The Painter's Paradox

In this video, we talk about the painter's paradox, that describes an object that can't be covered with paint but can be filled with paint. How is it possible to have an object with an infinite surface area but a finite volume? Doesn't it make sense that the outside of an object always takes up less space than the inside?
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This video was animated using manim: github.com/3b1...
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[@ParamLuhadiya]

Rick and Morty S4 + new video from Vivek = 2 million IQ boost

[@vcubingx]

😍

[@vcubingx]

Join my discord server! discord.gg/Kj8QUZU at 4:45, ds should be equal to sqrt(1+(dy/dx)^2) dx, it's missing a dx over there. my bad :(

[@all462]

Use annotations, many youtubers use that if they make any mistakes.

[@vcubingx]

@@all462 unfortunately they removed that feature

[@Mew__]

You COULD also paint the outside of Gabriel's horn with a finite amount of paint, but you'd need to use finite "mathematical" paint: Say you have 1 liter of paint. - Paint the first square unit (let's say m²) using 0.5 liters of paint. - Paint the second m² using 0.25 liters of paint, so, spread more thinly. - Paint the third m² using 0.125 liters of paint, so, spread even thinner. - Iterate to infinity.

[@hoodedR]

The question implicitly assumes an even thickness of paint

[@TheNSJaws]

@@hoodedR The premise alone is rather ridiculous since we're painting an impossible object with mundane real-world paint. If we were to equally abstract the paint to fit out impossible object, we're free to assume our paint layers are all equally dense just as our imaginary horn equally stretches into infinity.

[@hoodedR]

@@TheNSJaws of course it is. But what's your point? Also damn this is an old comment

[@Nova-op1ob]

I just ran into this when doing calculus problems. I hadn't realized that it was a well known problem and am incredibly glad to have stumbled upon this video. Thanks!

[@meimoscoso]

Man, I found you because of your fractional calculus and your content is really good. I love it. I can see how 3Blue1Brown affectes your presentation. It is really clean and pretty.

[@vcubingx]

Thank you! Just curious, how did you find my fractional calculus video? I seem to be suddenly getting a lot of traffic to that video and I can't figure out why.

[@meimoscoso]

@@vcubingx It was recomended to me next to Dr Peyam's video of the half derivative of one while watching something for my accounting final.

[@alexcheng2498]

This video is really insightful! It leads me to see perspectives that haven't been unveiled to me beforehand. Thank you so much for educating me - I look forward to viewing more of your videos in the future.

[@vcubingx]

wow! kind words sir

[@moonshine7753]

Well, if I pour all of my paint in there all the internal surface is painted

[@WackoMcGoose]

And then if you dip the horn into itself, you paint the exterior surface completely as well! ...And in the process, squeeze out all the paint, because to fully insert the horn, the gap between horn_inner and horn_outer approaches zero, thus the interior volume of the paint approaches zero. And depending on if you're truly inserting the horn into itself, or merely inserting a second, identical horn into the first, you then make an infinite series. But if you do the two-horn approach, you could fill H1 with paint (interior of H1 painted as a result), insert H2 into H1 (exterior of H2 painted), collect the paint that just got shot out at infinite velocity and pour it into H2 (interior of H2 painted, H2 now fully painted), and finally, dip H1 into H2 (exterior of H1 painted, H1 now fully painted). Abstract math is weird.

[@lennart-oimel9933]

That's the thing I don't get, too... What's wrong here? Maybe "paint" is not defined enough for that abstract things?

[@tablet9325]

@@lennart-oimel9933 i think because infinity is mathematical, not physical

[@catsdogswoof3968]

Infinite paint

[@kbtstyler]

@@lennart-oimel9933 that's why it's called Paradoxon

[@billyandriam]

We saw this in our Calculus class.Thanks for the explanation!

[@yashagnihotri6901]

5:00 What if we consider an elemental ring , the reason i thought this is because your frustum can be further broken down into 00(infinite) rings. The small area can then be written as : 2(pi)(1/x)dx , now integrating the small area from 1 to 00(infinity) gives you the required result ! By the way , i really liked the frustum one , I never did it that way hitherto.

