Is this a paradox? (the best way of resolving the painter paradox)
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- Опубліковано 14 тра 2024
- The painter's paradox, a.k.a. Gabriel's horn paradox a.k.a. Torricelli's horn paradox has been done to death on UA-cam. So why do it again? Well, being all about some remarkable features of 1/x, this topic nicely complements the previous two videos that were also dedicated to 1/x. Now the first Mathologer trilogy is complete! Also, I thought of a couple of nice twists to make this treatment of the painter's paradox really stand out from the crowd (I hope :)
00:00 Intro
02:59 What's that horn?
07:09 A 700 year old trick
12:26 What's the volume exactly?
16:33 What paradox?
18:17 Fun fact.
20:12 Thanks
Must check out.
Wiki page on the horn: en.wikipedia.org/wiki/Gabriel...
(also has a nice discussion of the converse of the paradox at the end)
Wiki page on Torricelli: en.wikipedia.org/wiki/Evangel...
(the anagram that featured in the video also gets a mention on this page)
Very nice write-up of the history of the horn by Paolo Mancosu and Ezio Vailati www.jstor.org/stable/233514 tinyurl.com/3rff7ubb
Wiki page on Oresme: en.wikipedia.org/wiki/Nicole_...
(has a page from one of his books with some graphs. Somehow overlooked this one. Would have been nice to flash this in the video :(
Stanford Encyclopedia of Philosophy entry on Nicole Oresme (in particular, see the section on math) plato.stanford.edu/entries/ni...
Wiki page on the Method of shells: en.wikipedia.org/wiki/Shell_i...
Fun/intersting: The opposite of this is a vuvuzela, which is a horn with finite surface but infinite volume (as in loud)
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. And 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.
Music at the end: Young rich pixies - Year of life
T-shirt: TONEWAVE Music Tee Geometric Overtone Frequency Musician Sacred Geometry Tshirt www.etsy.com/ca/listing/21276...
Enjoy!
Burkard
The opposite of this is a vuvuzela, which is a horn with finite surface but infinite volume
CORRECT
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 :)
Very clever pun 😄
🤣
@@Mathologer well a vuvuzuela may not have infinite spatial volume, but it definitely has infinite loudness volume
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
The issue is that calculus is TOO easy, and tends to hide an intuitive solution.
@@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.
@@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.
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.
@@QuantumHistorian That is what I'm learning right now in Calc 2.
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.
Good one :)
They should have just made a perfect mathematical paint can as well
@@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.
So now all that paint is collecting at the center of the earth, collapsing into a black hole. Oh no!
@@kenhaley4 A warning to be careful what you wish for...
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'.
Because who paints a snowflake?
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.
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 :)
@@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 :)
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.
I use the subtitles on UA-cam 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.
It is now possible, with a mouse, to left-click the subtitle and drag it to any position on the video.
@@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
I'll call it a Parker Anagram.
Matt's automatic claim to anything similar to his square. I am jealous :)
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.
That's why we use deluxe mathematical paint only :)
@@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!
Also painters know that they need at least 2x the amount of paint actually required
Captain Obvious
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.
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!
Glad it was helpful!
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!
Well, I've got a very good excuse for always getting the German names right :)
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.
@@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.
Funny he pronounced correctly Torricelli but wrote Torrichelli in the description. Probably hearing the pronunciation led to guessing the writing.
@@GianniCampanaleOr maybe it was some Spanish influence....
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
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.
"The calculus explanation is probably the most popular and obvious one." for the people who know calculus ... :)
"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.
@@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
@@Vaaaaadim You proved that the phrase "thin enough" in this case means "not limited by the thickness of real world molecules".
@@MathologerI honestly thought that Torricelli had simply “rounded up” to 4, then counted both horns from your graph to arrive at 8 😅
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 :)
Glad that this video worked so well for you :)
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.
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!
Glad you think so and thank you very much for saying so :)
Matt Parker must have invented that anagram
With N letters in total and one letter wrong it's called a 'degree-1 Parker N-anagram'
Fantastic! Looking forward to another video from Mathologer.
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.
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!
Glad you enjoyed this explanation :)
You guys are amazing! Thank you for the amazing videos!
Glad you like them so much :)
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.
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.
Which leads to another fun conclusion: all finite ideal volumes contain infinite ideal area
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.
@@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.
Until it is one molecule thick?
Amazing video, as always. Keep up the great work!
Thanks, will do!
That T-Shirt is quite harmonious 😃
An ideal fit for this video :)
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)!
Torrent = 1mmHg
Yes, Torricelli rules :)
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.
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
A similar thing is Koch island. A finite surface, but an infinite circumference.
@@adb012That's the same? You just wrote them in the reverse order
@@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).
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!
Yes, the more you know the less the basic stuff amazes. But luckily there is always more wonderful stuff to be discovered :)
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!
"... but it makes a big difference" Yes the difference is infinite too (as far as maths is concerned :)
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.
In those circumstances... close enough is certainly fun enough. Brilliant video. Thank you.
