So we're gonna talk about domes. And
we're gonna talk about one of the
largest, and possibly one of the most
famous domes in the world,
here in the UK. So in London
St Paul's Cathedral;
it's a very large cathedral and it's
right in the center of London and it has this
really imposing dome - one of the biggest
in the world, diameter is
34 metres. The question becomes, how do
you build
the correct shape of dome so that it
will sit
under its own weight without supports,
without masts - how do you actually
create that? And that's a maths problem,
so that's what we're going to figure out.
The great fire of London,
1666, had just happened and St Paul's was
really badly
damaged and they wanted to rebuild this
grand cathedral. Christopher Wren, very
famous architect of the time, he was given the
task of rebuilding this cathedral and he
wanted this
big dome to kind of be the centerpiece
of the cathedral. And at the time he was
really good friends
with a famous physicist, Robert Hooke,
and they kind of worked together
to try and think about how can we have
the largest dome possible
that doesn't need any support? And
because Robert Hooke was a physicist and a
mathematician he thought about this
mathematically. And his amazing
insight was to actually use-
basically use nature to help us figure
this out. So what he did
was get- I think he used a chain but
it's the same with a piece of string;
and you just hold your two ends like so
so they're level, so my hands are about level.
And then you just let it form its shape.
So this forms a sort of U-shape,
kind of looks like a parabola or an
x-squared graph
perhaps, and this is actually sort of
well it's nature being lazy, is the
easiest way to describe it. Because
what's happening here is it's forming
the shape that
minimises the energy of the of the string.
So the potential energy from gravity
pulling it down is minimised in this
shape.
So- and if I move my hands it will
obviously adjust its shape,
still forming this kind of U-shape. And
Hooke's idea
was to say, well this is forming to
minimise the energy,
so if we flip it up- because when it was
hanging
it was minimising its energy, and the way
it was doing that is by spreading out
the weight
as much as possible. So suddenly when you
flip it upside down
you've got your dome where your weight
is spread out as
much as possible, and so that is going to
help you to build the biggest possible
dome you can
basically using nature as like a guide
or a blueprint of how to do this. And
this is basically- this is what they did.
So so if you actually look at the
original plans for St Paul's, or even
the drawings of the of the building
itself, it actually
uses this idea in this shape. So the
outer dome is a very big
hemisphere, this is 34 metres in diameter
right across the bottom here. And then
this kind of then comes down a bit,
down here is the inner dome which is
also a hemisphere,
like this. And this one's a little bit
smaller so it doesn't quite go to the
edge, the inside one is 31 metres. So when
you're inside
St Paul's you can look up and you can
see all the beautiful artwork on the
ceiling - that's the inner dome,
not the outer one that you can see from
outside, they're two separate domes. And
the inner one is a little bit smaller,
three metres less across. The sphere and
sort of the perfection of it and the
symmetry has symbolic meaning, so that
was why these had to be
spherical - this is kind of what Wren was
told. But, of course he wanted to
to make it strong, make it- you know,
use maths to help us build this thing. So
what they did
was on the inside bit, in here,
was actually create a third dome and
this is the shape
of the hanging chain, or the shape of my
piece of string. So this thing in here
is actually what's bearing all of the
load. So you have these kind of
spherical shells on the outside, and all
of the weight is actually supported
by the hanging string or the hanging
chain shape in the middle.
(Brady: That's doing the business.) - Yeah that's
the one that's actually
supporting the dome. - (Brady: It's like the star trek symbol)
Yeah I'm not sure it's quite this pointy
but it's definitely not- it's definitely
not a hemisphere. As we saw when I was
hanging it wasn't a sphere it was very
much more like a U-shape.
So this is supporting all of the weight.
Now, they figured this out; so this was the
idea, like literally Hooke did these
experiments was like, this is the shape
we need to build.
Now the only problem they had was this
is was in the 1660s, 60s-70s;
and to actually know this shape - this
hanging chain shape -
you need an area of math called calculus
of variations,
and that was not discovered or invented
until the 1750s - 100 years later.
