Is this a paradox? (the best way of resolving the painter paradox)

[@ktbbb5]

The opposite of this is a vuvuzela, which is a horn with finite surface but infinite volume

[@nanamacapagal8342]

CORRECT

[@Mathologer]

Actually a surface of revolution that has a finite surface area but an infinite volume does not exist. Check out the Converse entry on the wiki page on Gabriel's horn :)

[@unvergebeneid]

Very clever pun 😄

[@satyadas4519]

🤣

[@nanamacapagal8342]

@@Mathologer well a vuvuzuela may not have infinite spatial volume, but it definitely has infinite loudness volume

[@andrewmirror4611]

My answer to the question on why people focus on calculus is because calculus is indeed a simple solution and thinking with calculus is way helpful for cases where the "neat" solutions don't work

[@godfreypigott]

The issue is that calculus is TOO easy, and tends to hide an intuitive solution.

[@arunprasad1022]

​@@godfreypigottIssue? I think it's a good thing. If that's the case then calculus only helps people see the beauty of mathematics. And I don't get how that is an issue. I am sorry if my reply was a bit offensive I didn't mean that.

[@toorero]

@@arunprasad1022I wouldn't say so. Because they use the brute-force "easy" calculus solution they neither see the beauty of mathematics nor do they get a good intuition in maths in general.

[@QuantumHistorian]

Not to mention that learning how infinite series work and how to prove which converge and which don't is taught at the same time (or slightly after) basic calculus.

[@GameBoy-ep7de]

@@QuantumHistorian That is what I'm learning right now in Calc 2.

[@mandolinic]

Some years ago, a company made a perfect mathematical paint as described. Unfortunately they couldn't get it to market because the paint leaked past the molecules of every container they put it in.

[@Mathologer]

Good one :)

[@TheEternalVortex42]

They should have just made a perfect mathematical paint can as well

[@mandolinic]

@@TheEternalVortex42 They tried, but the can was infinitely long, and very awkward to transport. Shops complained that since the bar code was halfway down its length, it required an infinite amount of time to scan at the checkout. Customers complained that it wouldn't fit in the back of the car. However, the Hilbert Hotel chain did put in an order for an infinite number of litres to paint their guest rooms, but since the only available colour was grey, an infinite number of guests complained that the rooms were too dull, so the order was never renewed.

[@kenhaley4]

So now all that paint is collecting at the center of the earth, collapsing into a black hole. Oh no!

[@mandolinic]

@@kenhaley4 A warning to be careful what you wish for...

[@incognidoqueen]

It is interesting to me that people find the 'infinite surface area and finite volume' combination to be so paradoxical. People seem much more willing to accept/reconcile the existence of 'infinite length/perimeter and finite area'.

[@unvergebeneid]

Because who paints a snowflake?

[@TheEternalVortex42]

It would be more paradoxical to have the converse (finite surface area but infinite volume or finite perimeter but infinite area). Of course this is provably impossible.

[@guillaumelagueyte1019]

It might be because of paradoxes like this one, but really it isn't much of a problem, you can paint an infinite surface with finitely much paint, you just have to use less and less as you progress :)

[@carly09et]

@@TheEternalVortex42 lol, take a circle, you have the disk, (the interior); the exterior. the disk has finite area and circumference, the exterior has infinite area... Now just invert(swap) the interior with the exterior ... :) the interior now has an infinite area. It is just a choice of metric. Contrived :)

[@mrosskne]

The actual paradox is that the horn has finite area on the inside and infinite area on the outside, which is actually impossible and is a genuine paradox regardless of how you feel about it.

[@ricardoguzman5014]

I came up with an interesting anagram a few years ago. We've all heard that saying "Rome wasn't built in a day". If you rearrange the letters in the phrase, it explains itself: "I want years to build, man". The interesting thing is: the apostrophe in the word "wasn't" in the first phrase becomes the comma after the word "build" in the second phrase, so every symbol is accounted for perfectly.

[@Mathologer]

Yes, I also like this one. And the apostrophe comma switch is a very nice touch :)

[@MathFromAlphaToOmega]

I have to say that you're one of the few people I know who actually pronounces other mathematicians' names correctly. I appreciate that you take the time to get them correct - it not only shows respect for those mathematicians, but also saves me from having to hear someone say "Weierstrass" with a W or pronounce "Weil" as "Wile" again. Thanks for paying attention to those little details!

[@Mathologer]

Well, I've got a very good excuse for always getting the German names right :)

[@TheEternalVortex42]

Some sounds are extremely difficult for non-native speakers to get. For example no one that isn't a fluent Czech speaker pronounces Dvořák correctly. And this is similarly true for Slavic or Asian languages, they are just too different from English for native speakers to do "correctly". So we're basically just deciding between various approximations.

[@MathFromAlphaToOmega]

@@TheEternalVortex42 Fair point, but I don't think it's unreasonable to expect someone to pronounce Weierstrass with a V sound or Oresme with a silent S. Those aren't really things that depend on the speaker's native language.

[@GianniCampanale]

Funny he pronounced correctly Torricelli but wrote Torrichelli in the description. Probably hearing the pronunciation led to guessing the writing.

[@WK-5775]

​@@GianniCampanaleOr maybe it was some Spanish influence....

[@DavidBeddard]

This has been a very helpful video. I'd been pondering how any island has an infinite coastline if measured to infinite precision yet has a finite land area, and had become stuck. This video came at a fortuitous time. Thanks for another great video!

[@Mathologer]

Glad it was helpful!

[@nanamacapagal8342]

CHALLENGES! 2:21 "Why is everyone using calculus - or being scared of it - when it's not required?" The calculus explanation is probably the most popular and obvious one. What can I say, calculus is robust. 15:49 "I hope you can forgive me for my little lie" Honestly, I thought something was off when you advertised the volume to be 8. I went through the calculus at the beginning of the video and got an answer of pi, checked wolfram and also got pi. But your reasons are perfectly valid, plus you gave a fix anyway to actually make the volume 8. You are forgiven. 18:06 "Let me know how this calculus-free exposition fares for you" Ties in REALLY WELL with your trilogy, and is a really nice way to finish off the series of this magical shape. Bonus As noted by someone in the comments, real world painters understand the amount of paint they use depends on how thick they apply it. One drop of paint is technically enough to cover the entire planet earth if it's spread thin enough, so it comes as no surprise that only 8 liters of paint can cover an infinite horn. Surface area and volume just work like that.

