So today we're revisiting the
painter's paradox; briefly mentioned on the channel before, but this time we're gonna- we're gonna do the maths. We're gonna crunch through it and we're gonna show this amazing paradox that you can have in this infinitely long horn which you can fill with paint but you can never paint the surface, because the surface area is infinite. And we're going to crunch through the equations, the maths, to see how it all works and exactly why this is what happens. [Music] Okay, so to form this- this infinite
horn - called Gabriel's horn - what we actually do is create a shape which is called a volume or a surface of revolution. And we're going to draw the graph of y equals 1 over x. So as x goes really small it shoots off to infinity; and as x gets bigger and bigger and bigger we've got 1 over that so that's going to go and decay down to
zero. And then what we do first of all is we're gonna just chop it here at 1. So this is at x equals 1 so y is also 1 here as well. So now we've got this, still infinitely long, graph and then we're gonna take it and we're going to
rotate it around the x-axis. It's basically like a big vuvuzela from the World Cup in 2010; bit of an old-school reference but it's one of those big long tubes where you're blowing down. The idea is of course this is infinite. Now this is is Gabriel's horn- (Brady: I'm assuming that comes
from the archangel Gabriel?) Yes so I think it does. I think this is called Gabriel's horn because the archangel Gabriel was often pictured with a really long horn; but this is infinite, it is important to remember that this horn goes forever. But of course it's going to be infinitesimally small as it- like, you're going to no longer be able to see it it's going to be that small as you carry on. (Gabriel must have a very small mouth) That's very true, yeah, if he's blowing in the small end,
exactly. First of all, let's figure out the volume of this shape.
- (I would have thought it) (would have an infinite volume because) (it never gets to a point where there's) (not a little bit of hollowness?) That would be a perfectly reasonable thing to assume; because it goes forever there's always something there, so surely its volume is also infinite. (It can always take just a
little bit more paint?) This is why I wanted to go through this, right? Because the paradox itself, it's called a paradox because if you try to think about it using our intuition it kind of doesn't make sense - that's literally why it's a paradox. But then, of course, we do have the mathematical tools to give us the definitive answer that almost like can't be argued with. So you might not believe what you get but you can't really argue with the maths. And that's what we're going to hopefully figure out. Now there's a formula for a volume of revolution, and some of you may know this, you do learn it at school, but we're going to kind of think about where it comes from by actually looking at the shape. So we will end up with the same formula, but let's think about like intuitively how do we figure out the volume of this? So we're going to need to do an integral, just like when an integral underneath a curve gives you an area underneath that curve; so here we're going to do an integral to get the volume. The way we normally do the area under a curve when you do an integral is you imagine splitting it up into loads of really thin strips and then you add together all of those strips as their width goes to zero - that's basically calculus. This is what Newton and Leibniz did, they came up with this whole concept. We do the same thing here but we're going to slice through and get little pieces of the horn, so 3d pieces. It's going to kind of look like a cylinder. It's not going to quite be a perfect cylinder, because there's going to be a little bit of a slant on the edge. But if you make these strips really thin it's going to get closer and closer to a cylinder. [Music] So what we do is we say this slice along the x-axis is going to be width dx, so this is where it's going to come into our integral, this is a small bit along the x-axis.
