So how do you measure the coastline of a
country? Let's draw a little bit of
coastline here, make it nice and- so
imagine this is somewhere in the north
of Scotland or something like that.
(Brady: Which side's the sea?) - This is the sea here, all right?
You could use a really big ruler and just kind of
line it up and go around and see how
much we've got - so I'm going to use a
ruler that's about this long. There's one
lot of ruler there, there's two, three,
four, five, six, seven, eight, nine. So this
coastline is nine units long, nine of
those rulers long. But I'm not
getting into all the nooks and crannies
and the lumps and bumps and stuff right
there. So let's use a smaller ruler, let's
use a ruler that is it is half the size.
So I'm just going to do that just by eye
trying to work out the- so 1, 2,
3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22. I
did it by eye but I was trying to make
the the brown rulers half the size of
the orange rulers. So two 9s are 18; we've
ended up with 4 extra short rulers. So by
measuring with the shorter ruler we've
got a longer length of coastline. This is
always the case though, the smaller I make
the ruler the the bigger the answer. I'm
getting into more of the little nooks
and crannies, that's the important thing.
So the more kind of lumps and bumps you
get into, the the bigger the answer you
get. So the question then becomes how how
small does that ruler need to be? So
let's have a look at a few examples of
of different scales. Let's zoom in on a
little bit of this coastline here, so as
if we're flying along in a plane and so
this bit of coastline might look like that
as we zoom in. Let's zoom in even further
so we're looking a craggy sort of rock
edge, what might that look like?
And that craggy rock edge might look
something like that. So we want to zoom
in even further, maybe a little rock
that's jutting out from from that edge so we're
we're crouching down, we're eyeballing a
bit of rock and maybe that bit of rock
looks something like that.
We get our
microscope out and we have a look even
closer at the sort of fine grained
detail of the rock, and maybe it looks
something like that. The problem is it's
lumpy and bumpy all the way down. There's
no obvious point where we say, okay, this
is the size of our ruler. So we need a
ruler that kind of got into all the sort
of molecular lumps and bumps to get a
true answer; but is that true answer? It
would be a meaningless answer, it would
be it would be, you know, hundreds
thousands millions of times greater than
than what we've got at this scale
depending on how low you go down.
Suddenly we're measuring the coastline
of Britain say to be millions of
kilometers and that doesn't feel like a
useful answer. So we present it to some
mathematicians to see what they think
and so, let's have a look then at a sort
of mathematically perfect coastline,
isn't that beautiful?
Brady, I know you like to know
where the sea is so here's the sea. But
our mathematical coastline needs to
mirror the real-world coastline so it
needs to be lumpy and bumpy. So let's put
a little lump in, like that, so now we've
got a lumpy bumpy coastline. But we need
to be able to zoom in and for it to
continue to be lumpy in the same way
that the real coastline is. So now each
of these straight lines let's put a
little bump in all of these as well. But
again we need to put a zoom in and for
it to still be lumpy and bumpy so now
every straight line that we've just
created needs to have a lumpy bit in it
as well. This is the last one I'll draw.
Okay, so to be like the coastline of Britain
we need to keep doing this, keep adding
lumps and bumps all the way down forever;
or at least you know to some to some
high degree. So how do we measure the
coastline of this mathematical drawing,
or how do we measure the length of this?
Okay so we can notice that this small
section here is the same as the whole
thing but shrunk down. So if we take that
whole thing and shrink it down that's
that's what you get.
This section here is a third of the whole
thing. So the whole- in other
words the whole thing is three times
bigger than this one section. But let's
look at it another way, this bit here is
the same as this bit and this bit and
this bit. So we've actually got 1, 2,
3, 4. So this- the whole thing is
three times bigger than that section and
it's four times bigger. So how can it be
that the whole thing is three times
bigger and four times bigger? Is there a
number that you can multiply by three
and four and get the same answer?
Well there's zero, so 0 times 3 is 0, 0
times 4 is 0. So the coastline is zero
metres long - that doesn't work because we
can see it. The other option is - it's not
really a number - but it's it's infinity,
it's infinite. So you know 3 times
infinity is infinite, 4 times infinity is
infinite. So the conclusion is that in
this mathematical case at least, so long
as those lumps and bumps carry on
forever it's infinitely long. And that's
in a sense true for the coastline of
Britain. Coastline of Britain isn't
infinitely long because you get down to
the molecular level but to all intents
and purposes it has no length, it's not
something we can we can give a number to
it in any meaningful way. The guy that
did this- well this is called a Koch
Snowflake, but the guy that came up with
all the stuff is Benoit B. Mandelbrodt.
The B stands for Benoit B. Mandelbrot by
the way; and he invented this thing
called fractal maths. And so although we
it's kind of we're left in this position
where, okay we can't measure the
coastline of Britain, we now have some
mathematical tools for describing it. So
we can talk about the lumpiness of
Britain. So all the fractal-ness of Britain
and it's- I forget what it is now it's like
0.2 lumpy. Or whatever - there's
some way- there's some way now we have
of describing the fractal nature of our
own coastline and any coastline. So yeah,
that is how trying to
the coastline of a country leads to some
some really interesting mathematics.
(You didn't tell me how we can do it though,)
(or the answer is we can't do it?) - The answer
is you can't do it, sorry about that.
I mean okay, well look it up. I mean the
thing is if you look it up you will find
either several different answers or
you'll find someone saying okay, you know,
we're going to agree that we use a ruler
that's an metre long or something like
that. - (Right, so you pick your)
(resolution?) - You pick you resolution, you get
a number, but it's kind of meaningless
because you pick a different
resolution you get a different number you know.
- (You must be really fun to walk along the beach with.)
I mean the other thing about- the other
thing, if you're measuring just the
length of each for example you could you
could arguably measure the straight line
distance between one end of the beach and the
other. You're not really getting the full
length of the beach but you're getting
some measure that's at least useful. In
the same way, you know, you could say okay
I'm going to say the coastline of
Britain is the distance from Land's End
to John O'Groats doubled. Of course it's far
too low, the actual coastline is much
longer than that, you know, or is it is?
There's no answer.