[@Mathologer]

"The calculus explanation is probably the most popular and obvious one." for the people who know calculus ... :)

[@Vaaaaadim]

"One drop of paint is technically enough to cover the entire planet earth if it's spread thin enough" I don't believe that is true. At least if we consider water, a drop of water is apparently ~0.05 grams, which would have ~1.67 x 10^21 molecules of water. And the surface area of the earth is ~5.1 x 10^8 km^2. Each molecule would have to cover ~3 x 10^-13 km^2, and if that area is in the shape of a flat circle it would have a diameter of ~0.00062 meters. But the diameter of one water molecule is ~0.000000000275 meters. Even if we pretend a water molecule takes up the same surface area as a circle with the same diameter - it only ends up covering 0.0000000000196% of the surface area it was assigned. Paint is not water but the conclusion would be basically the same, you need much more than a drop of liquid to cover the earth.

[@nanamacapagal8342]

@@Vaaaaadim by paint i mean mathematical paint that can run infinitely thin, and not physical paint which cannot. the task of filling up the horn requires that the paint runs infinitely thin anyway but you get my point

[@P_Ezi]

@@Vaaaaadim You proved that the phrase "thin enough" in this case means "not limited by the thickness of real world molecules".

[@Muhahahahaz]

@@MathologerI honestly thought that Torricelli had simply “rounded up” to 4, then counted both horns from your graph to arrive at 8 😅

[@LarsHHoog]

I use the subtitles on KZbin to compensate for my hearing loss and I kindly suggest that you watch this video with the subtitles to see the issue of having crucial information/animation in the lower parts of the video. I hope that this is seen as the constructive feedback that it's intended as.

[@jonquil3015]

It is now possible, with a mouse, to left-click the subtitle and drag it to any position on the video.

[@LarsHHoog]

@@jonquil3015 maybe so but that seems less feasible when watching on a TV via Chromecast from an android device. From an accessibility perspective it's better if the content creator take closed captioning in consideration

[@cxpKSip]

It's a shame that the KZbin subtitles don't support LaTeX.

[@eruwa1550]

You're definitely late to the party, and yet instantly my favorite video on the subject. Somehow, I'd never seen the harmonic sum approach and I loved how it fit in your 1/x trilogy. Bravo

[@titan1235813]

Dear Prof. Polster, your videos are always so clear, so entertaining, so well done, and so energetic and with such good vibes, that I thank you to infinity ♾️. Please, keep'em coming!

[@Mathologer]

Glad you think so and thank you very much for saying so :)

[@colekinyon2267]

I've always been intrigued by this paradox, calculus-involved approach or not. The visual aesthetics of your videos are really sleek as well. Thank you Mathologer for posting such interesting content in such approachable and easy-to-grasp ways :)

[@Mathologer]

Glad that this video worked so well for you :)

[@bernhardkrickl3567]

I'll call it a Parker Anagram.

[@Mathologer]

Matt's automatic claim to anything similar to his square. I am jealous :)

[@dcterr1]

Great explanation as always! I love how you manage to derive results without using calculus that seem to require them, and this is a good case in point!

[@Mathologer]

Glad you enjoyed this explanation :)

[@landsgevaer]

Given the 1/x shape, I was kind of expecting the squeeze and stretch method to be used again...: If you squeeze the width and breadth (i.e. diameters) of the horn by a factor 2, and stretch its length (i.e. axis) by a factor 2, then its volume halves but the resulting horn is the same as the original with the part between x=1 and 2 (in the graph) cut off. Similarly, if you squeeze width and breadth by a factor (1+eps) and stretch the length by (1+eps), then the horn has a factor (1+eps) less volume, but equals the original with a tiny eps-thick disk missing. That disk is like a thin cylinder and has volume pi*eps. So if the whole horn has volume V, then we get V = V/(1+eps) + pi*eps which simplifies to V*eps = pi*eps*(1+eps) or V = pi*(1+eps) which in the limit for small eps becomes V = pi QED But then you would admittedly have missed Torricelli and his anagram. Maybe I may suggest my username which is an anagram of my real name...? 😋 Unfortunately, that trick doesn't work for the area for reasons that the reader may ponder (but neither does Torricelli's).

[@Mathologer]

I thought I mix it up for a change :) But, yes, as you can imagine the squeeze and stretch method also gets my seal of approval :)

[@frechjo]

As always, I really appreciated the history bits that one comes to expect from this awesome channel :) But I especially liked the closing bit about ideal mathematical objects vs reality. Seems to me that often, a lot of otherwise great mathematical content tend to get a bit lost in idealism, and lose sight of that difference. It's a small detail, but it makes a big difference. Thank you for all the work that goes into these videos!

[@Mathologer]

"... but it makes a big difference" Yes the difference is infinite too (as far as maths is concerned :)

[@AR15andGOD]

Same with quantum mechanics. Wave function collapse, various quantum states, etc. It's all sci fi mathematical abstractions that take away from what's actually going on. In the case of collapse, there is no collapse. The wave function is pre-collapsed before we look; the "possibilities" of the function are only mathematical to explain the behavior of the function.

[@joyboricua3721]

That T-Shirt is quite harmonious 😃

[@Mathologer]

An ideal fit for this video :)

[@sumedh-girish]

yea he would look overly-toned without one of those...

[@hskim099]

I appreciate your intuitive teaching style. As you pointed out, other people tend to go deep into pedantic calculations, or keep gushing about how cool the paradox is.

[@benanddadmechanical6573]

But since the horn is an ‘open’ surface as 1/x never ‘closes’ by crossing zero then wouldn’t your paint just run out the other end?

[@Mathologer]

What end :)

[@P_Ezi]

First you have to stand it up with the pointy end on the earth. Since it is infinitely tall, you will have a difficult time reaching the big end in order to fill it.

