Gabriel's Horn and the Painter's Paradox
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- Опубліковано 15 жов 2024
- In this video, I explained the construction of Torricelli's horn also called Gabriel's horn using the function y=1/x. I explained the paradox of finite volume but infinite surface area of the horn.
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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.
I have to look him up. Sounds like an interesting scientist and mathematician.
I love scientific/mathematical history
@@sphakamisozondihe also did great work with fluid mechanics
"...inventor of the barometer" and remembered with the unit for measuring atmospheric pressure, the "Torr". (760 Torr == 1 atm)
Torricelli's theorem in physics is named after him
@@mikefochtman7164 bro, I feel so stupid, the Torr is named after him.
When a mathematician keeps the soul genre more alive than soul singers
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.
6:00 “Oh I dropped my pie!” That’s never a good thing when you drop your pie! 😂😂
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.
If your pour π units of paint in the horn, that is enough. It will just take infinitely long to finish the job. What you have to remember is that we use “mathematical paint”, which can be spread as thin as you want because it does not consist of a finite number of paint molecules. So you can always make the layer of this paint thinner and use the part that you took away to cover more of the horn.
In fact, even the smallest drop of this special paint would suffice!
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.
"One thing I don't trust... is infinity."
* *_nervously looks at the Prime Newtons logo in the top right corner_* *
😅
Thanks professor, for explaining the paradox so elegant style.
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.
Go for it, Mr Prime Newtons! 😊
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
I thought of this too!
Volume = Area * Height. You are basically saying that 0
I don't know if this guy is just a calm person and super passionate about what he is talking about
or if he is straight up high af
really nice video btw
The reason there is not enough paint is because it leaks out the infinitely small hole at the end of the horn
gabriel must be having a great time swallowing all that paint
Your narrating style never fails to keep me inspired.
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.
This topic made me drop Calculus two 😮 you made it so much more interesting!
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
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).
You know, I thought of doing that but again I wanted to show the direct comparison test because of some students who need to use it on their midterm soon. Your suggestion is actually, my preferred option.
@@PrimeNewtons Fair point 🙂
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.
But there's no reason to use "a layer of constant non-zero thickness". The layer can be 'infinitely thin' and yet the results are the same.
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.
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.
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
I wish if I Met you while I was young but anyway I m learning alor from you and becoming strong and stronger after any video
❤😅
Thanks for making such content.
This breaks the paradox, but it only bends the mind because we are comparing incomparable units (2d m² vs 3d m³)
We can't say that volume is greater than surface area and vice versa.
if we cover the thing with a layer of paint of thickness h then the volume needed to cover will be equal surface area * h
so no matter how thin the layer of paint is, we will not be able to cover the horn but will be able to fill it
so there is a way to compare the different units
@@daco-shitpost check my second comment (basically we cant say pi m³ is "more" than infinite m² and vice versa)
@Th3OneWhoWaits I mean, if you had an infinitely long string with width dx and tried to cover a slab with a finite area, you would be able to do that wouldn't you? Although now I think about it said string would really be in 2 dimensions (well 3, but 2 relevant)
@@Th3OneWhoWaits what i said is basically implied, however thin the layer of paint is, the "surface volume" will be bigger than the volume
You've blown my mind
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.
Amazing miracles of math 💥.
Thanks for the video!
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.
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 😂😂😂
Great video as always! I never realized Math could be this philosophical
…and entertaining!
The basis of math is philosophy
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.
That makes it even more paradoxical.
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.
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?
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.
if you start with pi paint in the horn, the inside will be painted as well. Next fold the horn inside out.
Infinity is so weird, especially when bounds are involved.
One small drop of a paint would be enough to paint all the surfaces in the Universe if you manage to make thickness of the paint layer equal to 0.
So you can fill the inside of the horn with Pi units of paint but there will be some side area that doesn't have paint in it? How can it be filled then? So it's both filled with paint and not filled with paint at the same time! This is I guess the paradox!
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)
Why is it a problem?
@@sobolzeev because the surface boils down to the first, while the volume boils down to the second. So the surface diverges and the volume converges, which is the basis for the paradox.
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?
My question is: What is the interior surface area of Gabriel's Horn? There is enough paint to cover that area.
