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

0:07 Welcome to another Mathologer video.

0:08 Are you familiar with this strange infinitely long horn over there.

0:12 It first made the news 400 years ago and it

0:16 turned its discoverer Evangilista Torricelli into a mathematical superstar.

0:20 Why?

0:21 Well, Torricelli's horn has a very paradoxical property.

0:25 Suppose you've got 8 litres of paint.

0:27 Then you can fill the horn to the brim.

0:30 Let's do it:) In other words, the volume of the horn is 8 litres.

0:36 Now comes the weird bit.

0:38 The surface area of this shape is– infinite (ominous voice).

0:42 Finite volume and infinite area?

0:43 How is that possible?

0:45 Also, when we fill the horn with our 8 litres

0:48 of paint we are also completely painting the inside surface, right?

0:51 In other words, when it comes to this strange horn,

0:55 it is possible to paint an infinite surface area with a finite amount of paint.

1:00 Let that sink in for a moment.

1:03 Sink in, um, no pun intended.

1:05 Ok, the pun was intended:) We are painting

1:08 an infinite surface area with a finite amount of paint.

1:12 This weirdness is commonly called the painter's paradox.

1:16 Actually, I just discovered a second painter's paradox.

1:20 I call it the YouTube painter's paradox.

1:23 There are lots of YouTube videos dedicated to explaining

1:25 the painter's paradox and when it comes to actually

1:28 taming the volume and the surface area of Torricelli's

1:31 horn there are two approaches that YouTubers take.

1:34 The first approach consists in unleashing

1:36 the full force of calculus on the problem, derivatives, integrals, the works.

1:41 The second approach involves the vigorous waving of hands,

1:45 proclaiming repeatedly how amazing all this is

1:48 and really not explaining anything of essence at all,

1:51 possibly because our YouTuber thinks

1:54 that their audience cannot handle the calculus.

1:58 Where is the paradox in all that?

2:00 Well, it turns out that you don't need any calculus whatsoever.

2:03 To get the volume and area of the horn under control,

2:07 the only thing you need is a simple algebraic trick

2:10 that a mathematical monk came up with 700 years ago,

2:14 a trick that all Math YouTubers will almost certainly be familiar with.

2:18 Pretty strange right?

2:20 Why is everybody using calculus or is being scared

2:23 of calculus since there is a much simpler way?

2:27 Especially on YouTube where simple is king?

2:31 Why do you think that is?

2:33 Leave your answers to these puzzling questions in the comments.

2:36 Okay, so today's mission is to give you

2:38 a complete calculus-free explanation of the painter's paradox.

2:41 Even if you're a pro and have one

2:44 of those horns rattling around somewhere in a cupboard,

2:47 stick around for I've got some really nice

2:50 twists to this popular tale lined up for you.

2:54 Here we go.

2:57 How is this particular horn built.

3:02 Very simple, take 1/x.

3:05 and focus on the part of the curve to the right of x= 1.

3:11 Now simply spin the curve around the x-axis.

3:15 The resulting surface of revolution is our horn.

3:17 Actually, we get a second copy of the horn by also spinning

3:21 the part of our 1/x curve to the left of x=1 around the y-axis.

3:28 There.

3:29 Let's save this second horn for later.

3:32 Now, let's first figure out what the surface area of this horn is.

3:36 My last two videos also dealt with some magical properties of 1/x and today's

3:41 video actually completes this first Mathologer

3:44 trilogy in the history of the channel.

3:46 If you watched one of these two earlier videos

3:50 you may be able to guess what comes next.

3:53 What comes after x= 1?

3:55 Yep, x= 2 AND at x= 2 1/x takes on the value 1/2.

4:02 Extend to a rectangle like this.

4:04 This rectangle is 1 unit wide and so its area is base times height,

4:09 1 times 1/2 equals 1/2.

4:11 So, the area is equal to its height.

4:13 At 3 we've got 1/3.

4:15 Extend to another 1 unit wide rectangle.

4:18 Again the area is equal to the height 1/3.

4:22 And so on.

4:23 This means that the area of the grey

4:26 staircase is the sum of all these rectangle areas.

4:29 1/2 plus 1/3 plus 1/4 plus 1/5, and so on.

4:33 The experts among you will immediately recognise this infinite sum.