It was really a mind opener. Thank you very much for wonderful insights
You're very welcome
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! 🙂
Thanks. Another great video on a Sunday afternoon with Coffee. Lovely.
Glad you enjoyed it
This channel deserves many millions of subscribers, it's very underrated. Unfortunate, except for the current followers! :)
Glad you think so :)
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).
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 :)
Excellent presentation
I like the little mesh of curves the nodes of the harmonic series traces out when stacked up vertically on that shirt. 🙂
Whenever I see that Mathologer has put up a new video, I know it's going to be a fine night.
That's great :)
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.
Yes, I also like this one. And the apostrophe comma switch is a very nice touch :)
beautiful as always ❤️
Glad you approve :)
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.
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.
If those were the easiest math disciplines that you studied, I wonder which were the more difficult ones?
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.
@@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.
@@nHans I don't think I can learn Galois theory to save my life. 😂
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
That took me a minute (dead fast, for me). Nice illustration of the Harmonic Series.
I love your videos!
What a great video (as usual)! I can't wait for the fifth video of an increasingly inaccurately named trilogy of four.
You may have to wait a bit :) Definitely the next video won't have anything to do with 1/x.
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.
I was Eagerly Waiting for Videos..And Yayyyy Hopefully I Have learned a new Intuition ❤❤
Hope you enjoyed it!
Loved this!
I don't always use tricks, but when I do I prefer 700 year old ones.
Definitely, tricks age well with time :)
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.
Sure, but clear enough as is I hope :)
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 😅
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!
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)
Paradoxes involving infinity are quite intriguing! Some other good examples are Hilbert's Hotel and the Tarski paradox.
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.
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.
So, well explained, thank you for the choice to keep it simple. I don't mind the fib at all!
That's great :)
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.
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)
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.
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.
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.
Same for me, I had to relearn how to do algabraic manipulations after two decades of ignoring math.
If you wanted to have a horn with volume 8, you could extend the "bell" towards the y-axis, to (if my crazy calculations are correct) pi/8, you will have a horn with volume 8. BTW, love the harmonics on your shirt!
I do some stretching at the end to arrange for a volume of 8:)
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!! ❤
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
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.
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.
@@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.
@Bodyknock I knew I could trust the comment section for demanding a further explanation of this argument! Thank you for asking
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!!
I'll keep on going, promise :)
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.
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 ? :)
Amazing video as always! You've got to love the geometric solutions, they once provided, when the the algebraic counterparts were invisible to these noble minds:) [Music at the end?]
Young rich pixies - Year of life
Enjoyed thoroughly
Glad you did :)
Thanks for a next beautiful video.
Glad you enjoyed it :)
Beautiful!
"close enough is fun enough" haha that's a nice saying
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!
Glad this worked for you :)
Good stuff!
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?!
Thanks for video , Please Mr can you make video about Euler-Compretz constant and the continued fraction for it 🙏
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 UA-cam 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.
Lovely video, as always. I did notice that you added an 'h' to Toricelli's name in the video description. In Italian, that would alter the pronunciation to a 'k'-sound.
Something to fix. Thanks for pointing this out to me :)
This man wears a unique shirt for every video and every video I wait to see the shirt. I appreciate that.
I own 300+ maths t-shirts. You still have not seen many of them :)
how do you design your shirts?@@Mathologer
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.
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.
Good pronunciation , Thank you for the Video . I get it also as a grown up I refuse to say that Gabriel's horn is paradoxical , ...... If I can paint it , it's finite .
Actually I got Torricelli wrong :( The c should be pronounced as a k. Somehow did not even occur to me to double-check.
Oh, I do love that T-Shirt!
Yes, it's a great one. Sadly I don't remember where I got it from :(
Found it: www.etsy.com/ca/listing/212767891/unisex-tonewave-music-tee-geometric
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?
What end :)
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.
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?
@@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 🤔
@@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.
The paradox may be simply due to denominating the paint by volume but the painted surface by area. One could say something similar about a humble cube of paint applied to an infinite succession of squares, all of the same side length: Successively "paint" each square, remove the volume of paint that was "used up" but since it's a box of height zero nothing changes, and repeat.
Anyways, nice video. I had seen Toricelli's paradox before but I like this presentation.
That's great. Yes, since this one's been done to death I was aiming for something cute very much off the beaten track. Still whenever I cover something that has been done to death that almost always means a significantly reduced number of click from the outset :(
This is one of the reasons why i love matgs !!! Gabriel horn paradox .
Still remember being very taken by Gabriel's horn the first time I heard about it :)
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.
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 :)
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.
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
@@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.
@@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
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
Good one :)
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!
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.
@@Mathologer I think so, but I'm just throwing that out there...Great video, btw!
There are some YT math channels that regurgitate proofs they learned elsewhere or are hand waiving because they dont understand effective teaching methods. Original content and methods and visual techniques are what make great creators who share their brilliance, logic and hard work and make genuine contribution. Which is what we know we get from your channel. UA-cam is saturated with a lot of borderline content now as well. But the paradox is resolved with the subscribe button and like button and the hope of the UA-cam algorithm properly doing its job.
