paint infinite amount of surface of the horn
Is not paradox. It's the fact that if you decrease the dimension of any volume. You can get a whole lot of area, mathematically infinite amout of area
Like a microscope slide. You can drop a very tiny bits of liquid. And when you put another glass slide over that drop, flatten the drop from 3d to nearly 2d, the tiny drop can fill the whole area,
Gabriel horn also squeeze the radius of paint closer to one dimension. And so the pi volume of paint gain closer to infinite amount of length along the horn
The point is, you can never paint it with 2D paint. But with 3D paint, you can collapse any amount of paint into infinitestimally thin for infinite amount of area. So you can always paint infinite area, if the shape is finite volume
The way I see it is that the object can't exist because of the size of atoms.
The hole of the horn would get so small as you go off to infinity that the hole is smaller than an atom.
Also the viscosity of the paint would make it so you could almost fill up the horn even if it was infinite in length.
As the pain drains down the horn, you would need to continually top it up a tiny amount as it falls down the horn for eternity.
You would be filling the horn forever but you could do it with a finite amount of paint.
But at the same time you would need an infinite amount of paint to paint the surface.
The object can't exist because as far as we can tell the universe is very large but finite... and it has a finite duration into the past.
Incidentally, if you assume that you can fill the horn with a finite amount of paint (although you'd have to take an infinite time to do that), then you can also paint it with the same amount of paint (or less!) - you cannot assume that paint will have zero thickness and viscosity when dripping down the horn but some thickness when painted on its surface. If it is infinitely stretchable, then it is infinitely stretchable when filling _and_ painting.
A surface area has no thickness, which is the same as having volume = 0, so any amount of paint will cover an infinite surface, if you set the thickness to zero.
No. It has to be infinitely thin, which is thicker than zero.
@@tsaszymborska7389 only in the the real world, not in the world of calculus.
As below stated, but a bit different: you don't need infinite volume of paint to cover infinite area.
It can be painted by any finite (continuous) volume of paint, if the paint spreads perfectly or in other words you can take infinite number of slices from the given volume.
It would be an interesting question whether you can cover this with a Menger sponge formed blob of paint, where the volume is zero, but the surface area is infinite :P
The same thing happens with fractals that are drawn on a sheet of paper. But those are worse because they are clearly bounded and don't go off to infinite locations.
A Koch Snowflake, for example, has an infinite perimeter but a finite enclosed area.
We can derive the area from the volume by subtracting 2 volumes of a horn inside another horn and taking the limit as the 2 horns converge. So if the volume is finite - then the area is definitely finite.
That's not how area is defined. If you assume that you can derive an area by starting with a volume and approaching zero, you can also prove that pi=4, which is a contradiction.
@@killerbee.13 It would get rid of that pesky quadrature problem though. There is grandeur in this view of life... (with apologies to C. Darwin)
One cannot paint the outside of the Horn if it is - quite reasonably - assumed, that the thickness of the paint layer is pretty much the same everywhere. But at the inside this is not possible, since from some point on there is not enough space for any predefined thickness. The layer has to get thinner and thinner, and at maximum can fill the volume. Of course, the entire configuration is not realizable with anything physical.
Imagine you have some paint that can be spread infinitely thin. You start pouring to fill the horn. You must wait an infinite amount of time before it is full.
It kinda makes sense. basically its the thing that you make volumes from infinite surfaces (same thing as a line is the infinite amount of points)
basically in theory you can paint ANY infinite surfaces with a finite amount of paint. in other word, the surface of the painted area can get close to 0
I completely understand the mathematics behind, but it still trips my mind how a finite volume could have infinite surface.
Is it there some other examples of finite volumes with infinite surface?
not exactly what you looking for but most fractal has finite area but infinite perimeter, including koch snowflake. 3D fractal like Romanesco broccoli is the example that u looking for.
Inspired by @Albert Lau's answer below, look up 'Sierpinski Sponge' or 'Menger Sponge'
"Don't mess with infinity."
Infinity is like some Tony Soprano-style mob boss. "You can't do that!" "Hey! I'm infinity! I can do what the f-k I want, OK?"
The way I resolve the "fill it and you already painted it" paradox is that any shape with non-zero volume automatically has infinite surface area.
Maybe the problem is in our intuition of painting = surface area. In reality we don't paint by surface area. We paint by volume. We measure the surface area of a room in litters of paint! That' just weird and it might be the root of the paradox
Mathematically, it should be paradoxical as well. Presumably, the horn has zero thickness. Therefore, if I can fill it up with a finite amount of paint, then its walls have been painted. On the inside, but that is irrelevant, because the inside surface is as big as the outside surface.
Love this video comments section.
No thats not the resolution... the right one is: mathematical paint can be spread infinitely thin. So any finite amount of mathematical paint can cover an infinite area.
