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 these very special, limited edition, gold print ones. They look even better in real life than they do in pictures. And as you can see they're based on the videos we've been posting this week on Numberphile. If you'd like to check out all the t-shirts, including the gold ones, I'll put a link down in the video description. The gold ones are actually only available for a 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. Repetition, repetition, repetition. Repetition, repetition, repetition. Repetition, repetition, repetition. Woah, big deal, what's so good about that?
If I wasn't playing persona 5 right now would you have still posted this? I just answered that question in class