4:37 Except for a missing 1 right at the start,

4:40 it's the famous harmonic series which every math demon knows adds to infinity.

4:45 In the second part of the video I'll use that 700 year old trick twice

4:51 and the first time will be to show that this infinite sum is equal to infinity.

4:55 In any case, for now just trust me that the grey staircase has infinite area.

4:59 But now since the horn encloses the staircase as it does,

4:59 it is clear that the surface area of the horn must also be equal to infinity.

4:59 Easy, right?

4:59 In any case, for now just trust me that the grey staircase has infinite area.

5:02 But now since the horn encloses the staircase as it does,

5:08 it is clear that the surface area of the horn must also be equal to infinity.

5:13 Easy, right?

5:13 Okay, getting there.

5:15 Now what about the volume?

5:17 That's trickier, but for our paradox we

5:19 don't have to calculate the precise volume.

5:22 We just have to show the volume is less than infinity, right?

5:27 So, how can we see that the volume is finite?

5:29 Here's a nice trick.

5:31 Extend the staircase by one step to the left.

5:34 There.

5:34 Move the whole staircase one unit to the right.

5:39 There, alines again perfectly along 1/x,

5:42 but now the staircase contains the curve,

5:44 rather than the curve containing the staircase.

5:47 Now spin the first rectangle, well,

5:49 really that's a square, spin that square around the x-axis.

5:53 That creates a cylinder.

5:54 What's the volume of this cylinder?

5:56 Well the volume formula for a cylinder is circular base times height,

6:01 pi r squared times height.

6:03 You remember that from school, right?

6:05 Now what's the radius and what's the height?

6:08 Well, obviously the height is 1.

6:11 And so is the radius.

6:13 And so the volume of the cylinder is pi times 1 squared.

6:17 Just pi.

6:18 Spin the second rectangle into a cylinder.

6:20 We calculate its volume in exactly the same way.

6:24 The only difference is the radius and so we just have

6:28 to replace the 1 by 1/2 to get the new volume.

6:32 And so on.

6:33 And so the volume of this infinite funnel

6:37 of cylinders is just the sum of the cylinders.

6:41 Now, the horn is completely contained in the funnel of cylinders and so

6:45 the volume of the horn must be less than that of the funnel.

6:49 This means that if we can show that the infinite sum in the brackets is finite,

6:53 then it follows that the volume of the horn is finite, too.

6:57 Is that clear?

6:58 If the sum is finite, then the volume of the funnel is finite,

7:07 and so is the horn contained within.

7:13 Slick, hmm?

7:16 Okay, so showing that the surface area is

7:19 infinite and the volume is finite boils down

7:22 to showing that the sums of these two

7:26 pretty infinite series are infinite and finite respectively.

7:30 And the only thing we need to prove both facts is

7:34 that 700 year old trick that I keep going on about.

7:37 Well, actually we also need this second fact here.

7:40 1+ 1/2+1/4+ 1/8 and so on, the sum of the reciprocals

7:44 of the powers of 2 is equal to 2.

7:48 You've seen that a thousand times before, right?

7:51 Why is that true again?

7:55 Well, here is a proof by animation:) All clear?

8:04 Good:) Okay, remember this for later 1+1/2+1/4 and so on is equal to 2.

8:10 Now we want to convince ourselves that the harmonic

8:12 series at the top adds to infinity.

8:15 The first recorded proof of this fact is, did I mention this before(:)?,

8:21 700 years old and is due to the mathematical monk Nicole Oresme.

8:26 Bear with me if all this sounds very familiar.

8:28 There will be a twist at the end.

8:30 In fact, Oresme also came up with a 2d

8:33 version of our paradoxical horn 300 years before Torricelli.

8:37 Just take the visual geometric infinite sum from just now–

8:41 and turn it into an infinitely tall tower, like this.

8:45 Then obviously the interior of this tower is of finite area 2.

8:50 On the other hand, its bounding curves are clearly of infinite length.

8:56 Right, that's a proper 2d counterpart of our horn

8:59 and is paradoxical for the same reasons as the horn is,

9:03 but is much easier to comprehend.

9:05 Unfortunately for Oresme,

9:07 nobody remembered his example by the time the horn first made headlines

9:12 and so Torricelli didn't have to share the fame for his discovery.

9:16 Oh, I should also mention that Oresme also came up with the first ever graphs.