That's not how paint works in the real world... that's why it "seems" to be a paradox.
But there are bounded things with finite volume and infinite area - like 3D variati9ns of the Koch snowflake. Still not possible to “make” one, perhaps.
Volume and surface area have different dimensionality, so are uncomparable. It's like comparing 1m^3 with 1m^2.
Pi being irrational is also a key factor here. If you filled the horn with pi to a trillion digits worth of paint, there would still be room for more because you haven't calculated an adequate number of digits. When taken to the infinite degree, no matter how accurately you think you calculate your volume of paint, there will still always be room for a bit more. So in a sense, pi is infinite, but along a different plane to that of an infinite series of ascending integers for example; it's true value would be at an infinitesimal point between 3.141592 and 3.141593 (with these bounds being increasingly limiting and precise for each decimal calculated). This obviously only works in the abstract, in reality there are perceived to be physical limits to how finely one can divide up the universe for measurement (though actual reality may indeed be infinitely granular, to excuse a contradictory sounding phrase). As explained in the video, it would be impossible to have an infinitely long horn, but what isn't explained, but that is equally important, is that it would also be impossible to have an 'exact' measure of pi as well. You would get to the point where you are counting individual particles of paint mixture but that would be the limit, and therefore still only an approximation of pi. The point is is that I don't think there really is a paradox, I think what may causes people to perceive a paradox is that it's seems easier to imagine the infinite length of the horn than it is to imagine pi being calculated to an infinitesimal point. As I say, 'I think'.
The rationality or irrationality of π has nothing to do with what is happening. You could repeat the reasoning with a square section horn, with integer ratios between area : side and perimeter : side and you would still end up with an infinite area bounding a finite volume.
A lot of people in the comment seem to confuse 'infinite quantity' with 'infinite decimal expansion' - the two are definitely _not_ the same thing. π is finite even if it has an infinite decimal expansion. ω (the smallest uncountable ordinal) is infinite, but it is arguably an integer.
@@dlevi67 I see what you're saying, I hadn't considered the volume being different. So really the 'paradox' arises when we try to visualise ANY number calculated to an infinite degree of precision, 1 being 1.000... for instance. The point being that if the horn, as you suggest, had a rational number as a volume, you would still never be able to construct it owing to the infinite area inherent in its design. So trying to imagine filling the horn with paint is futile, yes it would require a finite volume, but you would never, in the physical world, be able to fill it precisely enough to satisfy the equation.
@@dan3843 Yes, though I would say it in reverse: you can fill it* (get 4 units of paint, and it's going to spill out when it's full) - but you won't necessarily be able to use it as a super-precise measure even if in theory it's 'exactly' π units, depending on the precision of your construction.
* Or rather, you can't even fill it, because of its infinite length: paint will take an infinite time to reach the 'bottom' of the horn which is an infinite distance away. Infinity is nasty that way... 😁
If you think of pi as 3+0.1+0.04... if you added an amount of paint for each term you would be adding a finite amount of paint over an infinite number of steps. I’m not sure if that means anything but because pi never ends it seems like it fits with the infinite surface area.
It's nothing to do with π having a finite or infinite decimal expansion. You would get the same counterintuitive result with a square trumpet, but the perimeter/side (and area/side) ratios in a square are whole numbers.
Paint is a 3D object. If we have a surface, we can ask a yes or no question: "is it paint-able?", by which I mean "is it paint-able using a finite volume of paint?" To get from the datum [surface area of our surface] to the datum [volume of paint required to paint our surface], we simply fix a *thickness* of paint and multiply the surface area by the thickness to get the volume. Of course, the volume we get depends on our choice of thickness, but crucially the *finiteness* of our volume does not, so the paint-ability is simply determined by the finiteness of the surface area.
That is all true, of course, unless you are allowed to vary the thickness at different points of the surface. But that's kind of a cop-out and against the spirit suggested by the word paint-ability. After all, allowing for variable thickness, one can paint the entire infinite plane with a finite volume of paint! With this concept, surface area is now made irrelevant. Let's call these varying-thickness paintings illegitimate.
The paradox of Gabriel's horn is that it would seem that filling the horn with the finite volume of paint (π, specifically, as determined in the last video) *induces* a painting of the interior surface of the horn. The problem is that this is an *illegitimate* painting. The thickness must vary smaller smaller and smaller as you travel towards Gabriel's mouth at infinity. So this induced painting is the sort of painting that is irrelevant to the surface area, and so the surface area being infinite is not a contradiction to this induced painting's existence.
Any *legitimate* painting of the interior will have a fixed thickness. No matter how small, this thickness will eventually be thicker than the horn, and so this legitimate painting will extend past the filled-in volume of the horn. The intuitive bounding argument that is at the core of the paradox is, in fact, based on a bounding that does not exist for any chosen thickness!
