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  • 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 want to tell you about a very famous number that you've heard about before.

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