[@drdca8263]

If we map the space so that a finite distance in that direction has image only a finite length away, then it can be closed up by adding a single point at the end. If the paint has any viscosity at all, it shouldn’t leak, I think. Or like, even without viscosity, if you start at the opening facing up, and assume gravity independent of altitude, and like, that the paint is incompressible, and of positive density (but infinitely subdivisible), and start pouring it in... Hm, well, if you’ve poured a certain quantity in, then the amount of force of gravity on all the paint is some finite quantity, and, uh... hm, the surface area at the bottom is pi (1/d)^2 where d is the height of the lowest part reached, uh, how much vertical force do the walls apply on the paint? I guess the force (or force per area) applied should be tangent to the walls? ... I’m not sure, but I wouldn’t be surprised if the rate at which the top surface of the paint moves down, asymptotically approaches zero? Though the bottommost part of the paint should always be accelerating downward at a rate of g?

[@PMX]

@@drdca8263 No mater how fast the paint moved down, or how much it accelerates, it would still take an infinite amount of time to reach a "point" infinitely away from the large opening 🤔

[@drdca8263]

@@PMX yes! It would be cool though, to be able to get some somewhat precise bounds on how quickly the height at the top would approach being constant. ... hm, well... seeing as the total weight is constant, and the surface area at the bottom is approaching zero, I’m not sure how to justify the conclusion that the pressure (or pressure gradient) near the bottom of the paint, doesn’t approach infinity? Which, if the pressure gradient approached infinity, then I guess so would the acceleration? And in that case, perhaps it could “reach the bottom” in finite time? I would hope that the force from the walls would partially counteract the weight, and so prevent the pressure gradient from going to infinity. But seems there would be work to do to show it.

[@guyhoghton399]

You can paint an infinitely long fence as far as you like with just one (finite) tin of paint. Assume the fence consists of an infinite number of identical finite sections. Use half the tin to paint the first section, then half of the remaining paint to paint the second section, then half of what remains to paint the third section, and so on for as many sections as you like. The thickness of paint on any section will be half the thickness of that on the previous section, so every section you painted will be covered in paint, yet there will always be some paint left in the tin at the end. It is similar to painting the inside of Gabriel's horn by introducing paint into it. The thickness of the paint at any point on the inner surface is equal to the radius of the horn at that point, being _⅟ₓ_ at point _x_ on the axis. Starting with a pot of _π_ volume units of paint, fill the section from _x=1_ to _x=2._ This will use half the pot, the paint thickness varying from _1_ to _½._ Then fill the section from _x=2_ to _x=4._ This will take half of the remaining paint at a thickness from _½_ to _¼._ Continue with the next section from _x=4_ to _x=8,_ and so on. Each section is twice as long as the previous one, but only requires half the quantity of paint, so you can fill/paint as far as you like but there will always be some paint left in the pot. Since the paint thickness varies inversely with length from the "mouth" there will always be some paint on the inner surface of the horn that you have painted so far, and you can carry on "painting" the horn like this indefinitely from the same pot of paint.

[@Muhahahahaz]

Exactly! Ultimately, the problem occurs when people assume that “painting” the outside is exactly the same as surface area - But they are unknowingly assuming a constant thickness of paint And while using a constant thickness may be possible on the outer surface, it is literally impossible on the inner surface - No matter how thinly you paint, the horn will eventually be too thin to contain it! (Unless you allow the paint thickness to approach zero as well, which is no longer a valid method for measuring the inner surface area)

[@kenhaley4]

I have explained this as follows with no reference to the horn: Take any amount of ideal paint (that is, paint that can be spread as thin as you want), say 1 quart. Spread it over a surface of 1 square yard of an infinite plane. It still has some thickness so, spread it until it covers twice the area. Now it's only half as thick, but it stil has some thickness you can repeat this, over and over--each time doubling the area this quart of paint covers. There's no end to this procedure, so the paint will cover the entire infinite plane.

[@ejrupp9555]

I tried to explain that the problem set up is flawed but you explain it without having to look at the problem. There is no paradox there is a problem with assumptions. It's a infinity / infinity issue ... so you can make up any crap you want.

[@TheVoidSinger]

Which leads to another fun conclusion: all finite ideal volumes contain infinite ideal area

[@StevenSiew2]

An atom of REAL paint has a FIXED volume, you cannot just half the thickness every time. Sooner or later you will hit the minimum size of the paint atom.

[@ejrupp9555]

@@StevenSiew2 Then so does the material of the horn and the problem doesn't even exist, because the infinitely long tail of the horn cannot be blown into. So it is not a horn when a gas atom cannot pass through it. There are no paradoxes ... just simple minds that believe imagination is has anything to do with reality or believe they can contradict them selves and be OK with it ... at infinity, the thing defined as having no end but somehow you can get to it as in AT it as to be at the end of the thing that doesn't have one by definition.

[@BarrieHughes]

Until it is one molecule thick?

[@robinwallace7097]

I saw many of these paradoxes in high school. At first I was amazed, then I realized that, indeed, math is an imperfect tool, not a discovery. ... but I do love watching them! Thank you!

[@Mathologer]

Yes, the more you know the less the basic stuff amazes. But luckily there is always more wonderful stuff to be discovered :)

[@txikitofandango]

Torricelli's argument about the volume of the horn equaling the volume of lots of circular discs reminds me of Archimedes' argument about the area under the parabola, except his argument also used mechanics like moment of inertia.

[@Mathologer]

Even more than his proof for the area under the parabola, there is the famous weighing argument that gives the volume formula for the sphere. Precursor of Cavalieri's principle en.wikipedia.org/wiki/Cavalieri%27s_principle

[@matindrie]

A similar thing is Koch island. A finite surface, but an infinite circumference.

[@BryanLu0]

​@@adb012That's the same? You just wrote them in the reverse order

[@adb012]

@@BryanLu0 ... You are right. I actually thought they were really the opposite. Let me delete my stupid post. I wonder if there is any shape with finite surface but infinite volume (which is what I thought I was saying).

[@GracindaMF]

This channel deserves many millions of subscribers, it's very underrated. Unfortunate, except for the current followers! :)

[@Mathologer]

Glad you think so :)

[@frankjohnson123]

Great video as always! Personally, I would have preferred if Torricelli's construction for the exact volume also included a top-down view (looking down the axis of the horn) after 14:34. This way we could see clearly how each point on a radius of the base circle was assigned its own circle to construct the cylinder.