- (And the thinner it is) (the closer those two circles are to) (being the same)
- Exactly, yes. And we are eventually, through the integral process, we will let this width go to zero. That's what- an integral is taking that limiting process rather than just a sum where you add them together. So this is the height of our cylinder. So the volume of a cylinder is just equal to the area of the face multiplied by its
height - by its depth. So we know the area of a circle, it's going to be pi times the radius squared. And then we're going to multiply by the height, dx. So the volume of this cylinder is pi radius squared times its height dx. So now if we just add together all of these cylinders and then let dx, the width, go to zero that will give us our integral. The final volume of Gabriel's horn is going to be the integral- so we're starting from 1, and we're going all the way down here to infinity, tending to infinity. So we're starting from 1 to infinity, that's our limits of our integration, we've got pi we've got the radius squared and then we integrate with respect to dx. So this is the formula that perhaps you may have seen before for a volume of revolution, but it comes from these little pieces of cylinders being added together. Now the radius; so if we look back at our picture, and I've drawn this really badly and not very symmetrically, but if this was- if this is the central point and it's the same on either side the radius is exactly that distance from the axis up to the curve. And you can see that depending on where you take your slice you get different values of the radius. You're gonna get bigger circles up here and really really tiny circles down here. So the actual radius, the distance from the line to the graph, well that's just given by the value of y. Because you're starting at y equals naught and you're going up to this point here which is y, some value. So the radius r is actually equal to the value of y. But y is 1 over x, because it was our graph that we had; so the radius is 1 over x. So we can now plug this into our formula and we've got an integral which - we can take the pi outside - 1 to infinity and then we've got 1 over x squared dx. Now that is an integral we can do; if you've done any high school calculus we've got x to the minus 2, so if I do that integral I get minus 1 over x with my two limits and then I plug in the two values. So it's now pi multiplied by minus 1 over infinity; well 1 over infinity, that's zero. So you get nothing from that. And then it's minus minus 1 over 1 so it's 1. So the volume of Gabriel's horn is pi. Which in itself is nice, it's always nice when pi just appears out of nowhere. So if we take the graph of y equals 1 over x, rotate it around the x-axis to form this infinite horn, the volume is equal to pi. Amazing in itself; but for our purposes we didn't really care what this was as long as it wasn't infinity. Because this is a finite number so I can go out now and buy pi units of paint. I can fill Gabriel's horn with pi units of paint and it will be completely filled with
paint and there's- can't fit any more, can't fit any less, that's just that's it. Now the surface area; same kind of idea, we we want to think about what happens when we have a small piece of the horn, and then now instead of thinking about its volume we want to think about the area around the edge, so the surface area of the shape sort of going around and around and around and around. So imagine like trying to wrap the whole thing in toilet roll, for example. If we were to slice through
the horn you get something that looks like this. Not quite a cylinder, it's got like a slant; and then you have the big circle at this end which would be coming from up sort of this end of it and then the smaller one down here - so you have something like this. And now for the surface area you ha- the reason you kind of- this prop really helps is when you open this out it's like a part of a circle, annulus would be the technical term but it's like a sector of a circle isn't it, with the middle bit kind of chopped away; like a pizza crust. It looks like a pizza crust doesn't it? The area of this, again if I put it together, is going to be the surface area of our shape. This particular shape is actually the net of something called a conical frustum, which is a rather ridiculous name but that's what it's called. (It looks kind of rude, but anyway-)
- It does, frustum, yeah. (A conical frustum-)
- Pizza crusts and conical frustrums. So this is the the shape and this- we do have a formula for the area of this particular shape. So what we need to know to figure this out is you need to know two things: this length; and you also need to know this length. So let's call this one A and this one B. (Do you not need to know an angle or any) (thing about the the nature of the curve?) No, it's literally just- it's just A times B. Because what kind of happ- so if you were to sort of extend it and then think of this as a sector of your circle with some angle theta, it just so happens that the thetas kind of all come out. You can go through it and you'll get that it is just A times B in this particular shape.