9:21 So, all you who hate graphing, this is the guy to blame.

9:25 Anyway, back to our infinite series and Oresme's super famous trick.

9:29 Let's do this on algebra autopilot accompanied by some funky music.

9:51 All clear so far.

9:58 The top sum is greater or equal to the bottom sum.

10:05 Now we add up the terms in the coloured

10:08 boxes and that's where the magic happens.

10:14 There, infinitely many 1/2s.

10:19 And obviously, the sum of these infinitely many 1/2s is infinity.

10:31 Okay, so far so good.

10:35 Well, the experts among you are probably yawning at this point.

10:41 But here is something you've probably not seen before.

10:44 Turns out you can take care of the second

10:47 series in exactly the same way, very nifty.

10:51 Here is the second series.

10:53 Just now we started by highlighting the reciprocals of the powers of 2.

10:57 This time we'll highlight the reciprocals of the squares of the powers of 2.

11:03 Make a copy.

11:04 Earlier we filled in the gaps going left like this.

11:09 Now, we'll go the other way.

11:12 Earlier the terms at the top were always the same

11:17 or greater than the corresponding ones at the bottom.

11:21 Now it is the other way around.

11:23 Just a spot check to make sure that this is really the case.

11:30 There 1/9th is smaller than 1/4.

11:33 1/25 is smaller than 1/16, and so on.

11:37 And that means that the top sum is smaller than the bottom sum.

11:42 Now let's add up the terms in the coloured boxes.

11:51 Aha, so what we've got here is our 1+1/2+1/4th sum from before which,

12:06 as we all remember, is equal to 2.

12:15 But then if our sum is less than 2 that means it is finite.

12:29 Well, okay, so we know that the volume of Torricelli's horn is finite which is

12:33 really all we need to be sure that our horn is as paradoxical as advertised.

12:38 But I also claimed that the volume is exactly 8

12:42 and that I can also show this without using Leibniz and Newton's calculus.

12:46 I still owe you that proof.

12:48 Actually, this is not my proof at all.

12:51 This proof is due to Torricelli the discoverer

12:53 of the horn and this proof is really beautiful.

12:57 Remember the vertical copy of the horn that we saved for later?

13:01 We'll use it now.

13:03 Actually we'll first ponder the horn that has been

13:06 extended by a cylinder at the bottom like so.

13:09 So, a horn with a mute:) Considered

13:12 as a solid shape we can think of this extended

13:15 horn as being made up of thinner cylinders with the y-axis as the common axis.

13:20 There, that's one of these cylinders.

13:22 That's another one.

13:24 And another one.

13:25 Another Another.

13:26 A lot of them.

13:28 Alright, let's figure out the surface area

13:31 of this thin cylinder without the circular caps, so just the area of the mantel.

13:38 What's that area?

13:39 Well, that's just the circumference of the base circle times height.

13:43 And what's that circumference?

13:45 Well if the radius is r– then the circumference is 2 pi r.

13:50 And what's the height?

13:51 Well that's the value of 1/x at r.

13:55 And so the height is 1/r.

13:59 All the r s cancel and so completely vanish.

14:04 Cool, the area is 2 pi.

14:06 But of course the same is the case for all of the other thin cylinders.

14:11 Same calculation, same result.

14:12 That's SUPER cool.

14:14 All the areas of all the thin cylinders have the same area 2 pi.

14:19 Okay, now to figure out the volume, Torricelli argues like this.

14:24 Put a disk of area 2pi here.

14:27 So the disk has the same area as the thin cylinder.

14:31 If you do the same for every one

14:34 of the thin cylinders you get this stack of disks.

14:38 So there is one circular disk per thin cylinder, both having the same area 2 pi.

14:45 The thin cylinders combine into our extended horn–

14:48 and at the same time the disks combine into our stack.

14:53 Therefore Torricelli says the volumes of both solids must be the same.

14:58 And so what's that volume?

14:59 Well, the stack is just a cylinder with base area 2pi– and height 1.

15:05 And so the volume is base area times height, 2pi times 1 equals 2pi.

15:14 Super pretty way of reasoning don't you think?

15:18 Predates Newton and Leibnitz's calculus but is only made rigorous

15:21 and extended to the famous method of shells as part of calculus.

15:27 Now what about the volume of the horn?