With these new techniques, this calculus
of variations,
as it was called; which was actually
worked on mainly by Euler and Lagrange,
so two very big name mathematicians of
1700s. So with calculus of variations
you can show the exact shape that this
needs to be,
and it's something called a catenary.
This is the exact shape of our hanging string,
but the dome itself in St Paul's is
like a very good approximation of this.
But that's nothing against
Hooke and Wren, they just didn't yet have
the mathematical tools they needed
to get the exact shape. So they were
basing it almost on like an
approximation and experiment-
(Brady: So they literally kind of like traced)
(something like-) - More or less yeah.
And it's really interesting, you can
actually look at the original drawings
and you can see that they are literally
just like kind of sketching what the
shape should be
based on the shape of a hanging chain. So
even though they didn't have the right
mathematical tools
they had the insight to actually put
them to work; and it obviously works! The
dome is still there today. So now that we
do have the tools,
the shape itself is something called a
cosh curve, or a hyperbolic cosine. So, with the
full tools
of calculus of variations you can show-
you can set this problem up,
hanging chain, I want to minimise energy.
And your answer
is this cosh curve. So then, what we want
to know is, well
what on earth is a cosh curve? And that's
when I'm going to need more paper.
Cosh of x. So this is the actual shape we want
for the- for the catenary, for the perfect
dome shape. The cosh function
is an exponential function; so
the starting point actually here to
figure out what's going on with cosh
is to consider the function e to the x.
So e to the x is a function which has
an expansion, which is 1, plus x, plus
x squared divided by 2 factorial, plus
x cubed over 3 factorial, plus dot dot dot.
And the graph of this function goes
through 1
like so at zero, and then it decreases.
Never quite gets to zero,
and then this bit increases. So this
would be y equals e to the x.
So as you can see, as x gets bigger
you're just adding together lots and
lots of powers of x
so the thing's going to grow really
quickly - this is exponential growth. Now
the cosh term
is very similar to the exponential, but
what you say is I just want the
even terms in this sum.
So I want x to the 0, which is 1.
I want x squared, I want x to the 4, I
want x to the 6.
I'm just going to take the even powers
of x and let's see what happens;
and that's basically how you define cosh.
It's the even terms
of your exponential. So you would write then:
cosh of x is
1 plus x squared - because of course 2 is
even - plus x to the 4, because 4 is even;
and we carry on in that way. You can also
think about
reflecting its graph; so when you have an
even function,
what that means is it's the same either
side of the y-axis.
So if we kind of take this bit of the
graph, reflect it, this is basically the
cosh curve;
which is the nice dome shape and the nice
hanging string shape that we had, it's really
y equals cosh of x.
Now cosh comes from the even terms, maybe
you're thinking; well what about the odd
terms? And obviously we have a function
for that,
uh so if we do the odd terms then that
is going to be
sinh of x! Just like this is hyperbolic cos,
hyperbolic sine; and this one is going to
then be x,
because that's to the power 1, x cubed
plus x to the five;
so it's just all your odd powers. And
sinh of x, it's the same up here
but this one goes through zero, so it's a
bit more like an x
cubed graph; it's got that kind of kink
almost like s shape to it. And this one
can be negative, because you've got-
if you sub in negative values of x
you're gonna get minuses from the odd
powers whereas negative values of x up
here give you
positives. So cosh stays positive but sinh
can go negative. So we have these two
functions
and then there's actually a shorthand
way that you can write
cosh as well, using the exponential
formula, is you consider
e to the minus x. So you just take your e
to the x formula,
but wherever there's an x you plug in a
minus x. So plus
x that becomes a minus, x squared so
minus one squared is just one
so I still get plus x squared over 2
factorial.
Then I'll get minus x cubed over 3
factorial
and I'll carry on. So the terms- the sign
alternates when you plug in minus x.
So now, if you take e to the minus x and
e to the x,
if we add them together so we do e to
the x
plus e to the minus x; I'm gonna add them
together and I'm gonna divide by 2.
Because I'm gonna get 1 plus 1,
then when I divide by 2 leaves me with 1.
I've got plus x
and minus x, they disappear. I've then got
two lots of the x squared term, divided by
2 so I just get plus
x squared two factorial. The cubic one
here's got a negative,
here it's got a positive they disappear.
