I was actually emailed by some
Numberphile viewers, Yong Zheng Yew and
Gokul Rajiv; they've been doing a
project about cutting your fingernails!
And they sent it to me and there was
some pretty cool maths in there. Which
I've sort of developed, thought it'd be nice to
talk about. What we're gonna talk about
here is what happens when you cut them
with a pair of scissors.
You know you might - is this gonna freak
you out now? So I got a cut like that, so I
cut from the bottom there, and I- you you
probably would try to cut to the top,
right; and then you'd do another one like
that, alright. And then and then I would
cut across the top like that and then
you carry on going, till you've got a nice smooth thing, right?
(Brady: Urgh that's gross!) - I don't know where they went flying off to.
Let's draw a picture of our fingernails. So we'll
draw it as a semi-circle, imagine it's got
radius one.
So the first cut, well we cut out a
triangle, right. This is a 45-degree angle.
So next cut, so you don't want a big
pointy fingernail - it's cut across the
top in such a way that each new side
that we create, they're all the same
length. Do that again, we do another cut;
the idea being that we've now got four
sides and that these are all the same
length. And of course if I do this enough
times it's going to start looking like
an arc of a circle, okay? And that's your
goal, that's where you want to end up at, right?
- (Brady: Because you don't want like weird-)
- Yeah you don't want stabby fingernails, right?
You want nice, smooth, beautiful
fingernails. So you want to reach a
circle so we do that by doing successive
polygons essentially. So you could carry
on with this ad infinitum, now assuming
you've got an infinite amount of time.
After a hundred cuts gonna definitely
look like a circle, okay. Now the question
is how big is this circle? Because we
never cut at this point again after the
first cut, these angles here are still
gonna be 45 degrees. And that's very
important, that tells us a lot. So I'm now
going to draw a line there which is a
right angle to this new circle that I've
created here. And I'm gonna do another
one coming from this point on the circle,
again at right angles. Now, what does this
tell me? Because this is hitting the
tangent to the circle at a right angle, this
is actually a radius of the new circle.
So I can kind of complete the new circle.
The other thing I note is this point
here, whilst it's also the centre of the
big circle, it's also the bottom of the
top circle. That's because of something
that you know about circles, is that if
you take two points diametrically
opposite, and you connect them via any
point on the circumference, they will
meet at 90 degrees. So you know the picture now.
From this we can very easily work out
how much we've cut off. Our original
circle had radius of one. A little bit of
Pythagoras tells you that this line here
is root 2. From there to there is
root 2, from there to there is one; so from
there to there is root 2 minus 1. So now
we can compare the ratio between the
size of our nail before to the size of
the nail afterwards. They're sort of
height before, I'll call H-before over
H-after. H-before was 1, H-after is root 2
minus 1; and it's not too difficult to
show that this is the same as root 2
plus 1- woah big deal, what's so good
about that? By cutting my fingernails - so the
the rank one I had before was bigger
than the one I have after by a factor of
the silver ratio, that's what- so the
silver ratio is there. Okay, so what is
the silver ratio? You know about the
golden ratio, right? So golden ratio you
think about with the Fibonacci sequence,
okay, that's one way of understanding
where the golden ratio comes from. So you
start off with zero and 1, and then you
add these two together and you get 1,
and then you add these two together and
you get 2, you add these two together you get
3, these two together you get 5;
carry on like that that. That gives you the
Fibonacci sequence. The ratio between
terms in the Fibonacci sequence
tends towards the golden ratio. So the
silver ratio is slightly different. So
you start off at the same point, zero and
1; you double this and you add that.
Gives us 2. Double this and add that,
so that's now four- plus one is 5.
Double this and add that, so that's 12.