[@Mathologer]

Sure, but clear enough as is I hope :)

[@Muhahahahaz]

Surely you mean bottom-up…? Otherwise, Mathologer would have to move his camera infinitely far away from the origin! I’m not sure he has time for that 😅

[@tmendoza6]

Fantastic! Looking forward to another video from Mathologer.

[@3Max]

I wonder if there are any similar fractal-like paradoxes? ("volume" slightly less than 3 dimensional finite, yet infinite "surface area" slightly larger than 2 dimensional? Any trade-offs here as the two dimensions approach each other?) Hmm, something for me to ponder!

[@Muhahahahaz]

Each additional dimension gives you infinitely more “space” to work with I would guess that you could always construct such a scenario, as long as the two dimensions are not equal. After all, 0.001 times infinity is still infinity! (Though I don’t recall the exact definition of a fractal dimension, and one would definitely want to prove this idea rigorously)

[@dojinho]

As interesting as always. I especially like the focus on the almost perfect (eerily so) anagram at the end. I came across this paradox when I studied mathematics a long time ago. The proof used, of course, calculus. I like this one, which relies on the same principles, but in disguise! 🙂

[@nHans]

Unlike mathematicians, real-world painters know that the area they can paint with a given volume of paint depends on how thick they slather the paint on the surface. Volume = Area x Height. They also know that they cannot spread the paint infinitesimally thin-otherwise just one drop of paint would be more than sufficient to paint everything. So no paradox for them.

[@Mathologer]

That's why we use deluxe mathematical paint only :)

[@nHans]

​@@Mathologer That's why I love mathematics! As an engineer, I have to deal with paints whose thickness is strictly greater than zero, while you mathematicians get to play with paints of zero thickness ... perhaps negative thickness as well. I bet you even have paints with complex-valued thickness!

[@MMichiganSalveRegina]

Also painters know that they need at least 2x the amount of paint actually required

[@oliviercomte7624]

Captain Obvious

[@douglasstrother6584]

During my career as a Physicist, the brass paid me to discover the Laws of Nature. During my career as an Engineer, the brass pays me to violate the Laws of Nature.

[@FF-ms6wq]

Amazing video, as always. Keep up the great work!

[@Mathologer]

Thanks, will do!

[@VideoNOLA]

Now we all know why atmospheric pressure is measured in TORR (a unit of pressure that is equal to 1/760 of one standard atmosphere. One torr is approximately 133.32 Pa)!

[@michaelgian2649]

Torrent = 1mmHg

[@Mathologer]

Yes, Torricelli rules :)

[@JamesWylde]

Random curiosity, at about 10:30 when you group fractions to make a series of 1 + 1/2 + 1/2 +... could I start to bracket 1/2's to get 1 + (1/2+ 1/2 + 1/2 + 1/2) + (1/2 + 1/2 + 1/2 ....) +... to get 1 + 2 +3.. or -1/12?

[@SpencerTwiddy]

Matt Parker must have invented that anagram

[@melainekerfaou8418]

With N letters in total and one letter wrong it's called a 'degree-1 Parker N-anagram'

[@hubbaba]

You guys are amazing! Thank you for the amazing videos!

[@Mathologer]

Glad you like them so much :)

[@Quasarbooster]

I wasn't very impressed with Gabriel's horn having finite volume, since it's only unbounded in one direction. However, you can create infinite copies of the horn, then rotate and fuse them into each other, such that their circular bases share the same point. If you make each horn point at a rational latitude and longitude, then scale the horn based on those denominators, you'll have a 3d solid with finite volume that is unbounded in **every** direction. Any other solid, no matter how small or far away, would be pieced by infinitely many horns.

[@yugiohsc]

Why would infinite copies of finite volume scaled by all rationals add to a finite total volume? Surely you’ll have to avoid some rationals

[@Quasarbooster]

@@yugiohsc here's a rough idea how we could do it. Let's just look at the horns on the equator. We will measure longitude to go from 0 to 1 (aka 360°). The horns will point in every rational angle p/q between 0 and 1. If each horn is scaled to have a volume of 1/2^q, then all the horns with denominator q will add to at most q/2^q. Adding up that amount over all values of q will give a finite result. So the horns on the equator have finite total volume.

[@Muhahahahaz]

@@yugiohsc”based on” those rationals. The rational isn’t the literal scaling factor, but it’s related by some function that makes the factors vanish appropriately

[@mattbuck4950]

About 5:00, it's not clear why the harmonic series being infinite means that the surface area is infinite. Is the idea that if you looked down on the horn from above then you would see more than H(x) surface area and thus A > H = \infty?

[@Mathologer]

"Is the idea that if you looked down on the horn from above " ... as we do :) "then you would see more than H(x) surface area and thus A > H = \infty"

[@itsasecrettoeverybody]

I don't understand why so many people are afraid of calculus. Calculus, geometry, analytical geometry and linear algebra were the easiest math disciplines I had. Anyway amazing video, I did maths years ago, I abandoned it for software development, but I love how I'm always learning something new with your videos.

[@flickingbollocks5542]

So sad that they hate maths especially when it is so easy for you... They probably enjoy sexual intercourse instead to make up for it.

[@nHans]

If those were the easiest math disciplines that you studied, I wonder which were the more difficult ones?

[@iang0th]

It's probably because Calculus is the most advanced subject likely to be taught in high school. and one that most people don't take at that, so it gets a reputation as being the "hardest" math. I know I had that impression before taking my first calculus class. People with perspective on the types of math that come later usually seem to consider it easy and intuitive.

[@itsasecrettoeverybody]

@@nHans differential equations, Galois theory, algebra I was okish but algebra II I found to be hard too, Complex analysis is not super hard but it's not as easy as calculus, real analysis I and II is almost at the same level as calculus but can be a little more tricker. Numerical analysis I and II was very easy too, it was what made me want to go to software development in the first place. I also had computer programming I, and was a breeze, So I took an optional semester in computer programming II, and numerical analysis III and IV. My course was in pure math, so those were optionals for me. But at the end I regretted not doing applied math. Topology I and II for the most part I didn't found particularly difficult too... My initiation was in this discipline. For the most part it wasn't a very difficult course. But I reproved in one semester of differential equations, one semester of algebra II and two times on Galois theory. Even doing Galois theory three times I don't remember anything about it. It's insanity. To be honest I only got the minimum grade because it was the last discipline for me and the professor took pity on me.