- (I'm surprised it's so) (simple but there you go, I believe you.) I didn't believe it and I went through this after about four pages of algebra, it works. So I'll save you the the- agony of redoing that. So, now let's go
back and think back to our curve. This bit B along the top, this is actually going to be this bit here isn't it? As they join together this bit like along here - there is a slant here. So this thing along here we're going to say this is about a straight line; again we're going to shrink this thing down. We say this is about a straight line and we're going to call this ds. We use a different variable because it's not horizontal this time, the slant here is important which is why this was curved instead of just being a rectangle which it would be for a cylinder. So the slant here is important. Again, it's a little bit technical exactly why the slant is important, it's to do with Taylor expansions and lots of algebra. But the slant here matters for the surface area. So we know that, we're going to call that ds. And then this bit, A, well this is that circle there; it's the circle around the bit of the horn. And of course that's the circumference of a circle, the circumference of a circle is equal to two pi times the radius and again we showed here the radius is just given by y, which was 1 over x, because the radius is the distance from the axis to the shape itself. So what we've got then is the piece of surface area here is going to just be given by- so SA for surface area. We're going to have to integrate, with respect to ds - and we'll worry about that in a moment - and then we just multiply ds by 2 pi r, which is A times B. So 2 pi- we could
write r but we know r is 1 over x so we can just write that as 1 over x. So this is going to give us our surface area. Now this pesky ds; so the way to deal with that, we look at what's going on over here. ds is- it's kind of a straight line, and then if I were to turn this into a triangle like this then we've got ds here, call that a right angle. So this horizontal direction, well that's along the x-axis. As I'm moving horizontally this is the x-axis; so this is dx, a small change on the x-axis. And this is vertical so this is dy. So we can now actually write good old Pythagoras' theorem, he's got our back; ds squared is going to be equal to dx squared plus dy squared. Okay, so then what if I square root? So ds is now the square root of dx squared plus dy squared - we only take the positive one because it's a distance, it's got to be positive.
I'm going to do this pretty rough and ready, you can do this properly with a limiting process. So what you do is, you kind of factor out the dx. So you say, right, well if I take the dx out here, so it's come out the square root so it was a dx squared inside so I'm taking out a dx, that leaves me with 1 plus dy by dx squared. So this is actually a derivative, I've kind of treated them like they were fractions. You can do this properly with limits, again I'm doing it rough and ready because we just want the result. So what we now got then is we can- this is now in terms of y and x, which is what we wanted because we know those. We know, for example we know that y is 1 over x, so dy by dx is minus 1 over x squared, differentiating that function. The other good thing that we've got from this is when we had ds we didn't know what our limits were, because what is s? It's just the distance along the curve. We didn't know where to integrate between. But now we've
actually turned ds into something with respect to x. So again, we're going to go from the limits on x which are from 1 all the way to infinity. So finally we've got the integral; we can take the two pi outside and then we've got the integral from 1 to infinity of 1 over x multiplied by ds, which is the square root of 1 plus dy by dx squared so that's- square that I get 1 over x to the 4 dx. So we get this pretty awful looking integral. Now, we're not going to do this integral. We could do this integral; you- you can use trigonometric substitutions, you can even throw in a hyperbolic function if you really want. But we don't even need to do that, because we kind of know what we're aiming for and we know that this, based on the paradox, this is going to go to infinity. So if we can show this is bigger than infinity then it's infinity. It's a really good tactic, when you have an idea that something is going to be infinity, show that it's bigger than something else that's really common that we know is definitely infinity. Because if you're bigger than infinity you're infinity. I've got this function here, the square root of 1 plus 1 over x to the fourth. And I know that x is between 1 and infinity, so it has to take a value between 1 infinity. So whatever value of x I plug in here this whole thing is always bigger than 1. Because if I- if I plug in x is 1 I get the square root of 2, that's bigger than 1. And then as x gets bigger and bigger and bigger this bit I'm adding above 1 gets smaller, but it never quite gets to 1. So this whole thing is definitely bigger than 1. So let's just replace it and say, well, so the surface area is bigger than 2 pi times the integral from 1 to infinity of 1 over x dx. So we've really just replaced that whole horrible
square root by 1. And this is still true, this is an integral we can do again, so this is just 2 pi integral of 1 over
x is the natural log of x between infinity and 1. The natural log of infinity is infinity. So we didn't need to do the horrible integral, because we can show that the surface area is greater than something that goes to infinity, so the surface area is infinite. Gabriel's horn, where we take 1 over x rotate around the axis to create this really infinitely long horn; we can fill
it with pi units of paint but we can never paint the surface because the surface area is infinite. And that is the painter's paradox. (And you're talking about
just one face of) (the surface too, you're just talking) (about the outer face.)
