So today's video I want to talk about a
subset of problems in maths called
orchard problems. Which are not really to
do with trees, but it's kind of helpful
to think about trees - so you can think of
apple trees or you can think about a
forest. Fruit trees, they're quite often
planted in deliberate plantation ways, so
in lines. Have you ever seen- if you're
driving down a road and there's a huge
vineyard, you get this weird effect when
you see somet- that you see - you look
down the vineyard and you see just a line
of trees and then suddenly you move a bit
and you see that this stretches on
forever. And as you move you get this,
like, coming in and out of phase and
it's quite a nice effect. Actually
there's a lot of maths to ask about these
things, like which trees can you see at
any one point? And which trees can you not
see? And can you see outwith the orchard
or are there trees in the way? And does it
matter where you stand? And that's the
sort of thing that often gets described
as orchard problems. And the one I want
to show you; it's a simple one with a
surprisingly unexpected answer.
Imagine you're standing at the edge of
an infinite forest, or an orchard. This is a
mathematical one so the trees are thin;
and, and all the trees are on coordinate
points. So if you imagine a coordinate
grid, they call this lattice points
in the trade, the points with integer
coordinates. So (1,1), (1,2). And
I'm just gonna put loads of trees on
this because this is gonna be where
we're gonna play for a while. We're gonna
play in the orchard for a bit. Let's
start, as in all good mathematical models,
by keeping it simple and later on we can
upgrade that. But so the simplest way is to
assume that they're mathematically thin,
they're points. Which is obviously not
what real trees do, but we'll ask
questions about that before we upgrade
the model to be more realistic. - (Brady: Doesn't that mean we)
(wouldn't be able to see them?)
Well kind of but if you look directly at it there
is a tree there. And actually that's what
I want to start asking about, which
trees can you see? So I'm gonna imagine
that for the sake of the rest of this
argument we're gonna stand at the origin.
And I could have done all the trees
going over here and down here, but we'll
just work with this quadrant and you can
extend it
in your own mind if you wish. And the
real question is, if all these trees are
out there and they carry on forever, this
is an infinite forest so it doesn't end
in those directions; there are a lot of
trees I'm gonna just put that out there.
There's possibly some level of infinity
number of trees. And I think you can see
some of them, although your caveat is
noted that mathematically thin things
are gonna be harder to see than real
trees. But if I- for example if I stand at
this point and I look directly
45 degrees into the forest I can
see that tree. Like, if I get it right I
can see that tree. Obviously if it's thin
maybe it's harder to see; but I also can't
see the trees behind it because
they're directly in line. So there are
some obvious questions which are not
interesting yet, but about visibility. And
so a lot of these problems are also
called visibility problems or lattice
problems; I like the more poetic
orchard problems. Here's the
question, and I'm gonna ask you this Brady: if
you're standing at this corner I'm gonna
claim that you can see some trees. For example,
well you can see that tree - if you look
in the right direction. You can also see
that tree, if you look along with that
direction. And you would then hide that
tree and that tree behind it. So some
directions you look you can see trees,
and I hope you agree with that. Is there
any direction, Brady, that you could look
standing here into the forest and not
see a tree eventually? And I'm interested
in your intuition about this.
(Brady: My intuition is...no) - Because eventually if
you go far enough you will hit one. Even
if you like carve out to miss the first
few of them, it goes on forever. Now I have the
same intuition about that - The answer
is not as simple as that. The other
standard answers to this question are
you look up, no trees up there - which is
great, it's hilarious. But if you look
along the axes there are trees on there
as well. So I think it's obvious, I'm
gonna claim it obvious, that you can see
some trees. Is it all trees or are there
gaps? And where are the gaps if there
any? And that's that's what we're
gonna play with because it's it's
surprisingly easy to answer. So the first
thing I'd to point out is that I know
which tree I can see based on what angle
I look on the line. Actually, the angle is
not so important as the gradient. So if
you're standing here and you look along
an angle of 45 degrees there then you
will see this tree and that's got a
coordinate of (1,1). Now actually the
45 degrees is pretty intuitive but it's
also a gradient of 1; and just to remind
you about a gradient it's like how far along
you go- for how far up you go for
every one along. Rise over run is the
gradient - this has got a gradient of 1 for
every one along you go one up. And this
line here; I'm less worried about what
angle it has - I could work it out but I
need some trigonometry - but I can tell
you the gradient of that line, the
gradient here, equals 2. And it's hit the
tree at (1,2). So I'm gonna claim
that if I know I'm gonna hit a tree I
can probably calculate the gradient of
the line I hit. So if I picked this point
he- is that gonna be a good one? Well that's 1, 2,
3, 4, 5; and 2 up - so it's the point (5,2).