[@itsasecrettoeverybody]

@@nHans I don't think I can learn Galois theory to save my life. 😂

[@PUMAMicroscope]

Very nice. I never heard of this before (IANAM) but as someone familiar with image and signal processing this is not strange to me - it is just a curvy analogue of the Dirac pulse. Consider a signal plotted along X as a rectangle with unit width along X and unit height on Y so it has a finite area (area = 1, finite). Then simply shrink the width of the rectangle while preserving the area: the top goes up and up and up. As the width tends to 0 so the top tends to infinity in Y. The area, however, is constant - finite area with infinite height. If we started with an area of 2 (units squared) then the height would become infinitely higher than the first case. A very nice and easy way to explain even to kids that 'infinity' is not some constant number and that there can be infinitely many infinities, all infinitely bigger or smaller than some other one. Also a good way to teach that if you know where an infinity came from - you can tame it (in the above examples, first infinity over second infinity = 1/2, not 'undefined'). Thanks for the video - excellent as always.

[@mmburgess11]

Some YT'ers are just old-school/long-written proofs aficionados and some are just not very good at showing, visually, these complex problems (or they don't have the imagination to do so). The former are probably too conceited and are mostly showing off and the latter are patronizing us because they believe we cannot handle seemingly difficult equations. Your Mathologer videos are a perfect hybrid of both camps without the conceit or the patronizing. Your videos are very beautiful and thought-provoking while at the same time making complex concepts easy to understand (especially for visual learners as myself). Many times, I feel like I'm in over my head at the beginning of your videos only to have that "AHA" moment towards the end when you bring it all together with your impressive animations. I cannot help but think that some of these past great mathematicians agonized over how to write down the images and animations they were seeing swirling around in their minds-eye in exactly the same way you have shown us time and time again. Also, you always give credit to these past geniuses much more often than other channels and so we also learn some history along the way. Thank you for all yo do and please keep it going!!

[@Mathologer]

I'll keep on going, promise :)

[@Bodyknock]

5:00 One minor thing, he mentions the area of the harmonic series rectangles is infinite and therefore the area under the curve is also infinite, but doesn’t clarify that the corresponding surface area of the revolution of both of those around the axis is a constant times those cross-section areas. (I.e. the curved surface area of a cylinder is 2πrh, and in the example here h=1 and r is the harmonic series). He does clarify this a little more for the next part involving the volumes.

[@Mathologer]

Actually the exact relationship between the area of the staircase and the curved surface is irrelevant for all of this. What's important here is: 1. The staircase has infinite area. (and now I am spelling out the obvious part :) 2. The area of a surface is always at least as large as the area of its projection/shadow onto the xz plane. Therefore since the shadow contains the staircase, the shadow and therefore also the surface itself has to have infinite surface area.

[@Bodyknock]

​@@Mathologer Thanks for the interesting point 2) above. 👍 So yeah, it doesn't matter that it's specifically a constant multiple of the projection, just that it's larger.

[@sp_danger1729]

@Bodyknock I knew I could trust the comment section for demanding a further explanation of this argument! Thank you for asking

[@paulhilgendorf1446]

I think the fact Gabriel's trumpet is infinitely long, makes it feel paradoxical. Because in my opinion, something like the Koch Snowflake extended in a 3rd dimension, to make a prism of finite volume and infinite surface area, intuitively feels less paradoxical. You can easily contain it within a cylinder. Yet, you can take it apart into infinite triangular prisms. Then you can stack those atop eachother to get a roughly trumpet-like object; which has the same finite volume, and has infinite surface area.

[@tomkerruish2982]

Bizarrely, the three-dimensional analogue of the Koch snowflake, constructed by erecting progressively small tetrahedra on an initial tetrahedron in the obvious manner, grows to resemble a cube... on the exterior, at least. It becomes a cube riddled with internal fractal nooks and crannies.

[@jamirimaj6880]

7:54 is the perfect analogy to think the paradox rationally: You can divide any shape by adding a number of lines, but it will still have the same area.

[@mitchtheronin1469]

Another fun fact you missed about the Letters. „Æ“ is the sound you make when you hear about the paradox for the first time and „O“ is the sound you make after someone explained it

[@Mathologer]

Good one :)

[@stapler942]

I like the little mesh of curves the nodes of the harmonic series traces out when stacked up vertically on that shirt. 🙂

[@GabeWeymouth]

That is why it's an "anagr." instead of an full anagram, I guess.

[@Mathologer]

Yes, I was wondering about this. Why would anybody think it's a good idea to abbreviate here by two letters one of which is compensated by the period. Weird :)

[@brandontylerburt]

The first time I learned about Toricelli's horn was in an elementary calculus class, and until now, I was unaware of the existence of its algebraic form. I'm not sure how typical my experience is or how it compares to those of KZbin mathematics influencers, though. It did seem miraculous and mind-blowing to me as a beginning calculus student-as did so many other aspects of the calculus-whereas I'm not sure I'd have appreciated the paradox as much without Newton & Liebniz.

[@NottoriousGG]

Simple is King.

[@Mathologer]

Certainly is !

[@dogslife4831]

Have we discovered a new paradox? 😛 Simple = King King = King Charles King Charles = Simple (paradox) 😅

[@zetacrucis681]

I never understood how this could be considered a paradox (apparent or otherwise). Finite area under an infinitely long curve in the plane is completely analogous, but we take those for granted. (Or a 2-D 1/x^2 horn that could fit inside or outside one of Oresme's towers.) Fun informative video from Mathologer nevertheless, as always!! ❤

[@Muhahahahaz]

Well, in Torricelli’s time Calculus hadn’t been invented yet! (At least not from the European perspective - Apparently they weren’t aware of Madhava’s work) Also, from a modern perspective, not everyone knows calculus, and even if they do - a 2D graph is somehow more immediately abstract than trying to imagine a 3D object But yes, ultimately there is an analogy with any number of dimensions. And the lesson is that filling up larger dimensions is harder, because they have infinitely more “space” to work with. Thus something diverging to infinity in a “small” dimension can totally converge to a finite number in a larger dimension

[@pyglik2296]

I think the paradox partly comes from the fact that you can't compare surface area to volume at all. If some object you measured has the same area and volume, it's the consequence of your units and not the object itself.