- Yes, this is just the outer face yeah. So if it's really really thin then the inner surface area would be the same, so it's almost like doubly infinite! (So if the inner surface
area is the same,) (when I poured in that pi litres of paint) (hadn't I not automatically painted all) (the surface area?) That's the paradox I think. I tried to think about that and there was like- I think that's the paradox. It's quite neat, if you're really into your your technical maths it's quite neat. Because it all relies on the fact that the volume looks like 1 over x squared and the surface area looks like 1 over x. That it's- that is the key. Because if if you've if you've done a reasonable amount of maths, perhaps university level, 1 over x squared- so when you're doing an integral you're adding together lots of pieces. And if you add 1 over n squared forever you get pi squared over 6. It's called Basel's problem, it's a very famous result. But if you add 1 over n forever it diverges, it goes to infinity. And so Gabriel's horn and the painter's paradox is basically encapsulating the whole behaviour of 1 over n squared converging as a series versus 1 over n diverging. Well I'm sure you won't be surprised to see you can take this fantastic interactive quiz about none other than Gabriel's horn here on Brilliant, which is today's episode sponsor. It's really comprehensive, as you can see. This is part of Brilliant's 'Calculus in a Nutshell', one of many courses on the site. It covers all the classics, have a look at that. It's fun, superbly designed, as you'd expect. If you'd like to check it out along with all the other mathematics and science and other courses on Brilliant go to brilliant.org/numberphile. There's 20% off a premium subscription by using that url. There's always great new stuff popping on the site, go and have a look: brilliant.org/numberphile [Extras] ...sort of come to terms with it,
is to say, well- because one of the arguments I've heard about this is to say, if I made it see-through then I could fill it with paint and it would look painted from the outside. And again that's paradox, it just literally blows your mind.
As a math hobbyist, I just throw intuition out the window whenever infinity is involved. I understand all the math in the video, but I can't fathom a shape that you can fill but not paint the inside of.
Here's a way I found to help with intuition: Imagine the rectangle R_t having one side of length 1/(t2 ) and one side of length t. As t gets larger, R_t's perimeter gets larger and larger, but its area gets smaller and smaller. So as t increases, it gets easier and easier to fill R_t with paint but harder and harder to paint its sides. Gabriel's Horn is like a way to make R_infinity an actual shape (and in one additional dimension). This doesn't make it totally clear or anything but it is how I like to visualize it.
The maths makes sense to me, but it still boggles my mind, if that makes any sense?
So this is a good example of a function that is square-integrable but not integrable?
Can there be a shape with the opposite properties? Infinite volume but finite surface?
Is it possible to do the fluid dynamics for what the horn would sound like?
As I've told hundreds of calculus students over the years, the supposed paradox is "you can fill it with paint but you can't paint it."
This is just playing with words, ill-defined words.
What does it mean to paint something? Think about it.
I've used this a few times as an Oxford Maths admissions interview question, but I guess I can't anymore now you all know the answer...
i still dont understand this because logically it doesn't make any sense. we show that it can be filled with a finite volume of paint. well, that means that the horn completely encloses that volume of paint, with no empty space left over. then wouldn't u agree its also true that the volume of paint on the inside is contacting the horn on the inside, because we just established there is no empty space inside the horn? if it werent not touching the paint, what else would it be doing? so here we have the paradox. how can the finite volume of paint be completely contacting every point on the inside surface of the horn, but at the same time not cover the inside surface with a finite amount of paint? why cant u just take away a finite amount of the paint from that pi amount that would otherwise make contact the entire surface inside, so that there is only the portion that contacts the inner surface left?