And actually I can work out which line
will hit it. Because it's got to go to
there and it will carry on, but because
I've got a line and I've got some distances
I can work out the gradient of this one
is the rise over the run, which is gonna
be 2/5. What's lovely is that I now
know which lines will hit which trees.
And if I find a tree I've hit I also know
the gradient of the line. I'm now
wondering if your intuition has been
adjusted from when I first asked - you
said you wouldn't hit- you'd always hit a
tree eventually, right? You worried
that you wouldn't? - (Brady: Well if we can make a)
(gradient out of- that's not made out of integers, )
(like pi-) - Pi is one of those massively annoying
numbers that turns up everywhere you don't
expect, but there's just one example of a
number now which would do what you've
just claimed to, it's not built of
integers or ratios of integers. And
this is where they get the word rational,
because of that ratio of integers. So
ratio-nal would be better
pronunciation except no one uses that.
So these are all rational numbers,
if I look along a rational gradient
I will hit a tree. So if you looked along
a line of gradient - you said pi?
Well pi is close to 3, so the gradient
is roughly 3. It would go- - (Brady: So I would just miss the 3-tree?)
You just go above it. So it's three point
one ish so you kind of head up there, it
would be something like this line. But it
would miss that tree and it would go up here; and it's
not obvious to me where it would go next,
or in fact whether it would hit a tree.
Unless I know something about pi, which
you've kind of mentioned that it is
irrational, it's not a ratio. Now if it
was a ratio it would hit a tree. Since
it's not a ratio
it can't hit a tree. - (Brady: So it's gonna just)
(miss the 22 over 7 tree?) - It's gonna be
really close to it, because it's
ridiculously close to 22 over 7, pi is
really well approximated by that
rational number - and there's some other
really good ones too; but it is gonna
miss because if it didn't miss it would
be a rational number. Now there are lots
of irrational numbers. They're pretty
unintuitive things to exist but we can
prove they exist, as James has done a
lovely video about root two that we
think that's the first one that anyone
realised - ah, no ratio. And you can prove
it, which means there are lines on this
diagram that you could look along and
you will never see a tree. And that's the
first answer, you think, alright so there are
some lines you won't ever see a tree
despite this forest going on forever. And I
kind of like that first result is like,
oh I can see lots of trees, in fact
infinitely many trees? But there are some
lines I can look and never see a tree. So
the real question is,
if you pick a direction at random are
you likely to see a tree or are you
gonna miss? And this is where my mind
starts sort of complaining about things,
because the answer is to do with how
many numbers there are. How many trees
can you see? Well it's the same question as
how many rational numbers are there?
Everyone says infinitely many. It turns
out that mathematicians have this nice
way of talking about infinity of
different sorts, so absolutely the
rational numbers have a size of infinity
which is- we call countable. James Grime,
when he talks about it, quite often calls them
listable, which is a nice way to think
about it. You can't actually count them
'cos you don't finish but you can list
them in a way that means you've captured
them all. So we really have to ask how
many irrational numbers are there?
Because if there's more maybe we're more
likely to miss a tree. Which is then, we
could ask questions about can you count
the irrationals? And again if you go and
look into the study of infinity, look up
work by Georg Cantor - or a nice video
by James Grime on an amazing maths
YouTube channel. He explains to you why
the irrational numbers are uncountable,
or unlistable, which means there are
infinitely many more irrational numbers
than rational numbers. Which means, if you
pick a direction at random in this
forest you probably miss all the trees.