[@Mathologer]

True, but finite volume infinite area is independent of the unit used :)

[@orterves]

6:57 given that pi is transcendental, does it mean that the volume is finite but it would take infinite time to fill?

[@Mathologer]

Yes, however, if trying this act of filling in some real-world sense it would take infinitely long to do so. However, this would not have anything to do with the fact that pi is a transcendental number. It does not make any sense to say that a number like pi "is infinite" :)

[@orterves]

@@Mathologer fair enough! Thanks for clarifying

[@zavgravian4172]

It was really a mind opener. Thank you very much for wonderful insights

[@Mathologer]

You're very welcome

[@colinpountney333]

I often use dimensional analysis to do reasonableness checks. Areas have dimension of length squared, volumes have dimension of length cubed. Comparing quantities with different dimensions is invalid. Hence there is no paradox.

[@Mathologer]

Well, units come into play when you compare finite area and volume for example. However, the distinction finite volume/infinite area is independent of units and so is definitely worth considering :)

[@gildardorivasvalles6368]

I was confused by your statement at the 13:13 mark "we can think of this extended horn as being made of thinner cylinders". It wasn't clear that you were referring to the volume of the extended horn being made of the thinner *cylinder* *surfaces*, not the volumes of the smaller cylinders --- which would have meant you were overfilling the volume of the extended horn. It was when you mentioned the method of shells from calculus that, after some thought, I understood what you meant: you're filling the volume more or less like an onion --- a mathematical onion, made up of an infinite number of infinitely thin layers. It wasn't immediately obvious what you meant, as it is a little bit ambiguous in the way you phrased it: I had to rewatch a couple of times to confirm I understood what you said. Of course, this is a wonderful proof (which I realized once I understood fully what you were doing), and I thank you for showing it to us. Loved your video, as always.

[@Mathologer]

Yes, I also pondered that at some point. Too many cylinders floating around doing different things. That's also why I talk about a "stack" of disks at some point and not about a cylinder :) Anyway, I ran this by a few people and the consensus was that the thin cylinder business shouldn't be too much of a stumbling block. One piece of feedback was that talking about cylinder surfaces instead at that point, the "surface" in "cylinder surfaces" would also have the potential to derail things in a different way.

[@gildardorivasvalles6368]

@@Mathologer thank you for your reply. Yes, what I said probably would have been confusing in a different way. In any case, it only took a little bit of thinking, and frankly that made it sink in better, and made me appreciate the proof even more, thinking in retrospect. Very enjoyable, definitely. After all, if one has to work for it, it feels more like one has earned it. Once again, thank you very much for the video 🙂

[@gheffz]

In those circumstances... close enough is certainly fun enough. Brilliant video. Thank you.

[@philipashmore]

Could the dimensionality of each measurement be a factor? The surface area is a two dimensional quantity, but the volume is a three dimensional quantity. How many surfaces can you fit into a three dimensional volume?

[@3Max]

This is a great question, and I also wonder if there are any fractal-like paradoxes with horns with volume slightly less than 3 dimensions, and surface area slightly larger than 2 (is that even possible?)

[@Etothe2iPi]

What a great video (as usual)! I can't wait for the fifth video of an increasingly inaccurately named trilogy of four.

[@Mathologer]

You may have to wait a bit :) Definitely the next video won't have anything to do with 1/x.

[@josephbrandenburg4373]

Some things I missed 1. How is there a relationship between the surface area of the cylinders (minus the endcap) and the stack of discs? 2. Why is the stack of discs 1 unit long instead if infinite in length? 3. Didn't he prove its volume was less than 2? Then how is it pi? 4. Why would anyone abbreviate "anagram" as "Anagr."?? You only save 2 letters?

[@valjones7537]

I taught 7th and 8th grade math, so my students and I had such a fun time discovering arithmetic ways of solving rich problems or, more often, answering a student’s question of why , that were traditionally relegated to algebra or calculus. Thus I think: It’s more about how we were taught to solve problems and whether we were encouraged to verify in other ways.

[@MolonFrikenLabe]

For volume we added units (in this case liters) but for area we did not. If we add units to the surface, we will soon reach the plank length in our calculations and at that point we can add no more area either.

[@Vincent-kl9jy]

Didn't we say the volume of the horn was less than 2, but also equal to pi?

[@Mathologer]

If you scroll back, there was an extra factor of pi (pi times the second infinite sum) kzbin.info/www/bejne/pl6vk62NotJlsM0si=e9c3vgMPrXMFmDSp&t=408

[@ib9rt]

You completely lost me at 5:00, unless I am missing something obvious. You compared the plane area of a series of rectangles with the surface area of a 3D volume. I did not understand how you could make a deduction from that? Unless you are saying that when projecting the surface area of 3D shape onto a plane surface, the area of the projection must always be less than the area of the 3D surface? And if the area of the plane projection is infinite, then the area of the 3D surface must also be infinite? Maybe some additional commentary might help viewers when making that leap?

[@Mathologer]

"Unless you are saying that when projecting the surface area of 3D shape onto a plane surface, the area of the projection must always be less than the area of the 3D surface?" That is what I am saying. I was wondering whether I should make more of a deal out of that but in the end decided that this is obvious enough in this case and that not many people would have any problem with this statement :)

[@jonquil3015]

@@Mathologer I agree with ib9rt that the absence of any qualifier was confusing. It would have worked better for me if the video included "some thought will show that..." or "since the projection of the horn surface ... it follows that".

[@delmotteo]

There's another paradox here, I think. As shown in the video, the volume of the horn is Pi. But we also know that the integral of 1/x from 1 to infinity is infinite. That means the area between the curve and the x axis is infinite. We also know that the volume of the horn can be obtained by rotating that (infinite) section area 360º. So, an infinite area rotating 360º generates ... a finite volume.

[@P_Ezi]

When you rotate a surface, there will be r^2 factors in the resulting volume. For most of the length, the r is extremely small.

[@Mathologer]

You can also see this reflected in the two infinite series :)

[@kevinmhadley]

It has been a long time since I have done higher level maths. It takes me a while to to get back to that mindset but, when I do, I enjoy these videos.

[@vangrails]

Same for me, I had to relearn how to do algabraic manipulations after two decades of ignoring math.

[@SaturnCanuck]

Thanks. Another great video on a Sunday afternoon with Coffee. Lovely.

