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