Double this and add that, that's gonna
give me 29; and I carry on like that with
that pattern. If I wanted to write down
what that pattern is, it's like this, Pn
is 2P(n minus 1) plus P(n minus 2). So
in other words I take the one before, I
double it; and then I add the one before
that. You can see this ratio between
consecutive ones will tend to the silver
ratio. So you do it the same way as you
derive the golden ratio. So I just take
this expression here and I divide
through by P(n minus 1). So that gives me
Pn divided by P(n minus 1) is 2 plus
P(n minus 2) over P (n minus 1). So at large n
this ratio becomes what I'm going to
call delta s. This is 2. And
this is 1 over the ratio. So that's 1 over
delta s. And if you solve that equation
you get root 2 plus 1. Okay so that's the
silver ratio. So there is an entire
family of these guys and they're called
the metallic ratios. This is the Pell
sequence. Actually there's some fun facts
about this: if the Pell number is prime
its index is prime, okay? - (Brady: Like its position?)
Yeah yeah exactly, its
position is prime. We'll call this P2.
Okay so this is the second Pell number.
(Brady: We don't count 0?) - We're not counting 0.
We'll call this one P3.
Okay so you can see that this is prime
and this is prime. This is prime and this
is prime. This one here is also a prime
number, and yet surprise surprise it's
the fifth Pell number, which is also
prime. So it has this cool property that
if the Pell number is prime its index is
prime. The other weird property of this
sequence is it contains very few powers
of anything. So but by that I mean, you
know, does it contain squares or cubes or
fourth powers - well very few. Actually the
only ones it contains in the sequence
are 0, 1 - which is obviously a power - 169 is in there,
okay, which is obviously 13 squared. There
are no others. There are no fourth
powers, there are no other squares, there
are no other 26th powers - nothing, that's it.
This Pell sequence gives you the silver
ratio. There are of course
generalisations of this that give you
the other metallic ratios; so you've got
the 3-Bonacci sequence, it's 0, 1, 3, 10,
33, 109. - (Brady: How does that work?)
You got to start off with these two,
you take this one multiply it by 3 and
add this, take this one multiply it by 3
and add this, take this one multiply it
by 3 and add this, right? And you go on like that.
(Brady: So is the Fibonacci sequence)
(multiplying by 1 and adding the one before?)
Yes, yes and all the other metallic ratios
come from multiplying by a different number.
- (Brady: See I thought it seemed arbitrary but)
(I hadn't thought of the Fibonacci)
(sequence as fitting this, but it does fit it.)
Yeah absolutely, absolutely. So in general
you can think about sort of N-Bonacci
sequence, which would have the rule
Pn is NP(n minus 1) plus P(n minus 2). So you take
one of the numbers, you multiply it by n, you add the
one before that, that gives you your new
number. So this one gives you the bronze
ratio - n equals 4, n equals 5; they
don't actually really have names. So I
think some people call it copper, nickel;
but I think we're free to call them what
we like actually. So the bronze ratio,
which we'll call delta B, the limit of the
ratio of these terms.
That's 3 plus root 13 over 2, okay, all
right. Which is roughly 3.30.
- (Brady: Very arbitrary.) - Yeah, but they're not, right, because
you can think about what you get from the nth one.
Okay, the nth one - the ratio that you
would associate with that, I'll call it
delta N, and that is N plus the square
root of N squared plus four, over two. And
you get that just by applying the same
method as we did for the silver ratio
here. Everything you think about the
golden ratio will generally have an
analogue here. So you can talk about
golden rectangles, you can talk about
silver rectangles, bronze rectangles. You
can talk about golden spirals, you can
talk about silver spirals, bronze spirals.
So, you know, so what are their properties?
Okay so we probably need some more
brown paper actually Brady. So the golden rectangle is the
following: so you take a rectangle, okay, and
its sides have the ratio of the golden
ratio. So if this one's got length one
this one will have length the golden
ratio, which is 1 plus root 5 over 2,
right? - (Brady: Is that not truly golden?)
