The Painter's Paradox 

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

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 :(

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.

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

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.

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!

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.

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

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

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

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

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

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.

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

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!

lovely work my friend, keep it up!

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

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.

Beautiful!

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

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.

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!

Thanks for this!

Very nice 👌

Great video 👍

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

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.

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?

I am obsessed with this damn horn

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 .

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.

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

you are so good!!

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.

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.

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

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?

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

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

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.

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

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.

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?

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

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.

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

Reverse fractal intesifises

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.

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

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

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.

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

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

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

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

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

frustum i hardly know him

i love u

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

bruh, youtube placed like 3 short ads and 2 skippable

Conscious snowflake?

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.

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

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

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.

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?

Shut up and get my sub lol

Please don't rush through the math so quickly.

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

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

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

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?