[@vcubingx]

Yeah both methods work

[@raterix2]

Actually, I don't think this are the same thing. You're saying that the side area of all the cylinders converges to the area of the horn, as the sides of the cilinders get smaller. You can turn this problem in 2D by thinking about the lenght of a curve. If we want to calculate the area under the curve, we use small rectangles. Let's say we also use small rectangles for calculating the lenght of the curve, by considering the lenght of each rectangle and summing them up. But this is clearly not correct because the sum of the lenghts of the rectangles is just the lenght of the interval, not the lenght of the curve. But if you just want to proove its divergence, i think its enough, because the thing that you're actually measuring is always smaller than the thing you want to measure(the length of the curve or the area of a 3D shape).

[@amalantony8594]

@@raterix2 Yeah, thank you for putting the effort to explain it.

[@Diaming787]

When you paint something, that layer of paint has a small amount of "thickness", because in reality, atoms has a finite size. This already takes up some volume of paint. With infinite surface area, you'll run out of paint. The paradox simply due to bringing real life things to the abstract universe of mathematics. Great video and I subscribed!

[@sieciechczajka6554]

In reality for such an object to paint you need small amount of paint. If for example you need one bucket for the first meter of painting surface then you need just few dozens to paint to the edges of universe. It depends of course of parameters, but still you need very little paint to paint unbelievably long horn

[@davidrubio.24]

It actually has nothing to do with "bringing real life things to [...] mathematics". This apparent "paradox" happens within pure mathematics (as the video shows).

[@Nothing_serious]

This is about an ideal object where atoms don't exist. No one is talking about a real object because an infinite length of flute is impossible in the first place. It's also impossible to fill it in real life because you'd need an infinite time to fill an infinite length of flute.

[@JAzzWoods-ik4vv]

Fill horn with finite volume of paint turn horn inside out Take that, mathematicians

[@steventhijs6921]

I'm so happy youtube recommended me this channel, your style of videos reminds me of 3blue1brown. Amazing content for such a small channel

[@JalynKee]

Before watching this video, I was blind, but now I see very enlightening experience 10/10 recommend

[@MathyMahdi]

Hi Vivek! Though I haven't learned calculus yet, I can understand what is happening. Quick question: how much time did it take to write out the Manim code for this video. Thanks!

[@ericbailey386]

Another consideration/explanation is the irrational nature of pi. You can think of the volume of the horn in segments that are equal to the digits of pi and can be sumed to equal pi. V1 = 3, V2 = 0.1, V3 = 0.04... and so on. Since pi has infinite digits, the number of volume segments required to equal pi is also infinite. Leading towards infinite surface area for a finite, yet irrational, volume.

[@__karthikkaranth__]

At 4:17, is it assuming that the radii of the two ends of the frustum are equal?

[@Lotrfan1991]

I am obsessed with this damn horn

[@Lotrfan1991]

and you just blew my mind by comparing it to fractals.

[@furrball]

you didn't mention l (lowercase L) was the length of one side of the frustum when you introduced the formula. Sure, it gets clear shortly after, but I've always wondered what's with mathematicians and their apparent inability to state what's what in formulas. Not that I'm mad at this, but it's a mystery.

[@natevanderw]

As a math teacher, I might mention that it is VERY easy to forget to introduce something like that lower case l. It doesn't happen every lecture but definitely happens in a few classes each semester in one way or another.

[@staristo2355]

Great stuff! Thanks for bringing this to us! Getting 3b1b vibes here -- which is a good choice. Subbed

[@RohanDasariMinho]

lovely work my friend, keep it up!

[@vcubingx]

thank you dear

[@hardik-prajapati]

Amazing video! btw, are you using manim to animate everything ? (the same engine used by 3b1b)

[@casimirronnlof7396]

Yeah he uses manim

[@jimtwisted1984]

Why use the frusram only on the surface and not also on the volume?

[@brenofilho3320]

you are so good!!

[@casimirronnlof7396]

Great video! I just have a small question. When you calculate the surface area of Gabriel's Horn, wouldn't it be possible using the disc method but with the disc's surface area/"circumference" (it is not really the circumference since the small change dx). So at a point, say x1, the surface area of Gabriel's horn would be 2πrh (surface area of a cylinder without the bottom and the face), and if we replace h with dx and r with the height at that moment, which is 1/x1, it becomes 2π(1/x1)dx. And then if we sum up all these cylinders'/discs' surface areas it becomes the integral which you compared your integral to, which goes to infinity. Is there any wrong in my logic or can you even do the disc method this way?

