Welcome to another Mathologer video. Are you
familiar with this strange infinitely long horn over there. It first made the news 400 years
ago and it turned its discoverer Evangilista Torricelli into a mathematical superstar. Why?
Well, Torricelli's horn has a very paradoxical property. Suppose you've got 8 litres of paint.
Then you can fill the horn to the brim. Let's do it :) In other words, the volume of the horn is 8
litres. Now comes the weird bit. The surface area of this shape is – infinite (ominous voice).
Finite volume and infinite area? How is that possible? Also, when we fill the horn with our 8
litres of paint we are also completely painting the inside surface, right? In other words, when
it comes to this strange horn, it is possible to paint an infinite surface area with a finite
amount of paint. Let that sink in for a moment. Sink in, um, no pun intended. Ok, the pun was
intended :) We are painting an infinite surface area with a finite amount of paint. This weirdness
is commonly called the painter's paradox. Actually, I just discovered a second painter's
paradox. I call it the YouTube painter's paradox. There are lots of YouTube videos dedicated to
explaining the painter's paradox and when it comes to actually taming the volume and the surface area
of Torricelli's horn there are two approaches that YouTubers take. The first approach consists in
unleashing the full force of calculus on the problem, derivatives, integrals, the works. The
second approach involves the vigorous waving of hands, proclaiming repeatedly how amazing all this
is and really not explaining anything of essence at all, possibly because our YouTuber thinks that
their audience cannot handle the calculus. Where is the paradox in all that? Well, it turns out
that you don't need any calculus whatsoever. To get the volume and area of the horn under control,
the only thing you need is a simple algebraic trick that a mathematical monk came up with 700
years ago, a trick that all Math YouTubers will almost certainly be familiar with. Pretty strange
right? Why is everybody using calculus or is being scared of calculus since there is a much simpler
way? Especially on YouTube where simple is king? Why do you think that is? Leave your answers to
these puzzling questions in the comments. Okay, so today's mission is to give you a complete
calculus-free explanation of the painter's paradox. Even if you're a pro and have one
of those horns rattling around somewhere in a cupboard, stick around for I've got some
really nice twists to this popular tale lined up for you. Here we go. How is this particular
horn built. Very simple, take 1/x. and focus on the part of the curve to the right of x = 1.
Now simply spin the curve around the x-axis. The resulting surface of revolution is our horn.
Actually, we get a second copy of the horn by also spinning the part of our 1/x curve to the left
of x=1 around the y-axis. There. Let's save this second horn for later. Now, let's first figure
out what the surface area of this horn is. My last two videos also dealt with some magical
properties of 1/x and today's video actually completes this first Mathologer trilogy in the
history of the channel. If you watched one of these two earlier videos you may be able to guess
what comes next. What comes after x = 1? Yep, x = 2 AND at x = 2 1/x takes on the value 1/2.
Extend to a rectangle like this. This rectangle is 1 unit wide and so its area is base times height,
1 times 1/2 equals 1/2. So, the area is equal to its height. At 3 we've got 1/3. Extend to another
1 unit wide rectangle. Again the area is equal to the height 1/3. And so on. This means that the
area of the grey staircase is the sum of all these rectangle areas. 1/2 plus 1/3 plus 1/4 plus 1/5,
and so on. The experts among you will immediately recognise this infinite sum. Except for a
missing 1 right at the start, it's the famous harmonic series which every math demon knows adds
to infinity. In the second part of the video I'll use that 700 year old trick twice and the first
time will be to show that this infinite sum is equal to infinity. In any case, for now just trust
me that the grey staircase has infinite area. 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. Easy, right?