Yes, this is exactly right. I see a lot of comments missing the point of the paradox because they are taking the question entirely literally when really what they meant was an analogy to real-world paint, where you do actually have a relationship between the volume of paint and the area it will cover.
If you started painting the outside of the horn with π volume of paint, starting at the wide end of the horn, how far along the x axis could you get before you ran out of paint, assuming you were applying an infinitesimally thin coat of paint?
I think you could paint the whole thing because you're not using up any volume of paint to apply an infinitesimally thin coat. So it's not a paradox at all. Intuitively, the paradox only arises because you think that painting an infinite surface area with a finite amount of paint is impossible, but it is possible if your coat of paint is infinitesimally thin.
In fact, you could paint the entire outside of the horn and still have enough paint left over to fill the horn exactly.
@@whitslack They didn't mention they talk about a paint that behaves mathematically in some regard but behaves physically in some other regard.
@@whitslack You cannot have paint infinitesimally thin though. Infinity is a mental construct, its not a thing and its not something we can apply to physical objects. Thats why we use limits to work with infinity, because you cannot work with it any other way because its not a thing.
@@StreuB1 Right, and Gabriel's Horn isn't a physical object either. But okay, let's talk limits. Let's say our paint layer has thickness T. What volume of paint do we need to apply to paint the exterior of the horn when we take the limit as T approaches zero? Zero. Painting the horn requires zero volume of paint, leaving the entire π volume units of paint remaining to fill the interior.
The horn quickly becomes narrower than a atom. That last atom of paint has more volume than the whole infinite spike of the horn - the spike has infinite height but subatomic width
Enter the world of the Planck Length!
Area is 2D, volume is 3D, you can't really compare those two... for example:
Imagine a sheet of A4 paper, which is 0,1mm thick. How big area will it cover if you have 1m tall tower of those papers? Roughly 3 tennis courts.
Now imagine a sheet of A4 paper, which is 1 Planck unit thick. How big area will it cover if you have same 1m tall tower of those papers? Well, it would be roughly 10 000 000 000 000 000 000 000 000 000 km squared, which is 10 lightyears by 10 lightyears square! *
Now imagine a sheet of A4paper with 0 thickness... ok, you get it, infinite area from finite volume.
* (i checked the calculation like 10 times as i couldn't believe it, but it is really 10 lightyears squared: 10^35 papers in 1m tower = 10^34 m2 = 10^28 km2 = 100 trillion km square = 10 lightyear square)
It might be an interesting thought experiment to consider what happens when the radius of Gabriel's Horn reaches the Planck length (1.6 x 10E-35 meters)
You could use any arbitrary small value in place of the Planck length, I would think. But there are two possibilities there. If it's a minimum length but you still have a truly continuous spectrum of longer lengths, you probably only need to change the bounds of integration. On the other hand, if lengths are quantized based on that minimum length, that seems like it would change the calculation and would also seem to depend on how the quantization behaves with angles and curves.
The answer is: "most likely, absolutely nothing special". There is no reason to think that the Planck length corresponds to any significant physical limit.
It's not based in the real world..Anything in the real world is finite.
So if you chop the horn off at some finite value of "x", to fit Gabriel's finite sized mouth properly, what would the ratio of SA(x):V(x) be? Because as it currently stands this ratio is infinite, for infinite x.
I, too, would have appreciated seeing this problem solved for a horn of finite length. I think it would help me understand the maths better.
I love that transparent Gabriel's horn comment. And the follow up comment why intuition fails. I have this book on the Monty Hall problem, which is a spectacular example of people's intuition failing. We sort of have to rely on intuition in life, and some amazing things have some out of it. But intuition is never proof, and because people can't seem to internalize this, we have an internet filled with falsehoods.
I don't see a comparison - Monty Hall is simply a miscalculation of the odds by the average person, rather than a paradox. Pose the game-show choice with initially picking one out of 100 doors and then open 98 goats and offer a swap and the common intuition of the average person obviously now matches the actual odds.
Consider a cylinder of finite height. Both volume and area are finite.
Imagine filling a glass with water. You can fill it to the brim. And the interior is wet.
Consider a cylinder of infinite height. Both the volume and area are infinite
It's like a bottomless cup. You keep pouring water into it and it never fills up. And since it never fills up not all of the inside will be covered by watder.
So there is no issue comparing volume to area or actually making these things. It's a thought experiment.
But why does Gabriel's Horn mix and match?
Is base infinity a thing? If 0.000...1 doesn't equal 0 does the volume become infinite like the area?
Couldn't you look at it from a point of view that to make the horn, you need a finite amount of material that you are then stretching infinitely thin. Kinda like the fractal sponges, the volume is bounded by the cube (for example) but every time you cut a hole you are creating new surface area.