[@casimirronnlof7396]

It basically is a direct way to get the integral you compared your integral to, but can you do this way?

[@vcubingx]

Yes, you can because the frustum essentially approaches a cylinder as dx gets smaller and smaller. As that happens the radii on each side approach the same value, which is why that formula 2pi*r*l can be used in the first place. Although I think it's a much better visualized using frustums rather than cylinders.

[@casimirronnlof7396]

@@vcubingx Alright thanks!

[@vcubingx]

​@@casimirronnlof7396 Oops, you replied but I wanted to say a bit more about this. The surface area of one of the cylinders using this method is (2)(pi)(1/x)(ds) (think about it, the cylinder is slanted in some places, so we use ds). So, to calculated the surface area, we just use integral from 0 to infinity of (2)(pi)(1/x)(ds), which gives the same answer. Thanks for watching!

[@casimirronnlof7396]

@@vcubingx yeah, no problem!

[@Stephen_Math_and_Science]

If I fill the volume am I not also covering the inner surface area?

[@SuperMaDBrothers]

If you want to paint the surface, you can just dump a finite volume in there, which would coat the entire surface (and much more). So the surface is finite? Contradiction?

[@morbideddie]

I don’t understand how that’s a contradiction. As the horn demonstrates a finite volume can create an infinite surface so an infinite surface can therefore be coated with a finite volume.

[@deidara_8598]

Another way to visualize an object with infinite surface area but finite volume is to think of a cake; As you cut up the cake its surface area increases while it retains its total volume. If you keep cutting up the cake into infinitely many peaces you will end up with infinite surface area but finite volume.

[@korigamik]

But why consider the surface area of the figure as very small frustums when instead we can just use the surface area of the cylinder and integrate with the simple integral?

[@captainhd9741]

I am not satisfied with this at all. Correct me if I am wrong but did you say filling in the volume can be done however painting the surface completely would be impossible? Well if you can fill the inside with paint then you have also painted the inside surface area which is exactly equal to the outside surface area

[@strayorion2031]

When I heard the paradox my solution was that it will arrive a point where the molecules of paint are going to be bigger than the diameter of gabriel's tumpet so, the paint will stop

[@prithwishguha309]

No bro it's ds = {√[1 + (dy/dx)^2]} dx ; you forget the dx

[@cottonlarry8470]

Create a gabriel horn and dip the whole thing into the paint. And dry it.... There u go. No more confusions nor paradoxes.

[@mohammadburhanmanzoor3806]

You could’ve just used the same infinitesimal disc for its surface area and integrate that surface area from 1 to infinity. Right?

[@noether9447]

Great video 👍

[@FandangoJepZ]

I can’t stop thinking, if you fill it with paint, you could fill it. The horn has no width, and the inside area SHOULD be the same as the outside right? Therefore you have painted all its surface area, albeit the inside but the area is the same

[@vcubingx]

That's what my initial thought was too. But think of a fractal - It has infinite perimeter yet u can fit it on finite area.

[@FandangoJepZ]

vcubingx but in this case, you would be able to paint the same amount of area as the outside. That’s just what seems to confuse me

[@NightWanderer31415]

4:37 you missed a dx outside the square root.

[@nosuchthing8]

But here is the paradox. If you tip the thing vertically and pour paint into it you can fill it to the brim. The volume is finite. But even after filling it you cant cover the inside surface. Because it has an infinite surface area.

[@morbideddie]

The thing about this paradox is that it effectively solves it’s self. The shape demonstrates that it’s possible to have a finite volume with an infinite surface area so why would the paint be any different? We need to paint a infinite surface but we can do it with a drop of paint if required.

[@toaj868]

As the radius of each cross-section decreases, the area of each cross-section decreases at a faster rate than the circumference.

[@offswitcher3159]

Hey, your last video was pretty long ago. I hope you are okay..

