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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?

• 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.

• 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.

• 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.

• 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