This paradox, as mentioned by Prime Newtons (excellent video, by the way), was devised by Italian mathematician and physicist Evangelista Torricelli (1608 - 1647), prior to the introduction of modern calculus, and originated a great debate at the time. Nowadays Torricelli is mostly remembered as the inventor of the barometer.

When a mathematician keeps the soul genre more alive than soul singers

The real paradox is this: All the paint ever made plus all the paint that will ever be made will not be enough to paint it. Now, let that sink in.

6:00 “Oh I dropped my pie!” That’s never a good thing when you drop your pie! 😂😂

"One thing I don't trust... is infinity." * *_nervously looks at the Prime Newtons logo in the top right corner_* *

The way I justify this paradox in my head is by remembering that ANY positive amount of “mathematical” paint can be used to paint an infinite surface, because we can just keep taking thinner and thinner “slices” of the positive volume. So the pi cubic units of paint we filled the horn with are more than enough to paint the horns surface. And you can see that, because we painted the inside of it by filling it! And since the horn has no thickness, the inside and outside are exactly the same surface area

The reason there is not enough paint is because it leaks out the infinitely small hole at the end of the horn

I had an instructor that hastily introduced Gabriel's horn to me and my class at the end of one of his lectures and I have never really stopped thinking of it since. It rinkles your brain, and if i ever need an example of why calculus is interesting then I turn to it. Wonderful video, wonderful topic.

This was super interesting, it feels more like a fun application of calculus rather than a lesson. I'm very interested in how the surface area function works and how one could derive it.

Infinity is an interesting concept. Recently I have been reading about the infinity of infinities per the works of Georg Canter where he showed using set theory that there are infinite infinities, all different, and all going on forever.

Your narrating style never fails to keep me inspired.

I mentioned this a few years ago in another video about this so-called paradox. The real issue is that people are trying to equate area and volume and they are not the same thing. Further, lets fill the horn with the finite volume of paint, pi cubic units of paint. Now the horn is full. That means the interior surface of the horn is completely covered with paint. Now examine the horn. Mathematically, the horn is infinitely thin, which means the outer surface is actually the same surface as the inner surface. And since the inner surface is completely covered, so is the outer surface, and it is easily seen that the paradox is eliminated. If the thickness of the paint on the surface of the horn gets smaller and smaller, the volume of the paint that is on the surface gets closer and closer to 0. The total volume of pi comes from all the paint, not only on the surface but inside of the horn also. Compare this with another problem. Consider a square with side length X. The area is X² and the perimeter is 4X. Now cut the square in half at a midpoint of one of the sides and place the 2 rectangles together. Now you have a rectangle with 2 sides of length X/2 and 2 sides of length 2X. The area is still X² but now the perimeter is 5X. Keep doing this, each time cutting the rectangle in half along the long side and placing the two halves together to form a longer rectangle, and the perimeter increases without bound but the area stays the same. Is there a paradox? No. Length and area are not identical things.

Go for it, Mr Prime Newtons! 😊

This breaks the paradox, but it only bends the mind because we are comparing incomparable units (2d m² vs 3d m³)

Thanks professor, for explaining the paradox so elegant style.

Theoretically you can actually paint it.For example, if you paint with 1 mm thick paint (0.001m) then the area it will cover is pi³/0.001 which is the same as 1000×pi³ and if you approch the thickness to zero then the area approches infinity. So you can paint it

The 2 d version of this is take the series 1, 1/2, 1/4, 1/8, and view them as area and perimeter. First place a area 1 box at 0,0 to 1,1 for the area 1. You can find the perimeter of that fairly easy (4). now add on a box on top with area one half from (0,1) corner to (1/2, 2) corner, each box is 1 unit high and (series) unit width, and keep stacking the 2D rectangles (perimeter is 6 at 2 'boxes)). The area of the infinite stack is 2, but the perimeter is (2*number of boxes + 2), number of boxes > = 1 is infinite.

Nice video. Of course the paradox is resolved once you see that filling the horn is equivalent to painting it with an ever shrinking density of paint. On a real finite horn, you would actually put less paint on the pointed tip otherwise the job would look real sloppy on the tip. So the paradox assumes something that is even unphysical on finite horns. It is a clever paradox but when you think of it in practice, the paradox is resolved. You can take the paint you use to fill the horn and spread it on the outside. So if the density of paint you use is smaller than pi/x^2, the horn is painted! This is what you do on a real horn has I said above.

Great Video 😊 Just one comment on the calculation of the surface area. Easier than the formula in the video (and maybe geometrically more intuitive) would be the idea of stacking circle rings with variable radius on top of each other, the circle radius being r = 1/x, so dS = 2 pi 1/x dx. This is a much easier integral that results directly in S(x) = 2 pi ln(x).

I don't see any paradox in this: if we assume that "painting" means applying a layer of any constant non-zero thickness, then it is obvious that the radius of the horn's radial slice becomes less than it at some finite x.

I remember this paradox from calc classes. I likened it to, "each unit of length keeps increasing the surface area more than it increases the volume" So while area diverges, volume does not.

Thanks for making such content.

You've blown my mind

if you start with pi paint in the horn, the inside will be painted as well. Next fold the horn inside out.

Everytime I learn from your video I augh and exited how you make maths funny I wish some of professors could do the same way to make learning funny that Gabriel's horn

The fact that the volume is pi is hard to wrap my head around. This is because 1/x is used to generate the harmonic series which is divergent but only includes the infinitely many positive integers. The intagral says for its volume says take thin cylinders and add them (another infinite sum) but these cylinders encompass all real numbers. But the real numbers is infintely many more than the integers. What Im working towards is saying that a sum that (in theory) encompasses a larger infinity is convergent and the sum that encompasses a smaller infinity is divergent. I have no doubt that there is a flaw somewhere in my thinking. But it is mind blowing none the less.