I really suspect it's not- it's not even a rectangle,
look at the state of it. If this were
really a golden rectangle what property
would it have? If I remove the largest
square then what's left over would be
another golden rectangle. Basically 1
over 5 minus 1 is actually equal to phi.
What's the silver version of this? You
draw a rectangle whose sides have ratio
given by the silver ratio, remove two
large squares, both of side 1, and what we
have left over is going to be another
silver rectangle. If you did it with the
rectangle whose sides were the
bronze ratio you'd have to remove three
and then you'd get one leftover which
was another bronze rectangle. In England
of course we have A4 paper. Now, you
may know something about A4 paper, the
the ratio between the lengths of the
sides is root 2. If this is 1 this would
be root 2. This is another kind of
rectangle actually, another kind of
interesting one. It's not just A4, this
is sometimes also called
a silver rectangle. But it's not our
silver rectangle, so we're not going to
call it a silver rectangle; because some
people would call root 2 the silver
ratio. Actually we're going to call it
the Japanese ratio, because it is big in
Japan this ratio, this root 2.
And these - what I'm going to call a
Japanese rectangle. You know, lots of
Japanese architecture uses this
particular ratio and these particular
rectangles. You also see it in Roman altars,
all these sorts of things. But we're
interested in the real silver ratio; so
how would you create it from this guy?
You would just remove the larger square,
kind of appropriate, this feels like origami
and you know, we're talking about a Japanese
rectangle. So this would be the larger
square here. And if I remove it, I'm gonna
do it quite crudely, I would get this.
This is a silver rectangle. - (Brady: This won't work in America)
Yeah, A4 paper is wrong in America, you have to be
in England for this to work - well, some-
other places that have A4 paper but
definitely this can work in England, not
necessarily in America. So this is a
silver rectangle. Now if I start removing
two larger squares from this I'll get
all the silver rectangles,
ever-decreasing. So those are the
rectangles that you can have, and
again there's a whole family of them
that you can talk about. But more
interesting than the rectangles, of
course, are the spirals.
Okay, now the spirals are great because
the spirals are the things that
everybody says appear everywhere. Now
people normally say, oh the golden ratio
is everywhere isn't it? It's everywhere.
Because they're talking there about the
spirals. But it's not really the golden
spiral that's appearing everywhere, it's
all these other ones. We start off with
a square. And we draw the circle that
goes between there and there. Now we
create another square which is a factor
smaller by the ratio we're interested
in, so we decrease it by a factor of
delta, okay? Where delta's either the
golden ratio or the silver ratio or the
bronze ratio; whichever one you're
interested in. And we continue our circle.
But now we create another square, but we
drop by the ratio again. And we continue
our curve around. And then we go on again,
this is gonna have 1 over delta cubed but
let me go around like this; and we create
a spiral. If this was the golden ratio
this would be a golden spiral. If this
was the silver ratio it's a silver
spiral, if it's a bronze- bronze ratio
it's a bronze spiral - blah blah.
These are all a family of spirals which
are called logarithmic spirals. So what
is the formula for these spirals? So
this particular one I can compare, you
know I can think of an angle, the angle
that I've gone round. So I'm going round
like this, with changing angles. I
can ask how does the distance from the
centre change? So the distance from the
centre I'll call r. Well you can sort
of see that if I if I were to go around
this way I gain a factor of delta every
time I go around a quarter turn. So
actually it turns out that r will go
like, delta 2 over pi theta. So 2 over pi
of course is- or pi over 2 - is of course a
quarter turn in radians. Ok, so this is the formula for this this spiral.
- (Brady: If it's pi over 2 why have you put 2 over pi?)
Ah because I need to scale it by the by the
number of radians, okay? And this is part
of a larger family, the logarithmic
spirals, which look like this. Ok so
theta times tan p.
So what is p here? p is called the pitch
of the particular spiral that you're
interested in, and that's very
characteristic of it.
Imagine I'm coming along this spiral and
I'm going along this direction, call that
alpha, and then I could draw a right
angle there and I'll call that p. And
that's called the pitch of the spiral.