I pretty much agree with everything you say. As an aside I hardly watch any UA-cam myself :)
That little titbit at the end explains why we sometimes use Torr as a unit of pressure . . .
Yes, but hardly anybody knows :)
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.
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.
@@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 🙂
I first ran into this in 1st year university. Needless to say it has provoked endless conversations and disagreements, some of which get somewhat heated (all in good fun though). So when I asked my Calculus prof about this (back in the 70's) he gave me an answer which made me seriously re-think about what we really mean by area when applied to a curved surface. He simply said the we may not understand area on a curved surface as well as we think we do. Because, as we all know, area is defined - square meter- on a FLAT surface like the rectangle. But when we apply that unit to a curved surface we have to distort it (stretch it) to make it lie on the curved surface - and then all heck breaks loose. And, as you correctly point out, we are dealing with an "ideal" paint so squeezing some molecule into the horn just doesn't apply here.
On the other hand, the volume of the horn is based on the flat definition of volume (3 dimensional - but still "flat" space). And that means the volume concept does NOT have to be distorted or stretched when measuring the space inside of the horn. The inside of the horn simple lives in flat space. Whereas the surface area lives in curved space.
And, yes, I know that as you make a small area on the curved surface shrink and shrink the more and more it starts to look and behave like a flat surface - but it really never does become flat - certainly not in the macro world.
BTW, I know Mathematics Teachers who are still convinced that: .9999...... = 1 is a "paradox". But it really isn't. Infinity is not a number nor a place, it's just weird. Thanks so much for all your work. I really do like your approach.
I still remember having similar discussions :)
Here's another thought. Imagine a long thin cylinder (which is what the horn basically turns into waaaay out there). Section off a chunk of length H.
V= (pi)(R^2) H
Lateral Surface Area = 2(pi)RH
Take the ratio LSA/V and simplify to 2/R
As R ---> 0 the ratio ---> inf. The volume decreases proportionally faster than the lateral surface area.
It's like the reverse of the square/cube growth thing (limiting the size of animals) because the dimension (R) is ---> 0
That is why it's an "anagr." instead of an full anagram, I guess.
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 :)
Your channel is amazing but here's something to think of when designing the animations.
For us needing the subtitles due to hearing loss the text lines occupy the lower parts of the screen, meaning that parts of the info/animations cannot be seen.
Try it out and maybe this will be a useful piece of advice - reserve space at the bottom of the video so that important information is not obstructed when using subtitles.
The content and quality is of utmost quality as always.
I am assuming that you have played with the subtitle settings. Viewed on a large screen with a relatively small font does make the issue you mention bearable, doesn't it?
@@Mathologer the text needs to be readable so text size is what it is and with illustrations/animations going almost all the way down to the bottom of the screen any text will obstruct info.
A Cow Orker once told me that “the juice of one lemon, if spread thinly enough, can cover the entire state of Texas.”
Cow Orker ... nice one.
Indra built the seamless cylinder, show me its beginning
I can show you the beginning if we declare the lip of the horn to be the beginning :)
Can't believe I've never seen the Oresme trick applied to the sum of 1/n^2. I guess it's much more commonly done via comparison to 1/n(n-1), which gives a telescoping series. (of course you need to treat n=1 differently so the overall bound is 2)
Sure, I actually also cover the telescoping approach in one of my videos on the Basel sum. However, here I was interested in showing the same in a very cute way off the beaten track that mirrors Oresme's super famous trick :) I (probably re)invented that :)
I think the calculus comes in handy when you want a rigorous proof for the area, which you hand wave slightly.
It's not entirely obvious that the area of the rectangles is smaller than the surface area of the enclosing horn sections. You need to know that the surface area of something is greater than the area of its shadow or something to that effect. Proving this probably requires some calculus.
Absolutely. I actually pondered for a while whether I should say some more in this respect but eventually decided that this is all obvious enough :)
I too felt that the rectangles approximated the area of the shadow rather than the actual area. And indeed a subtitle such as 'the actual area has to be larger than its shadow' would have helped.
However, I don't think any hidden calculus is involved in that step.
I was going to comment that this was the only part of your excellent video I didn’t feel convinced by.
@@Mathologer That reminds me of a story told on a Radio 4 puzzle show. Because it was radio I can't check the details, but it involved a university professor (probably Oxford) who in mid-lecture paused in front of the blackboard that he had filled with advanced mathematics and wrote down a relationship that he wanted to use, turned to his audience and said "That's obvious isn't it?" But then he frowned and proceeded to cover another blackboard in equations, after which he announced "I was right. It _was_ obvious!"
maths history revisited, great!
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.
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.
You can also see this reflected in the two infinite series :)
"The experts among yiu are probably yawning at this point" this line literally caught me mid-yawn
See I am getting really good at anticipating audience reactions :)
Great video! Not a fan of the 8 lie though; but, I am a fan of turning 8 on its side to equal infinity!
Where's that infinity hiding in the surface area calculation? The thickness of the film. An infinitely thin coat of paint is the same as no paint at all. Oh. I jumped the gun there. I see you covered that issue near the end. Good job.
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.