15:30 That's what we are really interested in.

15:32 Well, the volume of our horn that's just the volume

15:34 of the extended horn minus the volume of the cylinder at the bottom.

15:39 As you can easily check, the volume of the orange cylinder is equal to pi and so

15:44 the volume of the horn is 2pi minus pi which is pi.

15:48 But didn't I also say that the volume of the horn is 8?

15:54 Well, yes, I lied, the volume is pi:) Well,

15:58 I've long been dead set on eventually having that fun

16:01 animation of an 8 turning into an infinity sign

16:03 in one of my videos and this was the perfect

16:06 opportunity to sneak this animation in:) Gotto do this, right?

16:12 Having said that, if we stretch our horn vertically by a factor of 8/pi

16:19 we actually do get a horn with volume 8– and infinite surface area.

16:24 And so I hope you can forgive me my little lie:) Okay,

16:38 so we proved that our horn is really

16:41 as paradoxical as we claimed at the beginning.

16:43 And, of course, now that you know that this is the case,

16:46 you want one of those horns.

16:47 You jump on e-bay and– nothing to be found.

16:50 Sad:) Well, just in case you have not guessed yet.

16:53 That infinite horn is something that only exists in an ideal mathematical world.

16:59 It has 0 thickness and it gets slimmer

17:02 and slimmer as we travel along it to infinity.

17:05 In fact, eventually it will be slimmer than even an atom and so not

17:09 even a virtual atom-based counterpart of real

17:12 paint can completely fill this imaginary horn.

17:14 And when I said at the beginning that we are

17:18 painting an infinite surface area with a finite amount of paint,

17:21 then it is also important to realise that for this to work,

17:25 we not only need an ideal mathematical horn but also ideal mathematical paint,

17:30 paint that can be applied as THIN as we wish and that still

17:34 covers the surface we are painting no matter how thin it is applied.

17:38 You can also not buy this sort of paint in the paint shop down the street.

17:43 Well, and that's all there is to the painter's paradox.

17:47 In general, this kind of paradox is quite common with infinity.

17:51 Infinity is not something that exists in the real world,

17:54 it's simply a mathematical idea.

17:55 And, once you realise that, it should not come as too much

17:59 of a surprise that otherworldly creatures

18:02 behave different from real-world horns and paint.

18:06 Let me know in the comments how this calculus-free

18:08 exposition of the painter's paradox worked for you

18:11 and how it compares to some other expositions

18:13 on YouTube that you may be familiar with.

18:16 Let me finish with a fun fact that I stumbled across while

18:19 reading up on the history of Torricelli's horn and the painter's paradox.

18:23 Have a look at this book featuring some

18:27 of Torricelli's writings that was published after his death.

18:30 There that's Torricelli on the left.

18:32 Now let's have a close look at what it says underneath.

18:36 Well there is the Latin version of Torricellis name at the bottom:

18:41 Evangilista Torricellius.

18:42 And at the top is says En virescit Galiaeus

18:46 alter which is Latin for Here blossoms another Galileo.

18:50 Basically, people were really impressed by Torricelli

18:52 and thought of him as a second Galileo.

18:55 Actually Torricelli was a student of Galileos and apart

18:59 from being famous all over Europe as a mathematician,

19:02 he also made a name for himself as a physicist.

19:05 Among other things, he is also famous for inventing the barometer.

19:08 Anyway here comes that fun bit.

19:11 Have a look at the word in the middle.

19:14 Anagr.

19:15 What could that possibly mean?

19:17 Well, anagram of course:) The sentence at the top

19:20 is an anagram of the name at the bottom.

19:23 In other words, you can rearrange the letters

19:26 at the top into the letters at the bottom.

19:29 Cool.

19:29 I thought it would be fun to animate

19:34 this rearrangement and here is what I came up with.

19:42 That's so unexpected, don't you think?

19:45 What's also unexpected is that we are

19:48 actually not dealing with a perfect anagram.

19:50 That Galileus here is not quite right.

19:52 Turns out that the aeh in Galilaeus at the top

19:55 and the o in Torricellius at the bottom don't have counterparts:) Weird Hmm?

20:02 Anyway, I agree with whoever invented this almost anagram:

20:09 close enough is fun enough.

20:16 And that's it for today.

20:23 Until next time.