You should also discuss the 'Four Colour Theorem' 😀
My math skills are good enough to follow this. It boggles the mind, and yes it is correct.
Its mind blowing
Cultivating intuition is an important aspect of mathematics
Infinity | R has its own cultivation
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
Its based on the series understanding of integrals, and diverging series compared to converging ones. For something to have a finite integral sum even at infinity it must be a converging series. Any non-converging series will have an infinite sum. Take the simple 1/x^2 compared to 1/x. The former converges but the latter does not. Take 1+ 1/2+1/4+1/8 etc, the total at infinity will be 2. But 1+1/2+1/3+1/4 etc.. does not converge so it can always be made bigger than any given limit. Thus infinity will lead to infinity. However, thinking about it, the whole concept seems to rely on only taking whole number values for the series, which seems a slight arbitrary restriction.
video about mandelbrot set and his godly designe would be cool
i would need good computer graphics to make that look good. I am not there yet.
i hope u get what you need sir. highly appreciate your videos
So 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.
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
Remember: you cannot tear or crease the gabriel's horn.
@@cheeseman4828 I don't need to, I just need to get to aleph1 bring the top of the horn to me which makes it flat, and then go to aleph2 bring the top of the horn to me again, and push it back by 1, it should have more or less the same shape
also renumber all the numbers from aleph 2 to aleph 1 in reverse order so that it is esaier to do the math for it
You are amazing
"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."
But by filling it aren’t you necessarily painting the inside, which is equal to the outside surface area? This is just crazy.
You put a decreasing thickness of the paint when painting from inside
I believe in א.
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. 😄
He needn't. He might just give it a proper vibration.
black vsauce can you add some background musics so you can be more like vsauce
Nice video, and great presentation style. Not a critique of your video. I just don't agree with the results and there are some discrepancies.
1) Pi is not finite, and cannot be used in finite measurements. Only approximations of Pi can be used in finite measurements.
2. Pi is not a measurement of volume. Pi cubic units has no meaning for determining a finite volume. A unit of volume, liters, cubic parsecs or something, must be included.
3. Surface area is a 2- dimensional measurement while volume is a 3-dimensional measurement. The volume of an object can never be less than the surface area.
4. Each iteration of the equation(each disc) is not exact. An infinite number of infinitesimals, represented by dx, are discarded.
All your but one of your objections are valid and I agree. Pi is finite but with infinite decimal digits.
@@PrimeNewtons Thanks for the response. Pi can be tricky sometimes. You have a great presence on camera, I hope you rocket to stardom. Aloha
That is definitely a paradox. How the volume is finite but the surface area is infinite contradicts logic.
Why does it?
@@sobolzeev How paint can fill the interior and not be enough to cover the exterior when the wall thickness is infinitesimal seems improbable no matter what the math shows. That's the paradox.
@@BartBuzz No paradox. The exterior is covered by a paint layer of a uniform thickness. The interior thickness is decreasing to zero as it departs from the opening. The mouthpiece is not painted at all.
That's amazing
Dear Prime Newtons, today I think I got an answer to the Gabriel's Horn paradox by myself! We can always paint an infinite surface with a finite volume of paint matematically, because Mr Maths doesn't stop at atoms, Mr Physics does. I mean we can always make infinite thin slices of the paint and keep painting the surface. But at this situation the thickness of the paint will tend to zero and we will not even notice the color of it! It will fade out! Waiting eagerly for your response.
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.
Brill!
Hey professor pai is not a finite number.
Do you know what finite means?
I think he meant to say pi is not a rational number
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.
Ahhhhhhhhhh!
Well... Pi has infinite digits
answer=oo 1x
Mathematics is great
Infinity 😂😂
π is not a finite number so also the volume won’t be finite. I don’t see where the paradox is
Pi is less than for example 3.15
Pi is a real number, and all real numbers are finite. Pi is an irrational number, and thus does not have a finite number of digits
pi is finite, it is just irrational. Meaning you cant express it as a ratio of 2 other numbers
@@drekkerscythe4723... ratio of two integers, to be precise.
@@drekkerscythe4723*integers, otherwise pi = 2pi/2
if the horn never ends, then how can the paint fill the horn? There is no bottom and therefore it cannot be filled?