[@Mathologer]

Glad you enjoyed it

[@tonekjansen5597]

Kudos for the comments at the end of the section "What paradox?" As a scientist I get annoyed when things (volume and area in this case) are compared that have different units. Good to see you dealing with that.

[@johnny_eth]

1:00 to cover the whole surface with paint you need exactly 0 liters of paint since mathematically surfaces are two 2 objects with no thickness and hence no volume

[@Mathologer]

There is also that ... :)

[@cryme5]

Another simple way to obtain the volume is to cut the horn at 1+epsilon with epsilon positive. Then stretching it with factor 1+epsilon in the perpendicular directions and shrinking it by a factor 1+epsilon in the longitudinal direction shows that the volume of this truncated horn is V/(1+epsilon) if V is the volume of the origin horn. The volume of the clipping is bounded between π epsilon and π epsilon /(1+epsilon)², thus π(1+epsilon) ≥ V ≥ π/(1+epsilon) for all positive epsilon. It must then be that, having shown V is finite, V = π.

[@Mathologer]

That gets my seal of approval :)

[@green-sd2nn]

beautiful as always ❤️

[@Mathologer]

Glad you approve :)

[@dcterr1]

Paradoxes involving infinity are quite intriguing! Some other good examples are Hilbert's Hotel and the Tarski paradox.

[@multi-purposebiped7419]

I don't know whether anyone has pointed this out but it would take an infinite amount of time to fill the horn as the paint has to travel an infinite distance to reach the [never]end, but it can't travel faster than light.

[@reidflemingworldstoughestm1394]

I don't always use tricks, but when I do I prefer 700 year old ones.

[@Mathologer]

Definitely, tricks age well with time :)

[@AbuTorabMahi]

I was Eagerly Waiting for Videos..And Yayyyy Hopefully I Have learned a new Intuition ❤❤

[@Mathologer]

Hope you enjoyed it!

[@Ggdivhjkjl]

Which colour should the horn be painted?

[@Mathologer]

Definitely blue :)

[@steviebudden3397]

Great video. Cheers. One minor thing though, the title made me think that Torricelli was a 14th century monk and I got quite confused when you started mentioning Gallileo at the end. Of course it was Oresme that was the 14th century monk, but he doesn't get mentioned so much in your vid. Still a great video though. Nice one, keep 'em coming.

[@paulb75]

5:06 can someone spell this step out to me. To me it seems inadequate to say “because the horn contains this infinite area (in its volume) the horn's area must be infinite. Not saying it's wrong, just that it is not beyond doubt for me. This video shows our intuition about "containing" is off, so I'd appreciate a more explicit step here, if possible. Thanks!

[@Mathologer]

Actually, what I say is this "But now since the horn encloses the staircase AS IT DOES it is clear that the surface area of the horn must also be equal to infinity". What's important here is: 1. The staircase has infinite area. (and now I am spelling out the obvious part :) 2. The area of a surface is always at least as large as the area of its projection/shadow onto the xz plane. Therefore since the shadow contains the staircase, the shadow and therefore also the surface itself has to have infinite surface area.

[@paulb75]

Thanks Burkard: I see now. Appreciate you taking the time. 😊

[@walternullifidian]

Whenever I see that Mathologer has put up a new video, I know it's going to be a fine night.

[@Mathologer]

That's great :)

[@dimitriostrigkakis2052]

When we start our actual working lives and get out of uni we never meet people with the same teacher soul like mathologger. Its very depressing having to live like this. I thank my lucky stars for this channel

[@stevenvanhulle7242]

Question for Mathologer: can you explain to us why the following set of parametric equations gives a 6-fold symmetry curve, given the fact that 6 nowhere appears in the trigonometric parameters ofthe equations, whereas 7 and 17 do? x = cos(t) + cos(7 t)/2 + cos(-17 t + Pi/2)/3 y = sin(t) + sin(7 t)/2 + sin(-17 t + Pi/2)/3 with t = 0.. 2 Pi

[@Mathologer]

A very pretty curve. Looks like a perturbation of a hypocycloid with 6 cusps. How exactly the 7 and 17 go into this I cannot tell at first glance. However, given that we are probably dealing with something closely related to a hypocycloid, I would not expect a 6 to feature in the formulas despite the 6-fold symmetries. I'd be expecting a 5. Anyway, If you want to puzzle this one out maybe a good idea to start here en.wikipedia.org/wiki/Hypocycloid

[@valjones7537]

My gut kept wondering, “Where will the 8 come in?” So it made me laugh when you explained; thus, it was a joke and not a lie at all!! And a hook for us to keep watching. Clever you!

[@Mathologer]

Glad this worked for you :)

[@aliskprado]

I love your videos!

[@aomaik7639]

Thanks for video , Please Mr can you make video about Euler-Compretz constant and the continued fraction for it 🙏

[@christianmaxschafer8696]

There is another example for the paradox of finite volume vs. infinite area: imagine slicing a finite marble cube into thin tiles to cover a floor. If the slices are infinetly thin the floor can be infinite large

[@Mathologer]

The slices don't have to be infinitely thin. Just create infinitely many square tiles of thickness 1/2, 1/4, 1/8, etc. :)

[@Pullen-Paradox]

How long would it take to fill the horn assuming gravity and the viscosity of water? Control the flow in a reasonable way. Is the time finite? Assume that the paint tends to adhere to the horn and any other reasonable assumptions. For the horn that has infinite volume, is there any way to fill it in finite time? Again, make any reasonable assumptions that will allow an infinite volume of paint to be delivered. For GH, it may appear that the answer is that the horn can be filled in finite time. However, for this to be true, we must assume that atoms are infinitely small. If we do that, we might as well assume that the speed of the water is given by the formula of gravity (Torricelli’s law). All of these assumptions render both questions less interesting.

[@mathandtharka7362]

Excellent presentation

[@joeyoest1105]

I really enjoyed this video. I teach Calc 2 on occasion and use this example, but I’ve never had a good gut-level explanation to give to students to help resolve the paradox. (I’ll argue based on the mathematical properties and note the importance of units /dimensions and the issues with comparison that arise here.) However, the 2-d -> 1-d analogue is perfect, and I will use it next time I teach Calc 2. I even explained it to my wife (not a math person) while we were on a walk (so without drawing it) and she understood it and enjoyed the analogy.