In any case, for now just trust me that the grey staircase has infinite area. 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. Easy, right? Okay, getting there. Now what about the volume? That's
trickier, but for our paradox we don't have to calculate the precise volume. We just have to
show the volume is less than infinity, right? So, how can we see that the volume is finite? Here's
a nice trick. Extend the staircase by one step to the left. There. Move the whole staircase one unit
to the right. There, alines again perfectly along 1/x, but now the staircase contains the curve,
rather than the curve containing the staircase. Now spin the first rectangle, well, really that's
a square, spin that square around the x-axis. That creates a cylinder. What's the volume of this
cylinder? Well the volume formula for a cylinder is circular base times height, pi r squared
times height. You remember that from school, right? Now what's the radius and what's the
height? Well, obviously the height is 1. And so is the radius. And so the volume of the cylinder
is pi times 1 squared. Just pi. Spin the second rectangle into a cylinder. We calculate its volume
in exactly the same way. The only difference is the radius and so we just have to replace the 1
by 1/2 to get the new volume. And so on. And so the volume of this infinite funnel of cylinders
is just the sum of the cylinders. Now, the horn is completely contained in the funnel of cylinders
and so the volume of the horn must be less than that of the funnel. This means that if we can show
that the infinite sum in the brackets is finite, then it follows that the volume of the horn is
finite, too. Is that clear? If the sum is finite, then the volume of the funnel is finite, and so
is the horn contained within. Slick, hmm? :) Okay, so showing that the surface area is infinite
and the volume is finite boils down to showing that the sums of these two pretty infinite series
are infinite and finite respectively. And the only thing we need to prove both facts is that 700
year old trick that I keep going on about. Well, actually we also need this second fact here.
1 + 1/2+1/4+ 1/8 and so on, the sum of the reciprocals of the powers of 2 is equal to
2. You've seen that a thousand times before, right? Why is that true again? Well, here is a
proof by animation :) All clear? Good :) Okay, remember this for later 1+1/2+1/4 and so on is
equal to 2. Now we want to convince ourselves that the harmonic series at the top adds to infinity.
The first recorded proof of this fact is, did I mention this before ( :)?, 700 years old
and is due to the mathematical monk Nicole Oresme. Bear with me if all this sounds very familiar.
There will be a twist at the end. In fact, Oresme also came up with a 2d version of our
paradoxical horn 300 years before Torricelli. Just take the visual geometric infinite sum from just
now– and turn it into an infinitely tall tower, like this. Then obviously the interior of this
tower is of finite area 2. On the other hand, its bounding curves are clearly of infinite
length. Right, that's a proper 2d counterpart of our horn and is paradoxical for the same
reasons as the horn is, but is much easier to comprehend. Unfortunately for Oresme, nobody
remembered his example by the time the horn first made headlines and so Torricelli didn't have to
share the fame for his discovery. Oh, I should also mention that Oresme also came up with the
first ever graphs. So, all you who hate graphing, this is the guy to blame. Anyway, back to our
infinite series and Oresme's super famous trick. Let's do this on algebra autopilot accompanied
by some funky music. All clear so far. The top sum is greater or equal to the bottom sum. Now we
add up the terms in the coloured boxes and that's where the magic happens. There, infinitely many
1/2s. And obviously, the sum of these infinitely many 1/2s is infinity. Okay, so far so good. Well,
the experts among you are probably yawning at this point. But here is something you've probably not
seen before. Turns out you can take care of the second series in exactly the same way, very nifty.
Here is the second series. Just now we started by highlighting the reciprocals of the powers of 2.
This time we'll highlight the reciprocals of the squares of the powers of 2. Make a copy. Earlier
we filled in the gaps going left like this. Now, we'll go the other way. Earlier the terms at
the top were always the same or greater than the corresponding ones at the bottom. Now it is
the other way around. Just a spot check to make sure that this is really the case. There 1/9th
is smaller than 1/4. 1/25 is smaller than 1/16, and so on. And that means that the top sum is
smaller than the bottom sum. Now let's add up the terms in the coloured boxes. Aha, so what we've
got here is our 1+1/2+1/4th sum from before which, as we all remember, is equal to 2. But then if
our sum is less than 2 that means it is finite. Well, okay, so we know that the volume
of Torricelli's horn is finite which is really all we need to be sure that our horn is as
paradoxical as advertised. But I also claimed that the volume is exactly 8 and that I can also show
this without using Leibniz and Newton's calculus. I still owe you that proof. Actually, this is not
my proof at all. This proof is due to Torricelli the discoverer of the horn and this proof is
really beautiful. Remember the vertical copy of the horn that we saved for later? We'll use it
now. Actually we'll first ponder the horn that has been extended by a cylinder at the bottom
like so. So, a horn with a mute :) Considered as a solid shape we can think of this extended
horn as being made up of thinner cylinders with the y-axis as the common axis. There, that's
one of these cylinders. That's another one. And another one. Another Another. A lot of them.
