I want to tell you about a very famous number that you've heard about before. I want to tell you why it is what it is, and it's the golden ratio. A lot of people think the golden ratio is this mystical thing. And it is, but not for the reasons they think. But I want to do that, and I want to tell you why it's interesting. And I want to do that through a mechanism of flowers, and you may have heard a connection with Golden Ratio and Flowers before, but I want to show you why that connection is there. For the sake of this little video, I'll be the scribe But I'd like you to imagine, Brady,
that you are a flower. Your job as a flower is to deal with your seeds, which is kind of the job of everything living. You're gonna grow some seeds, and we're gonna model this flower in a mathematical way. This is not how flowers actually grow,
but there are connections. This is the centre. When you grow a seed, I'm gonna represent that by putting a little blob. Now that's a seed you've grown from the centre of your flower, and one option you could have is you've got to decide where to put your seeds. And I'm going to give you the option of only how much do you turn around before
you grow your next seed? So you put a seed down, and you can turn a bit, and put another seed down. Kind of growing it. If you don't turn at all, you're going to grow seeds out like this. The first seed goes there. If you don't turn, the next seed goes next to it, and the next one goes next to it, and you grow the seeds out there, and you're going to push seeds out. Actually, I'm adding them on the end, but it would grow from there and push the seeds out in a line. This is a really bad arrangement for a flower, I hope you agree, and I hope you weren't imagining this when you were thinking of a flower. [Brady] 'cos it's a waste of space. Yeah. There's a whole bunch of circle unused. So the obvious thing if I'm going to do this model, of like, if a flower could grow by putting a seed and turning a bit. What would happen if you turn an amount of a turn. So I'm going to talk about fractions of turns. This is a fraction of a turn of zero. If you do a new one here, and you turn half a turn each time, then if the first seed goes here, then I think if you turn half a turn, the next ones going to go there. If you turn half a turn again; keep going in the same direction, it's going to go there, and there, and this is also not exciting, but you can kind of see why the decision of turning a half has made two lines. And maybe I'm going to call these spokes, because, just to get you in the mood,
lets do a third. You can probably predict it pretty easily. Seed. Turn a third of a turn,
roughly there. Third of a turn,
roughly there, and you're going to see these three spokes sticking out pretty easily. Are you happy enough with this? I mean, none of these are good flower designs, but the consequences of choosing a number
has given you some patterns. So if I jumped, say,
to a tenth of a turn, would you care to predict
what you would see? [Brady] Ten spokes? Yeah. And so I don't think the the
spoke behaviour is very surprising. It looks like the denominator of this fraction of the turn is controlling everything. Now I think it's much less obvious if I told you what would happen with 3/10. So with 3/10, if you start here, you turn 3/10 of a turn, you'd skip a couple of the branches, and another 3/10, you skip a couple, you get here, 3/10 you'd skip a couple,
and go here, and if you keep going round, you'll end up not repeating yourself for a bit until all 10 are done. You also get 10 spokes. So there's this really nice thing in Mathematics called a conjecture. We pretty much have one here. Looks like it's the denominator of the fraction that's controlling the number of spokes. So here's a quick computer model of what we've been talking about. And we can check that with other tests; 4/11. You may want to predict what happens. [Brady] 11 spokes. You're correct, there are 11 spokes. If you type in some other numbers there are some surprises. If I do 11 out of 23, you do get 23 spokes, but there's some interesting behaviour happening in the middle. And that's actually, looks like theres kinda two spokes but they're kinda twisted, and that's because this number is quite close to 1 over 2. And it looks like what numbers you're close to also affect what happens. One surprise that you should watch out for, I mean, if I do 7/10, you know about tenths, you get ten of them. But then occasionally you catch yourself out, you do 4/10, and you think "oh, 10 spokes", but there aren't; it's 2/5ths. And flowers can cancel fractions. Or they can't actually, and what's happening is that 4/10 is better described by 2/5. So, you've seen lots of bad flowers. This is pretty, but it's not what flowers do. What's interesting is if you change this number very, very slowly; and you realise that a tiny change gives you very different behaviour. So this number is changing ridiculously