[@Mathologer]

That's great, glad you enjoyed the video. Since you mention units, I never quite get there in this video but of course one interesting observation is that both area and volume being finite or infinite is independent of what unit we use. Therefore it does make sense to compare finite/infinite volumes and surface areas of shapes. One interesting observation in this respect is that solids of infinite volume and finite surface area don't exist. On the other hand, that also means that areas and volumes scaling differently is not part of the resolution of our paradox which only depends on the volume being finite and the surface area being infinite. Are these things you tell your students ? :)

[@TheMichaelmorad]

This man wears a unique shirt for every video and every video I wait to see the shirt. I appreciate that.

[@Mathologer]

I own 300+ maths t-shirts. You still have not seen many of them :)

[@TheMichaelmorad]

how do you design your shirts?@@Mathologer

[@MolonFrikenLabe]

Professor, at 10:08 I believe this proof fails. I don't know this, but I'd like to show you what I have come up with and get your opinion. May I make a response video and tag you so I can show you my work?

[@3Max]

I was wondering about this too -- does it matter the "speed at which" we generate these 1/2s? eventually the "width" of your 1/2 will be larger than any finite number, (or said another way, the speed at which you create 1/2s will be slower than any small epsilon). But then again, there are still an infinite number of these 1/2s, even if slowly created, so I was a little confused.

[@stevenvanhulle7242]

How a finite volume can have an infinite area can easily be demonstrated by the following thought experiment: Take a cube with sides equal to 1. Then the volume is 1 and will remain 1 even if we cut up the cube. The area is 6 x (1 x 1) = 6. Cut the cube in half. We get 2 cuboids with dimensions (1, 1, 1/2). The area of each cuboid is equal to 2 x (1 x 1) + 4 x (1 * 1/2) = 4, so 8 for both cuboids. What we've done is adding two 1 x 1 sides at the cut. If you repeat the process we'll add two 1 x 1 sides for each of the new cuts, so the total area will become 12. After another iteration the area will be 20, and so on ad infinitum, where the area will be infinite, while the volume still equals 1. (This is also the reason coffee gets ground)

[@Mathologer]

Nice way of thinking about this. And, yes, that part of the puzzle is really quite trivial when you think about it for a moment :)

[@davidgillies620]

There's many examples of later proofs of a something being much simpler (or elementary in the mathematical sense of the word). Look at the Fermat two square problem which was first proved by Euler with considerable effort by a method of infinite descent, and now can be expressed in Zagier's "one sentence" proof. Or Bertrand's postulate: Chebyshev's proof is lengthy and that of Erdos is elementary (although not simple). Sometimes it's easier to just wheel out the big guns than looking for a clever trick.

[@Mathologer]

Sure, if your audience can handle the big guns :)

[@Stroheim333]

Sorry guys, but the painter's paradox only appears to be a paradox because it is presented in a too complex and confused way. We can all agree that if you have a certain volume of a substance, you can smear it out indefinitely only you make it infinitely thin! You don't need to talk about a horn or a confused painter.

[@tcmxiyw]

Another problem with the paradox is that different types of quantities are being compared. A priori there is no reason to believe that a volume (in units^3) is comparable to the surface area (in units^2). e.g. a square of edge length 5 has “less” volume than surface area, but one of edge length 7 has “more” volume than surface area. Apples and oranges.

[@petevenuti7355]

Question, how is the proverbial singularity at the bottom of a black hole any different? Isn't it just another Infiniti in a finite space just four dimensions?

[@Mathologer]

How is it different? It's totally different :)

[@petevenuti7355]

@@Mathologer I've always thought of that kind of gravity well like Gabriel's horn, descending to the point like moving forever into the future. If it is VERY different, perhaps you could make a video on it. I know it's been done by many other KZbinrs , but I would love to see your take on it!

[@Veptis]

Isnt your reactnalge trick jist changing if the first rextangle is included or excluded. I also don't think something like "smaller than infinity" makes any sense. Doesn't rhat just mean finite.

[@JuliusUnique]

"close enough is fun enough" haha that's a nice saying

[@blackholesun4942]

I tried to work out volume and got π. I was confused and let the video play. It was then explained that the volume we want is of part of the graph touching the y axis. Therefore is searched for volume under : y=1÷(x+1) . This gave π again. At this point I am confused and it is clear to me this is because I do not understand implication of 8 strech 🤔🤔

[@Mathologer]

If you stretch any 3d figure in one direction by a certain factor its volume is multiplied by the same factor :)

[@blackholesun4942]

​@@Mathologer Makes sense of course😅Thank you for the videoes as always! ❤❤ Also I must say, you have the best fashion in the mathematical youtube community 😁😁😁

[@nmrs4342]

Is there gonna be the Christmas video this year?

[@Mathologer]

Pretty sure there will be. Have not missed a single Christmas since the channel started and so ... :)

[@BernardGreenberg]

The snowflake curve is an easier-to-grasp example of infinite perimeter but finite area....one can surely rotate it...The vuvuzela remarks are great!

[@Mathologer]

Easier than my 1+1/2+1/4+....=2 based example? :) With the snowflake curve even just to see that it actually is a curve is not that easy.

[@BernardGreenberg]

@@Mathologer I think so, but I'm just throwing that out there...Great video, btw!

[@bigmouthfisheyes]

The expressions 6n +/- 1 produce all prime numbers greater than three, and many more composite numbers. If we knew exactly where the composite numbers would appear in these sequences, we could infer the location of all of the prime numbers. Am I understanding this correctly? Of what use would this be to anyone?

[@Mathologer]

Sure, once you know which of the numbers are composite you automatically know which of them are prime :)

[@MrPictor]

Your tee-shirt shows sound wave harmonics. Will you be calculating the sound of Torricelli's horn in the next video? 🎺😆 If not you, who will?!

[@bluerizlagirl]

Fractal shapes can enclose a finite volume but have infinite length (you can't see this on a screen due to features smaller than a pixel) so this isn't completely surprising ..... Is it something to do with the self-similarity property of the starting curve?

[@Mathologer]

Not to start with. Self-similarity can lead to both finite or infinite length. Self-similarity helps with actually calculating the length/area/volume of shapes though :)