[@jbiasutti]

How about trying to paint a cylinder of zero thickness and infinite length with a paint thickness of 1nm. The object has zero volume and zero surface area but requires an infinite amount of paint to paint. Gabriella horn is bigger than our non existent object so must also take an infinite amount of paint.

[@comic4relief]

volume of 1x1x1 cube: 1 surface area: 6 1:35

[@devsutong]

explanation is perfect but whats with the occasional stopping of your speech while youre explaining

[@phyarth8082]

Again infinity surface area is when integral of function 1/x in range from zero to infinity when we have infinity volume and infinity surface area from zero not, even fact that by filling Gabriels horn with paint we fully paint surface area, just this contradiction says that area and volume is finite.

[@jocabulous]

frustum i hardly know him

[@mikel4879]

The volume of it is just only approximated as being finite, because in the integral calculus of the volume you assume that a quantity can reach infinity, which in reality it never does ( it satisfies the mathematical calculation, but it doesn't comply with reality ). That's why it is called "infinity". Mathematics has two aspects: one is the logic process created by a human or artificial brain and another one is approximation. It is known what logic is and its limit. Approximation is due to the fact that infinity is real, but it can't never be reached in finite real steps of real dynamics of the Universe. As a reality, the Gabriel's Horn has a real infinite volume, exactly as its area. The same understanding can be used for fractals and for etc.

[@larrydaiyuusha8615]

bruh, youtube placed like 3 short ads and 2 skippable

[@vcubingx]

wait, really? wow, let me see if I can disable it

[@larrydaiyuusha8615]

@@vcubingx it seems not the creators fault, just youtube being youtube

[@devsutong]

all of this is because volume is pi.. and pi is uncountable

[@NotBroihon]

No

[@morbideddie]

We could easily manipulate the shape to give us a volume of 1 and have exactly the same problem. pi being there is a fun coincidence but it doesn’t mean anything in regards to the meat of the painters paradox.

[@aaronsmith6632]

Please don't rush through the math so quickly.

[@dimuthuperera7229]

How can an irrational number like pi prove that the volume is finite? Even though Pi isn’t a technical infinity, it is still irrational and doesn’t have a defined end.

[@morbideddie]

Yes but we know pi is less than 3.2 and more than 3.1. The fact it is irrational is not impacting on the problem as when we talk about an infinite surface area we are talking about the value, not the length of the decimal expansion.

[@prithwishguha309]

And it's dA = 2πrds not S = 2πrl bro;...Lol

[@triton62674]

This is a really well animated and presented video but I feel it's a bit misleading to say the volume is finite by evaluating a limit

[@allanjmcpherson]

How else would you do it? It's the only way to consider it, and the volume is definitely not infinite. You can take as much of the volume of revolution as you like, and the volume will always be finite, and importantly, less than pi.

[@Nellak2011]

If you can fill it with paint, its surface can also be painted.

[@jamestagge3429]

this is pure cow pies. i dont care what the math says, there cannot be a boundary unless there is something it "BOUNDS". If Gabriel's horn is infinite surface area and the volume it contains is finite, then the rest of the horn surface area supposedly infinite, i.e., that beyond the finite volume the other part of it encompasses, cannot exist so the horn surface area cannot be infinite. It can only be that corresponding to the area it contains. A boundary is a contingent phenomenon. It is not foundational. It is the product only of a volume or area whose extent terminates in a particular place in that that area is finite in quantity/scope. The boundary is not a phenomenon but rather the name given to the point of termination of the volume/area, i.e., its edge. Define a boundary and you have really defined a volume or area and YOU CANNOT DEFINE A BOUNDARY OF NO AREA OR VOLUME. Such a boundary cannot exist even theoretically. This is all bullshit. How people can find this piffle amazing and mysterious is beyond me. Anybody here who can prove me wrong?

[@GhostyOcean]

Great video overall, but I noticed a false claim at the end. 7:30 you say that the volume is finite because the radius keeps decreasing to zero, but consider the function 1/√x. Using the formula for volume of revolution you get int (π(1/√x)²)dx from 1 to inf =int (π/x)dx from 1 to inf, which diverges. In fact, any function f(x)=1/x^p, for p≤½, diverges when you take it's volume of revolution. It's obvious that if the improper integral is to converge its radium must go to zero, but the converse isn't true. Just because the radius goes to zero doesn't mean the improper integral converges.