Now each spiral has its own pitch. Okay
so we can write down what the pictures
are for the different spirals that we're
interested in. So for the gold, for the
golden spiral,
the pitch is actually, you can show it's
about 17 degrees. You can work that out by
comparing these two formally. And just
comparing the two you put in the golden
ratio here, you can extract what the pitch is.
It works out as about 17 degrees. For the
silver it corresponds, the pitch is about
29 degrees. For the bronze it's 37
degrees. For n equals 4 we get 42 degrees,
for n equals 5 we get 46 degrees, n
equals 6 - 49 degrees. Remember the
Japanese ratio which we said was root 2?
That has a pitch of about 12 degrees.
People always go on 'oh the golden ratio,
it appears everywhere in nature'. It
does appear in some places in nature. But
when they're talking about spirals it's
actually these family of spirals that
appears. For example, sometimes you'll
hear it said that the Milky Way sees the
golden ratio, okay, through its spiral.
It's not true. It does indeed see a
logarithmic spiral, with some average
pitch. It's about 12 degrees. So the milky
way is not a golden
spiral; it's much more like a
Japanese spiral actually. There's another
one that I quite like, is the peregrine
falcons Brady. I love peregrine falcons,
they're the fastest creature on the planet,
you even get them in England, right? So
there's something cool, an animal that
we actually get in England. It wins at
Top Trumps as well, you know predator
Top Trumps that my kids have got. The peregrine
falcon's one of the winning ones. So I
love a peregrine falcon. The peregrine
falcon knows about these ratios as well,
right? Imagine you're a peregrine falcon,
you're flying around.
Here's your prey, it's sitting here at
the centre. Now you want to keep your
eyes on the prey. But you can't move your
eyes, you don't wanna start moving your
head around, right? So you- that stays in
a fixed position. So you have to keep a
fixed angle relative to the prey. So that
necessarily means you have to follow one
of these logarithmic spirals. The angle it
has to look at is actually 40 degrees.
Which means the pitch of the spiral is
about 50 degrees, which is wh- almost
one of our metallic spirals; the - I'm
going to call it the n equals 6 one. So
I mean I don't know what this is, right,
so we've got gold, silver, bronze, copper,
nickel, aluminium? Anyway so it's
there right, it's pretty much there; I
mean it's a good approximation, right? So
people talk about seashells, that they
see the Golden Spiral. They don't really,
they see a whole- all possible pictures
they're seeing these
logarithmic spirals. And there's a good
reason why why for example seashells see
these these log spirals. They have a very
nice property, all of them, all of them;
the golden one, the silver one - all of
them. They kind of have what's called a
self-similar property. So you go round
and then you go round again and when you
go round again you're basically seeing a
scaled-down version of exactly the same
thing. And you keep seeing that as you go
round and round and round. And when you
think about growth in a biological
system, that allows you to grow very
efficiently if you follow that pattern
which is repeated; repetition,
repetition, repetition. You also see these
log spirals a lot in architecture, and in
art. Again there's probably a good reason
for that and it's related to this self
similar property. So basically when you
look at something, you know, your eye
performs visualisations on lots of
different scales and then correlates
them. When you look at this and you look
at it at lots of different scales, you're
seeing the same thing. And so that- maybe
there's some sense in which that that's
good because we're correlating a lot of
the same thing on a lot of different
scales. - (Brady: We're reducing cognitive load?)
Exactly and somehow we like that. So this
is why these spirals appear. - Did you know
you can get Numberphile t-shirts? There's
a whole range of them; most recently
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can see they're based on the videos we've
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you'd like to check out all the t-shirts,
including the gold ones, I'll put a link
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very limited time because we have to
make them to order. But either way, thanks
for watching, we'll see you in the next video.
[Extras] Repetition, repetition, repetition.
Repetition, repetition, repetition.
Woah, big deal, what's so good about that?