slowly, but even a small jump is giving us spirally shapes, and very quickly they stop looking like spokes. [Music] [Brady] What are you changing? Just the top number? The number is the fraction of the turn before I grow each seed. So this number that's here is 0.401 of a turn, and I grow a new seed, then that's already enough to stop it going in lines. And they're starting to bunch together, and this is is already looking a better, prettier thing; for a flower. It's also nicely hypnotic, if you need to hypnotise people. You start seeing things kind of turning one way, but also maybe turning the other way. You see spokes arriving and disappearing. And this is already a better flower. [Brady] You're using more of the space. I'm using it much more efficiently. Now, I'm not saying flowers are thinking about this, but somehow they do this efficiently, and we've got now an obvious question is: Is there a fraction of a turn that is an efficient one. What's really lovely about this is that you can see rational numbers arriving. You can see that I'm not at a third, but already, the number 3
is dominating everything. It's like hunting for big game, you can hear these animals
coming in the undergrowth. You can see it. This third is about to arrive. We're .329 now, and as soon as we hit exactly a third, we're going to get those 3 spokes. And it's really nice to see it arrive, and then disappear. So it's about to get there. As soon as we hit .3333, through as long as it carries on forever, you will see our three spokes. [SNAP] [Music] And then it's gone,
and we're into other numbers. If you put a number in for a fraction of a turn, and it is a fraction. i.e. has a denominator, it's going to give you spokes. And so maybe we're into familiar territory from many other discussions about numbers. Maybe you can suggest a number Brady that you could type in that
wouldn't give me spokes. [Brady] An Irrational number [Brady] Square root of 2 The square root of 2? What do you think you are going to see? [Brady] Kind of spirally, spiral-ness. [Brady] Oh It looks less like it's got spokes, but you can kind of count them. And it turns out that this is definitely an irrational number. I'm approximating an irrational number on a computer. But, this arrangement looks much better. So it sounds like you've hit upon the idea that maybe flowers need to turn an irrational amount of a turn. But there are other irrational numbers. I'm gonna type in 1/Pi, because Pi is a lot of peoples favourite irrational number. This surprised me. Think about what you might see. We know it can't produce spokes because Pi is irrational. Now they're not quite spokes, but they're slightly curved spokes, and there are in fact 22 of them, just so save you counting. I don't know if 22 rings a bell to do with Pi? But the older generation used to get taught that Pi was pretty much exactly the ratio 22/7. It's not quite, but it's unreasonably good, and you can see in this diagram that it's unreasonably good
because this is irrational, but it's really well approximated by something to do with the number 22. In fact, what I love about this diagram is you can see another approximation for Pi buried in the middle. There aren't 22 spokes in the middle, there are 3. 3 is a very well known
approximation for Pi. In fact, if I carried this diagram on really big, you'd see lots of rational approximations for Pi in the arrangements of
seeds in the flower. Or a mathematical flower. But what it also tells you is that Pi is not very irrational. It looks like root 2 is more irrational. So, actually, to obvious question which has turned up in lots of situations, and maybe in other videos from me, is that "is there one that's the most irrational". And there is. And I'll show it to you, and I'll show you why. So, here it is. If I jump to the square root of 5, minus 1, over 2, you get this. This is the golden ratio of the turn, and what's lovely about it is you can see spokes, but you can see them going in both directions. They're kind of crossing over both ways. And you can try and count them, and if you do, you get fibbonaci numbers. And if you go and look in the real world, this is the bit that a lot of people claim that this is what sunflowers do. So if I hide the seeds there, that's the arrangement of seeds in the head of a sunflower. That's not generated by a computer. This is a flower doing something to be efficient. And if I put the seeds back and hide it, the correlation between those placements is almost frightening. And what's lovely about it is that no other number works. So if I start this animating again, "ah, the spokes are obvious" And this kind of unravels and already you can see the spokes unravelling, and they're obvious spokes. You could count the spokes and figure out what rational number it's near. So the golden ration looks like it's the best one, but I want to show you on paper why it's the best one. And I'm going to do that by starting with Pi, because, eh,
