So today we're revisiting the painter's paradox;
briefly mentioned on the channel before,
but this time we're gonna- we're gonna do
the maths. We're gonna crunch through it
and we're gonna show this amazing
paradox that you can have
in this infinitely long horn which you
can fill with paint
but you can never paint the surface,
because the surface area
is infinite. And we're going to crunch
through the equations, the maths,
to see how it all works and exactly why
this is what happens.
[Music]
Okay, so to form this- this infinite horn -
called Gabriel's horn - what we
actually do is
create a shape which is called a volume
or a surface of revolution.
And we're going to draw the graph of y
equals 1 over x.
So as x goes really small it shoots off
to infinity;
and as x gets bigger and bigger and
bigger we've got 1 over that
so that's going to go and decay down to zero.
And then what we do first of all is
we're gonna just chop it here
at 1. So this is at x equals 1 so y
is also
1 here as well. So now we've got this,
still infinitely long, graph and then
we're gonna take it and we're going to rotate
it around the x-axis.
It's basically like a big vuvuzela from
the World Cup in 2010;
bit of an old-school reference but
it's one of those big long tubes
where you're blowing down. The idea is of
course this is infinite.
Now this is is Gabriel's horn-
(Brady: I'm assuming that comes from the archangel Gabriel?)
Yes so I think it does. I think
this is called Gabriel's horn because
the archangel Gabriel was often pictured
with a really long horn; but this is
infinite, it is important to remember
that this horn goes forever.
But of course it's going to be
infinitesimally small
as it- like, you're going to no longer be
able to see it it's going to be that
small as you carry on.
(Gabriel must have a very small mouth)
That's very true,
yeah, if he's blowing in the small end, exactly.
First of all, let's figure out the volume
of this shape. - (I would have thought it)
(would have an infinite volume because)
(it never gets to a point where there's)
(not a little bit of hollowness?)
That would be a perfectly reasonable
thing to assume; because it goes forever
there's always something there, so surely
its volume is also infinite.
(It can always take just a little bit more paint?)
This is why I wanted to go through this,
right? Because the paradox itself,
it's called a paradox because if you try
to think about it
using our intuition it kind of doesn't
make sense - that's literally why it's a
paradox. But then,
of course, we do have the mathematical
tools to give us the definitive answer
that almost like can't be argued with. So
you might not believe what you get but
you can't really argue with the maths.
And that's what we're going to hopefully
figure out. Now there's a formula for a
volume of revolution, and some of you may
know this,
you do learn it at school, but we're
going to kind of think about where it
comes from by actually
looking at the shape. So we will end up
with the same formula, but let's think
about like intuitively how do we figure
out the volume of this?
So we're going to need to do an integral,
just like when an integral underneath
a curve gives you an area
underneath that curve; so here we're
going to do an integral to get the
volume. The way we normally do
the area under a curve when you do an
integral is you imagine splitting it up
into loads of really thin strips and
then you add together all of those
strips
as their width goes to zero - that's
basically calculus. This is what Newton
and Leibniz did, they came up with this
whole concept. We do the same thing here
but we're going to slice through
and get little pieces of the horn, so 3d
pieces. It's going to kind of look like a
cylinder. It's not going to quite be a
perfect cylinder, because there's going
to be a little bit of a slant
on the edge. But if you make these strips
really thin
it's going to get closer and closer to a
cylinder.
[Music]
So what we do is we say this slice along
the x-axis is going to be width
dx, so this is where it's going to come
into our integral, this is a small
bit along the x-axis. - (And the thinner it is)
(the closer those two circles are to)
(being the same) - Exactly,
yes. And we are eventually, through the
integral process, we will
let this width go to zero. That's what- an
integral is taking
that limiting process rather than just a
sum where you add them together.
So this is the height of our cylinder. So
the volume of a cylinder is just equal to
the area of the face multiplied by its height -
by its depth. So we know the area of a
circle, it's going to be pi
times the radius squared. And then we're
going to multiply
by the height, dx. So the volume of this
cylinder
is pi radius squared times its height dx.
So now if we just add together all of
these cylinders
and then let dx, the width, go to zero
that will give us our integral. The final
volume of
Gabriel's horn is going to be the
integral- so we're starting from 1,
and we're going all the way down here to
infinity, tending to infinity.
So we're starting from 1 to infinity,
that's our limits of our integration,
we've got pi we've got the radius
squared
and then we integrate with respect to dx.
So this is the formula
that perhaps you may have seen before for
a volume of revolution, but it comes from
these little pieces of cylinders being
added together. Now the radius;
so if we look back at our picture, and
I've drawn this really badly and not
very symmetrically,
but if this was- if this is the central
point and it's the same
on either side the radius is exactly
that distance from the axis up to the
curve. And you can see that depending on
where you take your slice you get
different values of the radius. You're
gonna get bigger circles up here
and really really tiny circles down here.
