So I managed to escape the office today
and I brought you down to the river here,
this is the river Cam in Cambridge; and I
brought you down to the river because I
want to talk about pi.
So what does pie have to do with rivers?
Let me show you. So let me draw an
approximation of the river Cam here, or
any river in fact, here is my picture,
there you go, of a river.
What does this have to do with pi? Well,
we can measure how bendy this river is,
how much it meanders, by taking the
length of the river - that's the whole
length which we might call capital L - and
we divide that by the direct route, there
we go, as the crow flies. That direct
route, well it's going to be much quicker,
we're going to call that little l, so
we're going to divide by little l. That
figure is called sinuosity, so we'll
call that S for sinuosity - it's a measure
of how bendy the river is. So if the
river was completely straight then you
would have a sinuosity of 1. Or if it
was really bendy it would have a high
sinuosity; there's no limit to how
high sinuosity can be. But there are
some people who figured out that- well in
theory, the average sinuosity of
rivers should be pi. So if we took the river
Cam here and we looked at its total
length, which could be, I don't know, 10
miles divided that by the direct route
then we should get something like 3.14.
Now it might be bigger than that, it
might be smaller than that, but we look
at all the rivers in the world the
average should be about 3.14. So the
length of a river is just over three
times longer than the direct route. How
did these guys justify this? Well this
was published in a paper in Science in
the 90s. They were actually trying to model
rivers and how they meander and bend,
they used fluid dynamics to model the
flow of a river; and sometimes they they
became very bendy and the behaviour
became very chaotic. And then suddenly
all the rivers became straight again
because they would form something that
is known as Oxbow Lakes. So if I have a
river here, if it becomes very very bendy
like this, then what can happen is the
water will find a shortcut.
It'll go across this- it will cut off
this large bend and this forms an oxbow
lake. So this will dry out and it is no
longer part of the river. The river
becomes more straight by forming this
oxbow lake, so it becomes straighter. So
they found that the highest sinuosity
tended to be around 3.5, so it didn't
just go unlimitedly large. So high
sinuosities were about 3.5, low sinuosities
were about 2.7; so not all the
way down to 1, and it would tend to sort
of meander between those two values. So
it's the formation of these oxbow lakes
that control the sinuosity of a river.
Now how is this meandering caused? Well,
as the water flows through the river, on
the outside of the bend it starts to
erode here so it becomes more bendy; on
the inside of the curve it starts to
deposit soil, and this is how you form
the bends but these oxbow lakes then cut
off the bend.
They did try and justify- they tried to justify that figure pi in
the footnotes - it was footnote 14 of the
paper; and to justify it
they went to fractal geometry. And they
said, well the curves of a river are like
arcs of a circle. And the little wiggles
of a river, they're like little arcs of
circles, and if you add up all these arcs
together then the sinuosity you get
should be around about pi. I did
wonder for a long time if this was internet
folklore, I did have to look at the paper
to see if it was justified. They are using a
model here. So it was a fluid dynamics
model, they invoked this fractal geometry
to justify this pi figure; the rivers
they're modelling was unrestricted by
geography - in real life would you get
this figure? Well the theory
says we should, but in real life do we?
That's an interesting question. - (Brady: But I mean)
(the world's really well mapped now, has)
(no one checked this? Has no one fed in the)
(Amazon and the Nile and said, well, what's)
(the reality?) - I've tried to find the data,
I've looked for the data, I've tried to find
it. I couldn't find a nice set of data. I
think I found a paper about rivers in
Norway but I couldn't get access to it,
so I did try for you, I did try and find
out. So if anyone wants to have a look at
the sinuosity of their local river and
see if it's anywhere near pi that would
be interesting. It may not be near pi,
because this is an average we're talking
about. It could be hugely meandering, it
could have a huge sinuosity, it could be
very straight; we need to get all the
rivers in the world and have a look at
their data. Plus we do have restrictions
in real life.
[Preview] ...using pi. So the entire thing is going to
be pi based. We get the circumference and
an exact number of pies - give or take.
Can you do that again and this time talk me
through it as you do it? - You ask a lot
Brady, I'll I'll try. So 3, 1, 4,
1, 5, 9 - bend up to 9 - back to 2,
and then we've got-