it's a good place to start. But Pi is an irrational number that is apparently not very, irrational. And we kind of already know that it's approximated well by a rational number. But let me show you how you can
sort of quantify that. So I'm going to say that Pi is 3 plus "a bit". I don't think that's controversial. But what I'm interested in is writing this number Pi, which I can never really write down in full, which is why we have this symbol for it. Can I write it in a way which looks more like a fraction, instead of like a decimal. And so 3 plus "a bit" isn't very helpful, but I could say, since this bit is less than 1, (other wise it would be 4 plus "a bit", right.) Then I can say this is 3 plus 1 over "something". And I can find out what that something is on a calculator. I could take away the 3, and I get the "something". And then I can do 1 over that, or x to the -1, to find out what it is. And it actually is 7 point something. So I'm gonna write this, 1 over 7 plus "a bit". The words "a bit" are not sort of mathematically recognised terminology, but you get the idea. So I could carry on, I know this is 3 plus 1 over seven plus "a bit". And I could write that as 1 over something. And that's on my screen now, so I could take away the 7 and get the bit,
and do 1 over that. And I get 15 and "a bit". I can start writing 15 plus "a bit", and instead of doing the "a bit" now, I'm just going to go straight in with 1 over something. Take away 15, get "the bit". x to the -1.
Do one over it, and I get 1, and a bit. Take away 1, do the bit It's a very small bit this time. I'm going to get 292. And you can see that if this number is truly irrational, I can just keep going. And actually, this thing here is called the "Continued Fraction", for Pi. And something very obvious happens with Pi is that you get a very small number here, and then you get a
very large number here. In the trade,
they call it truncating, but if you chop the continued fraction at a certain point, you'll get an approximation. So 3 and a seventh is 22/7, is a good approximation for Pi. If you chop there, you get an approximation. If you chop there, if you chop it there. But because this number is massive, and it's 1 over that number, this additional bit is tiny. [Brady] Becoming more and more trivial. Well, actually, after that it
goes back to being some ones in the continued fraction. But that particular point means that you don't add very much accuracy at those two levels of truncation which means it was really accurate before, which means the step before that was ridiculously good. Which means Pi is well approximated quite early on by a rational number. Which is why I'm going to claim that it's not very irrational, and why when you saw the diagram of it, it looks like it had spokes. So, looking at the continued fraction, the question is, "What would the most irrational number look like"? It would be the one with the continued fraction that doesn't have any large numbers in it. So I'm going to claim that this is a pretty good candidate. Call it x, but the continued fraction would be, well, let's just start with 1, but then the continued fraction would go 1 over the smallest whole number. One. Then we'd have a one here, and a one here, and a one here, and this is something that a lot of people do recognise. It's an odd thing to ask what it is, but whatever it is, it will be badly approximated any time you truncate it, because these numbers are small. Now, just as a little heads up, I'm going to tell you that root 2 has a continued fraction as well. 1 plus 1 over 2, plus 1 over 2, plus 1 over 2, plus 1 over 2, and carries on like that. Which is incidentally why root 2 looked pretty good on our diagram. Because although these are not the smallest numbers, they're consistent, and they stay small. Where I want to get to with this video, you might know the answer, but I want to prove it, is: What is this number? I'm going to solve this. This is an infinite thing, but we can solve this surprisingly easy because it carries on forever. Let me point out something which I think is obvious when someone points it out, which is that this thing, is the same as the whole thing. It is x. Which means I can sort of grab that thing and call it x and I can rewrite this equation as x equals 1 plus 1 over x. And that looks much less scary. In fact it's a quadratic equation, which I'm gonna solve. I'm sure people watching this video would have their favourite way
of solving quadratics. I'm going to do maybe not quite the quadratic formula. I've seen a friend of mine, called Matt Parker, try this with a quadratic formula. There's a better way, I'm multiplying by x. To get x squared equals x plus 1. I'm going to rearrange it onto one side. I've got x squared minus x minus 1 equals 0. At this point a lot of people reach for the quadratic formula. That minus B plus or minus