So the actual radius,
the distance from the line to the graph,
well that's just given by
the value of y. Because you're starting
at y equals naught and you're going up
to this point
here which is y, some value. So the radius
r is actually equal to the value of y.
But y is 1 over x,
because it was our graph that we had; so
the radius
is 1 over x. So we can now plug this
into our formula
and we've got an integral which - we can
take the pi outside -
1 to infinity and then we've got 1
over x squared
dx. Now that is an integral we can do; if
you've done any
high school calculus we've got x to the
minus 2, so if I do that integral I get
minus
1 over x with my two limits
and then I plug in the two values. So
it's now pi multiplied by
minus 1 over infinity; well 1 over
infinity, that's zero.
So you get nothing from that. And then
it's minus
minus 1 over 1 so it's 1. So the
volume
of Gabriel's horn is pi.
Which in itself is nice, it's always nice
when pi
just appears out of nowhere. So if we
take the graph
of y equals 1 over x, rotate it around
the x-axis to form this
infinite horn, the volume is equal to pi.
Amazing in itself; but for our purposes
we didn't really care what this was as
long as it wasn't infinity.
Because this is a finite number so I can
go out now and buy
pi units of paint. I can fill
Gabriel's horn with pi units of paint
and it will be completely filled with paint
and there's- can't fit any more, can't
fit any less, that's just that's it.
Now the surface area; same kind of idea,
we we want to think about what happens
when we have a small piece
of the horn, and then now instead of
thinking about its volume
we want to think about the area around
the edge, so the surface area of the
shape sort of going around
and around and around and around. So
imagine like trying to wrap the whole
thing in toilet roll,
for example. If we were to slice through the horn
you get something that looks like this.
Not quite a cylinder, it's got like a
slant;
and then you have the big circle at this
end which would be coming from up
sort of this end of it and then the
smaller one down here - so you have
something like this.
And now for the surface area you ha-
the reason you kind of- this prop really
helps is
when you open this out it's like a part
of a circle,
annulus would be the technical term but
it's like a sector of a circle isn't it,
with the middle bit kind of chopped away;
like a pizza crust.
It looks like a pizza crust doesn't it?
The area of this,
again if I put it together, is going to
be the surface area
of our shape. This particular shape is
actually the net
of something called a conical frustum,
which is a rather ridiculous name but
that's what it's called.
(It looks kind of rude, but anyway-) - It does, frustum, yeah.
(A conical frustum-) - Pizza crusts and
conical frustrums.
So this is the the shape and this- we do
have a formula for the area
of this particular shape. So what we need
to know to figure this out
is you need to know two things: this
length; and you also need to know
this length. So let's call this one A and
this one B.
(Do you not need to know an angle or any)
(thing about the the nature of the curve?)
No, it's literally just- it's just
A times B.
Because what kind of happ- so if you
were to sort of
extend it and then think of this as a
sector of your circle with some angle
theta,
it just so happens that the thetas kind
of all come out.
You can go through it and you'll get
that it is just A times B
in this particular shape. - (I'm surprised it's so)
(simple but there you go, I believe you.)
I didn't believe it and I went through
this after about four pages of algebra,
it works. So I'll save you the the-
agony of redoing that. So, now let's go back
and think back to our curve. This bit B
along the top, this is actually going to
be this bit here isn't it? As they join
together
this bit like along here - there is a
slant here.
So this thing along here we're going to
say this is about a straight line;
again we're going to shrink this thing
down. We say this is about a straight
line and we're going to call this
ds. We use a different variable
because it's not horizontal this time,
the slant here is important which is why
this was curved instead of just being a
rectangle which it would be for a
cylinder. So the slant here is important.
Again, it's a little bit technical
exactly why the slant is important,
it's to do with Taylor expansions and
lots of algebra.
But the slant here matters for the
surface area. So we know that, we're going
to call that ds.
And then this bit, A, well this is that
circle there;
it's the circle around the bit of the
horn. And of course
that's the circumference of a circle, the
circumference of a circle
is equal to two pi times the radius
and again we showed here the radius is
just given by
y, which was 1 over x, because the radius
is the distance
from the axis to the shape itself. So
what we've got then
is the piece of surface area
here is going to just be given by-
so SA for surface area. We're going to
have to integrate,
with respect to ds - and we'll worry about
that in a moment -
and then we just multiply ds by 2 pi r,
which is A times B. So 2 pi- we could write
r but we know r is 1 over x so we can
just write that
as 1 over x. So this is going to give us
our surface area. Now this pesky ds; so
the way to deal with that,
we look at what's going on over here. ds
is- it's kind of a straight line,
and then if I were to turn this into a
triangle
like this then we've got ds
here, call that a right angle. So this
horizontal direction,
well that's along the x-axis. As I'm
moving horizontally this is the x-axis;
so this is dx,
a small change on the x-axis. And this is
vertical so this is dy.