So it'll be
no cubic term, x to the fourth, and then
we carry on.
And of course this is cosh of x.
(Brady: The evens.) - The even terms. Exactly as we
defined it here,
but now you can write it in this really
nice shorthand formula;
it's just e to the x plus e to the minus
x divided by 2.
(Brady: What's the variable here? If I want to go)
(home and build a dome)
(on my house what's the- you know, which is)
(not as wide as)
(St Paul's Cathedral - what's the thing I)
(need to change, like to get my graph?)
So if you are to draw the the graph of cosh-
so depending on how wide you want your
dome this is going to tell you the
shape. So if your dome's really narrow,
let's call it five metres, then you would
go up here
and you're going to get a very kind of
wide flat dome in that instance.
Whereas if you're going to like St
Paul's, which let's say maybe that's all
the way out to here,
and that distance is your 34 metres; then
you're getting this kind of bigger shape.
So all of them follow the same graph.
So the other thing that will change is
the length of the string,
and this is actually- this comes up in
the full
calculus of variations way of solving the
problem, you do have to input 'my string is of
this fixed length'.
So so the solution does depend on the
length of your string.
So if I have my fixed length of string
here then I form
this particular shape, but then if I say,
right well I'm only going to have this
bit of string, and they're still that far
apart, I've got a much shallower dome now.
But no matter what
length of string you have and how far
apart my hands are
all of these shapes that are being
formed are
cosh curves, they are all hyperbolic
cosine curves.
You've just got a variable in there
which is representing the fact that the
length or the distance is changing.
(Brady: So no matter what droop I'm seeing there)
(in the string)
(it is it is some variation of that same curve I'm seeing?)
It's still a hyperbolic cos curve, yes. In all
of these situations the answer is
the cosh curve. - (Brady: I'm just sort of seeing)
(it at different zooms?) - Exactly.
Yeah yeah so either at different zooms
or or you can-
because if I put them really close
together now it's a very different shape,
but that is still a cosh curve.
You could still write down the the
actual formula. So so the the
the full formula will be cosh
of something, where that something
depends on the length of your string,
depends on how far apart you are; so
there are variables that you have to
input
to get the exact answer, but everything
that comes out
is this hyperbolic cosine function.
Wren and Hooke
didn't- like cosh didn't exist until the
1760s in fact.
So it wasn't even a function. So they had
no chance of knowing that this was the
shape they needed to build this dome.
So they just had to take like best
guesses and
they they kind of did this through
almost knowing what it wasn't.
So through their experiments with the
hanging chains and the string
they knew it wasn't an x squared. So they
were definitely sure it wasn't the
parabola.
So then they kind of tried, well what's
the next simplest thing I could have?
Maybe it's a cubic? Maybe it's a y equals
x cubed. So if we were to plot
y equals x cubed; and we're just going to
add 1 to it so that it goes through
1 instead of zero,
just because cosh also goes through 1.
If we were to draw y equals x cubed plus 1
it would look like that here, and then it
would kind of go down like this.
What Wren and Hooke did was they reflected
it, so then we kind of extend it this way.
So it is believed that the
actual shape of this inner dome
at St Paul's is an x cubed curve,
that is what we what we believe at the
moment they actually based this on.
And it's a really good approximation to
to the cosh curve. A cosh curve on top of this;
it basically looks kind of the same,
very similar at the start, it does
eventually come through.
And then what happens at really large
values of x is, because the cosh
is an exponential function, exponential
functions
will always beat polynomials. This is why
exponential growth is such
an issue and such a thing because
exponentials will eventually grow faster
than a polynomial.
So it will eventually beat the x cubed.
So even though it sort of starts inside
and then comes outside,
eventually it comes back up up here and
becomes narrower because it grows faster.
So the the idea the thing that clearly
Wren and Hooke spotted was that, you know,
their drawing - the red curve - is based on
their experiments with the string and
the hanging chains,
and they clearly spotted that the x cubed
does a really good job of capturing that.
And so we believe that that is actually
what they used to build that inner dome
at St Paul's that's taking all of the weight;
allowing such a big dome to actually exist.
[Church Bells]
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