the square root... I'm going to complete the square, which is where the formula comes from. So I'm going to realise that if I write x minus a half,
squared, that would square to
give me the x squared, it would also give me
the minus x I need. Try it if you don't believe it. But it will create a quarter, which I don't want, from the half squared. And I've still got a minus 1 here. I'm just going to carry on, putting this stuff on the other side. I get x minus a half,
squared,
equals; combining these, I'm gonna get 5 over 4. And now I can square root it. This is the whole point of
completing the square. x minus a half equals plus or minus the square root of five because I square rooted the five) over 2 because that's the square root of 4 And I'm gonna do one more step, and this needs a box because, x equals Put this half on the other side, [Writing sounds] That's the same thing, and this is equal to Phi. Which is the Golden Ratio. And it is the most irrational number because of the way it builds as a continued fraction. Which is why it looks so nice when you stack it round a sunflower. And why it carves a path through an infinite orchard if anyone ever talked to you about that. That's furthest away from all the other points. And that's why Phi is a cool number, It's not because the Greeks designed the Parthenon to look like it, Because that was not true. Brady: There's a plus or minus there, Brady: I feel like you haven't done the job! Brady: I feel like we're still sitting on a fence! So which one is it? Let me show you on the calculator, if you tap this in, obviously this is going to give us an approximation. But if I do 1 plus the square root of 5, and divide it by 2, I'll get a familiar number. Which is 1.6180339,
and so on. Now, that is a familiar number, the Golden Ratio,
but if I did 1 minus it. 1 minus the square root of 5, which is the other option I had, and divide it by 2, I get -0.6180339, dot dot dot. And I actually get the same decimal expansion, it just happens to be negative. And this is all because of the property of the Golden Ratio, that if you take away 1 from it, you get 1 over itself. And that's actually built into this equation. And if you make it negative, you can get reciprocals of itself. So either of those numbers lay a claim to be the Golden Ratio, and when I did it on the sunflower earlier, I actually used 0.6180339, because that give me a fraction between 0 and 1. but they're kind of all Golden Ratio, or directly evolved from it. [Music] How about we check on on Brilliant's problem of the week? So the basic level. A vertex of one square is pegged to the centre of an identical square. The overlapping area is blue. One of the squares is then rotated about the vertex, and the resulting overlap is red. Which area is greater? What do you reckon? Fancy your chances? Over on intermediate, well we've got a Chess problem there about promoting a pawn. Or the advanced problem, if you're feeling a little bit dangerous. The centres of three identical coins, form the angle that's coloured green. What angle maximises the area of the blue, convex hull? And you've got a whole range of options. You really have to check out Brilliant. Go to brilliant.org/numberphile and check out their huge range of course, and quizzes. All sorts of great stuff. This is really gonna get your brain working. This is like, kind of going to the gym to make you smarter. Go to brilliant.org/numberphile and you can actually get 20% off a Premium Membership. Go and check them out. brilliant.org/numberphile And our thanks to them for supporting this episode.
I never quite understood the supposed relationship between Phi and sunflowers. This is great!
So any other ideas for a scale of "irrationality"? This one seems retrofit to the problem. It might be better to come up with some other definitions and see where they point...
I mean when I think of extreme irrationality I think of sequences of digits that have no closed form...random digits. Can't even make a decent series to express them.
Meanwhile his argument seems centered around that first tiny remainder...1/292, being so much smaller than the term before it. So maybe he is finding the slowest converging continued fraction? Is that equivalent to his definition?
Mathologer did a similar video on continued fractions and what it means to be the most irrational number.
This was beautiful. Thanks for sharing
I literally never even thought about the equation of x=1+1/x and how this equation is used to start so many lessons in math... And how it so eloquently works when plotting it in a diagram
just watched it and was about to post it here XD
God damn that is one fine video. I am a mere math enthusiast so I never knew about the concept of numbers being more irrational then others. The connection with how the sun flower evolved over time is even more fascinating now.
I'd like to see the flower patterns made from these numbers: http://mathworld.wolfram.com/ContinuedFractionConstants.html
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