So we can now actually write good old
Pythagoras' theorem, he's got our back;
ds squared is going to be equal to dx
squared plus dy squared.
Okay, so then what if I square root?
So ds is now the square root of dx
squared plus dy squared - we only take the
positive one because it's a distance,
it's got to be positive. I'm going to do
this pretty rough and ready, you can do
this properly with a limiting process.
So what you do is, you kind of factor out
the dx.
So you say, right, well if I take the dx
out here, so it's come out the square
root so it was a dx squared inside
so I'm taking out a dx, that leaves me
with 1 plus
dy by dx squared.
So this is actually a derivative, I've
kind of treated them like they were
fractions. You can do this properly with
limits,
again I'm doing it rough and ready
because we just want the result.
So what we now got then is we can- this
is now in terms of y
and x, which is what we wanted because we
know those. We know, for example we know
that y is 1 over
x, so dy by dx
is minus 1 over x squared,
differentiating that function.
The other good thing that we've got from
this is when we had ds we didn't know
what our limits were,
because what is s? It's just the distance
along the curve.
We didn't know where to integrate
between. But now we've actually turned ds into
something with respect to x. So again,
we're going to go from the limits on x
which are from 1
all the way to infinity. So finally we've
got the integral; we can take the two pi
outside
and then we've got the integral from 1
to infinity
of 1 over x multiplied by ds,
which is the square root of 1 plus dy
by dx squared
so that's- square that I get 1 over x
to the 4 dx.
So we get this pretty awful looking
integral.
Now, we're not going to do this integral.
We could do this integral; you- you can
use
trigonometric substitutions, you can even
throw in a hyperbolic function if you
really want. But we don't even need to do
that, because
we kind of know what we're aiming for
and we know that this, based on the
paradox, this is going to go to infinity.
So if we can show this is bigger than
infinity then it's infinity.
It's a really good tactic, when you have
an idea that something is going to be
infinity,
show that it's bigger than something
else that's really common that we know
is definitely infinity.
Because if you're bigger than infinity
you're infinity. I've got this function
here,
the square root of 1 plus 1 over x to
the fourth.
And I know that x is between 1 and
infinity,
so it has to take a value between 1
infinity. So whatever value of x I plug
in here
this whole thing is always bigger than
1. Because if I- if I plug in x is 1
I get the square root of 2, that's bigger
than 1. And then as x gets bigger and
bigger and bigger
this bit I'm adding above 1 gets
smaller,
but it never quite gets to 1. So this
whole thing is definitely bigger
than 1. So let's just replace it and
say, well,
so the surface area is bigger than
2 pi times the integral from 1 to
infinity
of 1 over x dx. So we've really just
replaced that whole horrible square root by
1. And this is still true, this is an
integral we can do again,
so this is just 2 pi integral of 1 over x
is the natural log of x between infinity
and 1.
The natural log of infinity
is infinity. So we didn't need to do the
horrible integral,
because we can show that the surface
area is greater than
something that goes to infinity, so the
surface area
is infinite.
Gabriel's horn, where we take 1 over x
rotate around the axis to create this
really infinitely long horn; we can fill it
with pi units of paint but
we can never paint the surface
because the surface area is infinite.
And that is the painter's paradox.
(And you're talking about just one face of)
(the surface too, you're just talking)
(about the outer face.) - Yes, this is just
the outer face yeah.
So if it's really really thin then the
inner surface area would be the same,
so it's almost like doubly infinite!
(So if the inner surface area is the same,)
(when I poured in that pi litres of paint)
(hadn't I not automatically painted all)
(the surface area?)
That's the paradox I think. I tried to
think about that and there was like-
I think that's the paradox. It's quite
neat, if you're really into your
your technical maths it's quite neat.
Because it all relies on the fact that
the volume
looks like 1 over x squared and the
surface area
looks like 1 over x. That
it's- that is the key. Because if if
you've if you've done
a reasonable amount of maths, perhaps
university level,
1 over x squared- so when you're doing an
integral you're adding together lots of
pieces.
And if you add 1 over n squared
forever you get pi squared over 6. It's
called Basel's problem, it's a very
famous result.
But if you add 1 over n forever it
diverges,
it goes to infinity. And so Gabriel's
horn and the painter's paradox
is basically encapsulating the whole
behaviour
of 1 over n squared converging as a
series
versus 1 over n diverging.
Well I'm sure you won't be surprised to
see you can take this fantastic
interactive quiz about none other than
Gabriel's horn
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[Extras] ...sort of come to terms with it, is to say, well- because one of the
arguments I've heard about this is to
say,
if I made it see-through then I could
fill it with paint and it would look
painted
from the outside. And again that's
paradox, it just
literally blows your mind.