In fact, not even probably - probability 0
that you'll hit a tree. You are 100% gonna
always miss the trees. Which means if you
stand on the corner, mathematically you
can't see any trees because every direction you
look there are no trees. Except you can
see some, because we've picked out some
but there would be infinitely many more
that you can never see, or lines which
never hit a tree. - (Brady: So if these trees)
(are sort of super super-thin,)
(well they're point thin-) - Yeah
(and I'm standing)
(in the middle of the forest I'm gonna be)
(rained with light not shade?) - Yeah. It's all gaps. And
this is completely irrational, to sort of
have a horrifying pun, these irrational
numbers they're so densely populated
compared to the rationals that
everywhere you look you're likely to hit
them and not these special rationals
which are really rare in comparison. In
fact so rare that they account for 0% of
all the numbers. But it still messes with
my head, that you can stand in a forest of
infinite trees and not see any of them;
which is where the mathematical model
comes into question, like ok because
they're infinitely thin you could say,
upgrade the model, you could say
well what if the trees are a little bit
thick? Or maybe if you if you think about
this as firing a laser beam in different directions
then you could, say what if my laser beam
is a little bit thick? And it's very easy
to start realising that eventually, when
the thickness comes in, you do get trees
blocking the way. And you go into a forest,
all you can see is trees, which is different
from what we just explained. But the
question is how far can you see into the
forest? And then suddenly we've got
quite a real-life question of, like, well
how far does this laser actually get
before it gets blocked by tree? It'll
on the thickness, it'll depend on maybe
the size of the orchard. And so those
questions become orchard problems and
the the famous one is Polya's orchard
problem, if you want to look it up. It's
to do with how far can you see, or could
you get out with the wood with your
laser, depending on the size of the
orchard? - (Brady: I'm guessing that if the)
(trees are one millimeter thick, suddenly)
(you would have zero gaps, on an infinite scale)
On an infinite scale you are
completely correct. That mathematically
thin is the only reason you can see
nothing, and as soon as there's any
thickness you can't get out at all. But
there are some other interesting
thoughts; it's like on the infinite one,
with the mathematical thin trees, you
could ask instead about the lines in
this forest you could ask about the
points. So which points can you see? Which
trees are visible in this forest? The ones
all behind the (1,1) are invisible if
you're standing at the corner. They're
doomed to be never be seen by any line
because they're blocked by this one. But
that point is visible, but that one's not;
so which points are? And maybe if you
picked a point at random out of all of
them, what are the chances you've picked
a visible tree? Talking about picking
random stuff from infinitely many things
is dangerous mathematically. Actually
what we're talking about is if you make
it finite, like N and N, and then answer
that question when it's finite and then
let N head off to infinity. It turns out
you still have a sensible answer. The the
chance - I'm gonna tell you this - the
chance of a visible tree, if you pick one
at random - and this blows my tiny mind - is
six over pi squared. Another instance of
pi turning up the place you least
expected. But this is because of
something called the Riemann Zeta
function, which I think there are other
videos available about. And it's because
of the fact that if you actually want a
visible tree, what you need is the tree
you've picked to have two numbers, the
coordinates here, which are co-prime.
Because if they weren't co-prime, if they
had common factors, they'd be hidden by
an earlier tree. So what we're actually asking is,
like, the chance of picking a visible
tree is actually an abstract
mathematical question - pick two numbers
at random, what are the chance they're co-prime?
The answer is the Riemann Zeta function,
with s equals 2, look it up; and it turns
out to be this thing. It's not obvious
why that's the case, but the Riemann Zeta
function is a treasure trove of
non-obvious mathematical stories. - (Brady: So if I)
(was allergic to apple trees, and I wanted)
(to walk out of this forest without going)
(near one, what is the best line to)
(walk along?) - I'm gonna disappoint you
first of all, you're gonna go near one.
But maybe there's a line which is least
near - it's still gonna be near I'm afraid.
And maybe you could argue
mathematically you're gonna get pretty
damn close, but anyway the idea that
there may be a line which is least
near means that there's a most irrational
number. And actually it is this line. For
reason is I'm not gonna go into now, but
there are some interesting familiarities
about it. So here's my little computer
screen of some some trees. This line is
at a gradient of precisely one plus the
square root of 5, all over 2. Which, if
it's not familiar to you, you've probably
heard it by different name, is called the
golden ratio. It's roughly 1.618,
obviously if I told you the full decimal
we'll be here for longer than you want to
watch a video. This golden ratio turns up
in lots of places, and it is the line
that dodges the most trees. It's not easy
to show in this case but- - (Brady: Well, there are lots of lines)
(that dodge all the trees?)
In fact infinitely many, but this one seems to be
furthest away, because we know something
about the golden ratio which is that
it's the least well approximated by a
rational number. Which means, in this case,
it's the line that's least near any tree.
And one thing you can see nicely about
this is, if you imagine this line in the
middle is the the golden ratio gradient,
and you imagine it was an elastic band
going off to infinity and you pulled
this end along; it would catch on these
trees. And if you pulled it up it would
catch on some other trees. And it turns
out that these numbers are familiar, this
is the coordinate (1,1). This is the
coordinate (1,2). This is the
coordinate (2,3). The next one it
hits is (3,5). Those numbers I'm
reeling off are all from the Fibonacci
sequence. In fact all the trees that
this elastic band catches on are
the numbers from the Fibonacci sequence.
And that's another reason why the
Fibonacci sequence captures, in its
ratios, the golden ratio. It's a very
intriguing number, which we should
explore a bit more in another video.
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