Amazing miracles of math 💥. Thanks for the video!

Infinity is so weird, especially when bounds are involved.

There's a beautiful poetry to the concept. Rev 13:10 feels appropriate [...] The endurance and patience of the saints will be tested here. Obviously, math isn't religion, but sometimes the hardest ideas in math feel like the trials of saints and martyrs 😂😂😂

One should note that the infinitely thin wall of the horn is already painted on the inside. And that is very close to the outside.

The main "problem" here is, that - given the bounds of 1 and ∞ - the following is true: ∫1/x dx = ∞ (diverges) ∫1/x² dx = 1 (converges)

My math skills are good enough to follow this. It boggles the mind, and yes it is correct.

Cultivating intuition is an important aspect of mathematics Infinity | R has its own cultivation

The horn’s surface is two dimensional, therefore the amount of paint (three dimensional) needed is zero (i.e., paint thickness is zero). The horn volume is made up of infinite number of concentric horns, each with zero volume. The proper integration over the INFINITE number of concentric horns of volume ZERO is equal to pi. Conclusion: the horn’s surface can be painted using the available paint in perpetuity. The more interesting point is having a finite volume with infinite surface. But that can always be achieved by modifying the volume surface such that the volume is bounded by the original shape (e.g., introducing grooves on a surface of a sphere). For this particular problem, the ratio of the incremental surface area (ring) to volume (disk) as a function of x is proportional to x. Therefore, for a finite integrated volume, the integrated surface area diverges.

First, the surface area computation involves an error. Since a rotational body is an integral of an infinite number of discs, the body volume being an integral of the surface of the discs, the surface area of body is the integral of perimeters of these discs, 2π∫rdx = 2π ∫ dx/x. However, the integral is still divergent. Second, here is no paradox at all. If you cover the horn by the paint from outside, you put an equal-thickness layer of paint on every piece of the surface. So the total volume of paint is proportional to the surface area, hence infinite. However, when you fill the horn with paint, the farer is the cross-section from the opening (or the closer it is to the Archangel Gabriel lips, if it makes any sense), the smaller is the radius of the cross-section, and hence the thickness of the layer of paint on the wall. If you choose to think as a physicist, the mouthpiece (and some infinite part of the horn) will in fact remain unpainted since no molecula will fit into such a small radius. Just imagine: the paint fills in some part of the horn only (some billions meters, if our scale is in meters), and then for an infinite distance the horn remains empty! Alternatively, instead of sticking big fat molecules of paint to the horn, you might choose to decorate it with an electric charge, to enjoy clicks on Gabriel's lips. First way is, to cover the horn from outside by an infinity of minuscule balls of equal static charge. Then each piece of the horn surface will get the same charge. So the total charge will be proportional to the surface area, hence infinite. Alternatively, you can put a single ball with a finite (though, maybe, big) charge at the opening of the horn, and let the electrostatic field to spread inside the horn. Its strength decreases inversely proportional to the distance, so the example is relevant. Then the charge on the far part of the horn will decrease with the distance, so far parts of the horn surface will get less and less charge, with no click at all on Gabriel's lips.

About wanting to know the Infinite. Since the quality of the Infinite is Infinite, it cannot be experienced on the level relative life. One will need to go beyond the field of the relative. The finest relative aspect of life are thoughts. So, if one is able to take the mind beyond the field of thought, one can experience the Infinite. The process that takes the mind beyond the realm of thought is called Transcending. There are a number of techniques, methods that can give this experience. Most of them are difficult, but there is one that is easy and natural.

Great video as always! I never realized Math could be this philosophical

That's amazing

So cool!

You should also discuss the 'Four Colour Theorem' 😀

Why is it not correct to just integrate 2*Pi*y for the 2nd value? He is calculating the length of the curve - or what's wrong with my thinking?

Its mind blowing

You are amazing

When I painted my room recently, I thought about the thickness of a layer of paint. On the bucket there is a relationship between volume and area, i.e. litres per square meter. If the layer of paint was thin (more than infinitesimal), would the surface still of Gabriel's Horn be infinity?

If the end of it is in Gabriel's mouth, he's gonna have a hard time getting any air through the mouthpiece, since it will have a radius of dx. 😄

video about mandelbrot set and his godly designe would be cool

to get infinity near you just replace x by 1/x in the x-axis but in this new coordinate you will never reach the number 0.

"Perhaps it's proof that irrational numbers are, in fact, infinite. As you can't reach the last digit of PI... A volume of PI would be filled with ink indefinitely."

still didnt get how the volume approaches pie and the surface area infinity. feels like we are just another step from converting the surface area to a number. cause both "growth rate" decreases

If it is a topological garbie's horn, can't we just fill it with Pi paint, invert the shape and refill it with Pi paint there for you will have both sides painted ? Note: you may spill paint every where by inverting the shape

I believe in א.

Brill!

My question is: What is the interior surface area of Gabriel's Horn? There is enough paint to cover that area.

That is definitely a paradox. How the volume is finite but the surface area is infinite contradicts logic.

Ahhhhhhhhhh!

But by filling it aren’t you necessarily painting the inside, which is equal to the outside surface area? This is just crazy.

black vsauce can you add some background musics so you can be more like vsauce

Well... Pi has infinite digits

Infinity 😂😂

I can solve this paradox, volume is finite, but filling that volume with paint will take infinite amount of time, so you still cant fill it with paint.

answer=oo 1x

Mathematics is great

But there is no information about how thick the surface layer should be. If it is an infinitesimal number of uniform thickness, then the amount of paint needed is essentially infinity/infinity which still theoretically could converge.

Hey professor pai is not a finite number.

π is not a finite number so also the volume won’t be finite. I don’t see where the paradox is