Interesting
Its also interesting that pi is transcendental. Could you even have precisely pi amount of paint?
Doesnt matter. Just add a bit more.
You can fill a coffee cup right? The volume is pi r squared times the height.
After some point the endless values of pi are irrelevant in the real world. In this case when smaller than the molecules on the cup.
Definitely not, no.
It is quite similar to Koch snowflake, Minkowski island and other fractal figures, which have finite area, but infinite perimeter. But those are 2D, and this is 3D.
Mathematical paint would need volume of 0 because it's a surface.
The thickness is finite. If the surface were finite you'd need a finite volume of paint.
@@AnonYmous-rw6un yeah that's the problem, we are comparing the volume of paint to an area. if you make the paint surface have a finite thickness and it's proportional to 1/x, (instead of constant) then we get the finite volume of paint. If the paint's thickness is constant, indeed we need an infinite amount of paint, most of it would cover an atom thick tunnel at the back (that's why I suggest making the thickness proportional to 1/x). No paradox there, just wrong question to ask. But in a purely mathematical context, a painted surface is just a surface with a specific color, so 0 thickness.
@@AnonYmous-rw6un The "thickness" is 0 -- if we give the paint thickness, then it would always take less than Pi volume to fill the horn, i.e., you would never be able to fill the skinniest part of the horn.
So, we can't say the paint volume is Pi and also say that the paint has any thickness -- it just doesn't mathematically work.
isn't the Koch curve the same thing? finite area infinite circumference. To me it seems one can "stretch" a finite blob of paint to an infinite long shape , assuming the paint is continuous
But no such paint exists. Akin to explaining Santa with his ability to move infinitely fast.
Doesn't the inside surface need to be painted as well? Isn't that infinite also?
You’re painting the inside surface by filling the trumpet with paint.
Yes that the paradox
@@E1craZ4life no that's the paradox. You can fill the thing with paint but it wont cover the inside surface all the way.
You said that pouring in paint, it could never reach the bottom. One reason would be that under relativity, there is a limit to how fast it can travel.
So I have another puzzle - under Newtonian physics, if the paint travels fast enough to fill the horn in finite time, is its kinetic energy finite or infinite? And is its momentum finite or infinite? A tiny amount of paint has to travel really fast, is that enough to make the total kinetic energy and/or momentum infinite?
For reference, energy = 1/2 mass times square of velocity, and momentum = mass times velocity.
You can fill up with paint until the inside diameter is less than the size of the paint molecules, then it's full because they can't get through. So although it would still go on to infinity after that point, the filling is finite :)
But the volume is finite in any case, so the filling takes a finite amount of liquid... that's not the 'paradoxical' (or better, counterintuitive) aspect of the object.
What about the ocean’s coastline? Ok so the area of the sea is finite. But because when you zoom into the coastline at ever greater resolutions it tends to infinite length. Yet the finite ocean area is held inside this infinite length.
In that paradox, the mathematical paint isn't defined. And we're trying to intuitively compare a surface and a volume. Which doesn't really make sens.
I was wondering if there was a point where the volume matches the surface area, like a cut off point before the surface area outpaces the volume, but Wolfram alpha says that that is at either 1 (so zero length horn) or at a≈0.2847, so negative lenfrh.!
i agree...there is no paradox if only finite length horns exist in reality....just shows a limitation of maths in describing reality.
But you can still make a horn long enough for the amount of paint needed to paint it on the inside to be greater than the amount needed to fill it.
Guys, it is not a paradox, if you have any given non zero volume of paint you can paint with it infinite area if you will use infinitely thin layer of the paint. It is as simple as that. You have to agree, that the only reason why you think you couldn't paint the trumpet is because in normal life paint needs to have some thickness, right?? pi/x when x->0 is equal to innfinity. Problem solved.
A kind of a paradox could be that an infinite area can be "closing" a finite volume, but it is not so much paradoxical any more. Eg. infinite sequence of non zero numbers may have finite sum. Is it still a paradox for you?
The not being able to physically make it is definitely not the only problem the horn would hit Planck thickness at a finite length then you couldn’t even conceptually make the horn any longer.
(If you start off with the opening of the horn at 1m it is much longer than the observable universe)
Long before you get to that sort of a length you run out of things you could make the horn from (unless you can make it out of spacetime directly or something)
if you make it out of atoms it would only be about 30 light seconds long if you start of at 1m mouth radius.
How often does our old friend the Zeta function pop up in these videos?
There is one more thing that you miss that you are comparing area with volume.. which is not right as dimensions are not same
You can put one Gabriel's Horn into another one (identical), filling the gap between them with paint ;)
Isn’t this only a paradox because we define 1/∞ as 0, whereas it’s actually just a very small number.
The result would be the same either way.