[@natevanderw]

Yes, what he said at 7:30 was a mistake indeed.

[@sunnyguancbr]

very intriguing and well presented topic, 10/10 would recommend

[@newtonraphson99]

Excellent work, dude. Your videos are masterpieces. I thank you too much for your effort. I hope you make a video about Lagrange Multiplier Optimization, please. Greetings from Mexico. :D

[@vcubingx]

Thank you! I'll look into visualizations for Lagrange multipliers and see if it's video worthy

[@newtonraphson99]

Thank you so much! Your channel it's gold.

[@RobertMilesAI]

Excellent stuff. Is there a reason you can't use the circumferences of the disks, rather than using frustums? Also, I think the gain on your microphone may be set too high, it sounds like you're clipping sometimes.

[@vcubingx]

Yes, you can because the frustum essentially approaches a cylinder as dx gets smaller and smaller. As that happens the radii on each side approach the same value, which is why that formula 2pi*r*l can be used in the first place. Although I think it's a much better visualized using frustums rather than cylinders. And sorry about the microphone issue. I noticed that a bit after uploading, but re-recording would've taken a couple of hours so I left it in. I'll make sure to have my microphone settings right next time.

[@vcubingx]

The surface area of one of the cylinders using this method is (2)(pi)(1/x)(ds) (think about it, the cylinder is slanted in some places, so we use ds). So, to calculated the surface area, we just use integral from 0 to infinity of (2)(pi)(1/x)(ds), which gives the same answer.

[@triton62674]

Robert Miles follow's 3B1B subreddit!

[@U9191-e6s]

Check 1/x is it having infinite arc length? And a finite area under the curve !

[@natevanderw]

Indeed it does not have a finite area under the curve.

[@RF-fi2pt]

Well, and this object, finite value at 2 dimensions contains an infinite at 1 dimension: Circle have finite area to one given R (although the precision is given by the π decimals). One swirl line starting at center until that R have 1D Length infinite, as the line diameter is infinitesimal. The Integral from 0 to R of 2πr, gives exactly πR^2, but trying to see the integration process as that increasing swirl see the 1D line Length going to infinite. At Gabriel Horn is the same. 3D finite Contains a 2D infinite .

[@massimoconti5431]

Pi is finite in the sense of it is less then 4, but it has infinite decimals. The surface follow these decimals. Both never end. The shape will become so thin very shortly for so long, toward to infinity. You can't complete to paint an infinite surface. You can solve the painting problem if you make the horn 📯 transparent, it will look painted. It is a pratical issue releted to convergence. Even the sum of infinite items never ends, but you calculate the limit.

[@thomaskolb8785]

Just dip the bloody trumpet in a bucket of paint and it is covered inside and out. Also, you never described how the horn sounds. OK, thanks, I'll show myself out.

[@arnavgarg2606]

Absolutely astonishing! You really have presented a beautiful solution to this paradox. very interesting

[@catsdogswoof3968]

Reverse fractal intesifises

[@Amanda-cd6dm]

Conscious snowflake?

[@matron9936]

Beautiful!

[@gaurangagarwal3243]

Shut up and get my sub lol

[@vcubingx]

Ily

[@shubhambibekar1919]

Very nice 👌

[@non-inertialobserver946]

Btw the computer voice read "Evangelista Torricelli" wrong. You should've made the italian google translate voice say it.

[@vcubingx]

oh I see. and also username checks out @Dillon Berger

[@RunnerRunnerRun]

Hey, great work on this; but I would advise you do a little quieting and post-processing on your audio - or buying a pop filter if you don't already have one! there's just a bit of audio clipping.

[@vcubingx]

Hey thanks, this was a problem in this video because I switched video editors, but I've solved this in newer videos!

[@tonyxin4186]

i love u

[@vcubingx]

I love u too

[@NovaWarrior77]

Thanks for this!

[@karthicks2518]

Volume is 3D, while surface is 2D. Then, cant we create infinite slices from a finite volume to create infinite 2D surface? Translating to the painting example, assuming molecules of a paint are finite 2D slices, can't a finite paint bucket hold an infinite paint molecules? Why cant we paint infinte surface with finite volume?