Alright, let's figure out the surface area of this thin cylinder without the circular caps, so just
the area of the mantel. What's that area? Well, that's just the circumference of the base circle
times height. And what's that circumference? Well if the radius is r– then the circumference is
2 pi r. And what's the height? Well that's the value of 1/x at r. And so the height is 1/r. All
the r s cancel and so completely vanish. Cool, the area is 2 pi. But of course the same is the
case for all of the other thin cylinders. Same calculation, same result. That's SUPER cool. All
the areas of all the thin cylinders have the same area 2 pi. Okay, now to figure out the volume,
Torricelli argues like this. Put a disk of area 2pi here. So the disk has the same area as the
thin cylinder. If you do the same for every one of the thin cylinders you get this stack of disks.
So there is one circular disk per thin cylinder, both having the same area 2 pi. The thin cylinders
combine into our extended horn– and at the same time the disks combine into our stack. Therefore
Torricelli says the volumes of both solids must be the same. And so what's that volume? Well, the
stack is just a cylinder with base area 2pi – and height 1. And so the volume is base area times
height, 2pi times 1 equals 2pi. Super pretty way of reasoning don't you think? Predates Newton and
Leibnitz's calculus but is only made rigorous and extended to the famous method of shells as part of
calculus. Now what about the volume of the horn? That's what we are really interested in. Well,
the volume of our horn that's just the volume of the extended horn minus the volume of the
cylinder at the bottom. As you can easily check, the volume of the orange cylinder is equal
to pi and so the volume of the horn is 2pi minus pi which is pi. But didn't I also say
that the volume of the horn is 8? Well, yes, I lied, the volume is pi :) Well, I've long
been dead set on eventually having that fun animation of an 8 turning into an infinity
sign in one of my videos and this was the perfect opportunity to sneak this animation
in :) Gotto do this, right? Having said that, if we stretch our horn vertically by a factor
of 8/pi we actually do get a horn with volume 8 – and infinite surface area. And so I
hope you can forgive me my little lie :) Okay, so we proved that our horn is really as
paradoxical as we claimed at the beginning. And, of course, now that you know that this is the
case, you want one of those horns. You jump on e-bay and – nothing to be found. Sad :) Well,
just in case you have not guessed yet. That infinite horn is something that only exists in
an ideal mathematical world. It has 0 thickness and it gets slimmer and slimmer as we travel
along it to infinity. In fact, eventually it will be slimmer than even an atom and so not
even a virtual atom-based counterpart of real paint can completely fill this imaginary horn. And
when I said at the beginning that we are painting an infinite surface area with a finite amount of
paint, then it is also important to realise that for this to work, we not only need an ideal
mathematical horn but also ideal mathematical paint, paint that can be applied as THIN as we
wish and that still covers the surface we are painting no matter how thin it is applied. You can
also not buy this sort of paint in the paint shop down the street. Well, and that's all there is to
the painter's paradox. In general, this kind of paradox is quite common with infinity. Infinity
is not something that exists in the real world, it's simply a mathematical idea. And, once you
realise that, it should not come as too much of a surprise that otherworldly creatures behave
different from real-world horns and paint. Let me know in the comments how this calculus-free
exposition of the painter's paradox worked for you and how it compares to some other expositions
on YouTube that you may be familiar with. Let me finish with a fun fact that I stumbled across
while reading up on the history of Torricelli's horn and the painter's paradox. Have a look at
this book featuring some of Torricelli's writings that was published after his death. There that's
Torricelli on the left. Now let's have a close look at what it says underneath. Well there is the
Latin version of Torricellis name at the bottom: Evangilista Torricellius. And at the top is says
En virescit Galiaeus alter which is Latin for Here blossoms another Galileo. Basically, people were
really impressed by Torricelli and thought of him as a second Galileo. Actually Torricelli was a
student of Galileos and apart from being famous all over Europe as a mathematician, he also made
a name for himself as a physicist. Among other things, he is also famous for inventing
the barometer. Anyway here comes that fun bit. Have a look at the word in the middle.
Anagr. What could that possibly mean? Well, anagram of course :) The sentence at the top is an
anagram of the name at the bottom. In other words, you can rearrange the letters at the top into the
letters at the bottom. Cool. I thought it would be fun to animate this rearrangement and here
is what I came up with. That's so unexpected, don't you think? What's also unexpected is that we
are actually not dealing with a perfect anagram. That Galileus here is not quite right. Turns out
that the aeh in Galilaeus at the top and the o in Torricellius at the bottom don't have counterparts
:) Weird Hmm? Anyway, I agree with whoever invented this almost anagram: close enough is fun
enough. And that's it for today. Until next time.
