(host, echoing) A number so perfect... we find it everywhere... Mystery.

Sacred geometry.

Sacred geometry.

A mathematical property hardwired into nature.

(echoing) Secret, secret, secret, secret, secret, secrets.

The golden ratio.

The golden ratio.

The golden ratio.

Whoa.

What's the answer?

What's the answer?

What's the answer?

Mystery.

Sacred geometry.

Sacred geometry.

Sacred geometry.

The golden ratio.

Secrets.

The golden ratio.

Wait, wait, wait, wait.

Hold on.

I mean, really?

Is there actually one special number that underlies everything from sunflowers to seashells, everything from pineapples and pinecones to the pyramids and the Parthenon?

I mean, a number that can link beauty and art, music and the human body?

One number that links nature's order to the rules of mathematics?

Well, some people think so.

But like Uncle Carl says, "Extraordinary claims require extraordinary evidence."

So let's take a closer look at what the golden ratio is really about.

I mean, after all, the universe is a strange place, full of surprises.

[mysterious music] Hey, smart people.

Joe here.

Which of these rectangles is the most perfect?

Give them a look.

Which one just feels most balanced, the most beautiful?

Did you pick this one?

That's a golden rectangle.

And many people believe that this shape is the most aesthetically pleasing quadrilateral that there is.

This one, not so much.

This one, ew.

Gross.

Gah, get that away.

In a golden rectangle, the ratio of its long side to its short side is exactly this.

This is the golden ratio, abbreviated as phi or "fee," depending on how you prefer to pronounce your Greek.

These numbers after the decimal point, they go on forever without repeating.

Like the better known pi, phi is an irrational number.

It's irrational because it can't be written as the ratio of two integers.

Five, that's rational because we can write it as the integer five over the integer one.

The number 0.75, also rational.

We can write it as three over four.

Even 0.3333 infinitely repeating, that's rational because it can be written simply as one over three.

But what about the diagonal of a square whose sides are one unit long?

The Pythagorean theorem tells us that the diagonal has a length of the square root of two, which is a number, but one that can't be written as a ratio of two nice and tidy integers.

It's irrational.

Likewise, phi also can't be written as a simple integer ratio.

And an ancient Greek named Euclid was one of the first to notice that.

This Euclid guy was big into geometry.

In fact, most of the geometry that we learn in school is named after him.

It can be a pretty big deal to get a whole section of math class named after you.

So around 300 B.C., Euclid wrote a book called "Elements," a collection of most of what was known about math at the time.

And until the 20th century, it was the best-selling book ever other than the Bible.

Euclid noticed that there was one special way to divide a line where the ratio of the whole to the longer segment was the same as the ratio of the longer segment to the shorter one, and that ratio is phi.

Well, Euclid called it the extreme and mean ratio, which sounds like what happens when I make a bad tweet.

The names phi and golden ratio, they didn't show up until almost the 20th century.

Anyway, the Greeks and mathematicians of that time, they didn't think of numbers like we do as these strings of digits from zero to nine.

To them, phi was this ratio.

Just like to them, pi wasn't 3.14159 et cetera.

Pi was just the ratio of a circle's circumference to its diameter.

This is literally the golden ratio.

And you can do some weird stuff with it.

The ratio of the long sides of this triangle to its short side is, you guessed it, phi.

This is a golden triangle, also called a sublime triangle.

And the angles of that triangle are 72, 72, and 36 degrees.

Now, if I divide one of the long sides according to the golden ratio and make a smaller triangle there, it's another golden triangle.

Same angles and all.

And that other triangle we just created, the length of these sides to the base is one over phi.

We call this squatty shape the golden gnomon.

And if I take one golden triangle and stick two golden gnomons on the side, I get a regular pentagon.

Yeah, we're just getting started.

Let's overlap two golden gnomons and add a smaller golden triangle on the side.

You make a pentagram.

[rock guitar music] And going back to our golden rectangle, if we put another golden rectangle here and another inside that and another and so on and so on and draw a curve through all these shapes, we get a shape called the golden spiral.

If that looks familiar, it's probably because you've seen an image like this before on the Internet, and we will be getting back to that very soon.

Oh, there's even more strangeness.

If you multiply phi times itself, that's the same as one plus phi.

Take one over phi and that's the same as phi minus one.

This is a weird number.

Okay, fine.

So, what?

Phi is weird.

There's infinite numbers, so some of them are gonna be a little strange.

What makes phi special is that it shows up in a bunch of really unexpected places that are pretty far off from geometry class.

Or at least people claim to find phi in a lot of unexpected places.

And this is the really interesting thing about phi, because as cool of a number as it is on its own, to achieve this almost mythological status-- I mean, this is, like, the Elon Musk of numbers.

And many people say that because we find it in so many places, it can't just be a coincidence.

It must be a sign of some deeper secret about the universe.

Well, where does the real story of phi end and the myth begin?

Well, if there's one person responsible for the mythological status of phi, it's this guy, Leonardo of Pisa, aka Fibonacci.

Around the year 1200, Fibonacci was responsible for bringing Hindu Arabic numerals into common use across Europe.

These are the numerals that we use today, zero through nine.

And merchants quickly realized that doing arithmetic with these was way easier than Roman numerals, which is what everyone in Europe was using at the time.

So to teach people how to use these "new numbers," which had actually already been in use in Asia for, like, 1,000 years, Fibonacci wrote a math textbook, "Liber Abaci," which just means "The Book of Calculation."

This book was full of math problems to teach people how to add and exchange currencies and divide and multiply with these new numbers.

And tucked inside of chapter 12 was this weird problem about rabbits, doing what rabbits are known to do, that would end up making Fibonacci famous.

Imagine you have a pair of rabbits in a field, one male and one female.

No rabbits die or get eaten.

Starting from the second month she's alive, every female reproduces each month, making a new pair of rabbits, one male, one female.

So how many rabbits will they produce after one year?

You can pause and take a minute to work it out if you want to.

But it ends up looking like this.

You might notice something special about the number of pairs of rabbits.

Each month, the number of pairs is equal to the sum of the previous two months.

And after 12 months, you'd have 144 pairs.

This is the famous Fibonacci Sequence.

You can carry it on forever.

Just add the previous two numbers to get the next and so on until the end of the universe, or until you get bored.

The reason that we're talking about the Fibonacci Sequence in a video about the golden ratio is because as you go on in the Fibonacci Sequence, the ratio between numbers gets closer and closer to phi.

In fact, any sequence of numbers that follows the Fibonacci rule, adding the two previous to get the next, trends to phi.

Like this set, the Lucas numbers.

Follow the pattern and carry it on, and the difference between the terms all trends to phi.

Yeah, I know.

It's weird.

But Fibonacci never made that connection himself.

A guy named Johannes Kepler did a few hundred years later.

The same Kepler who figured out the math that explains how planets move.

Pretty smart guy.

It's after that, when the Fibonacci Sequence and phi got together, that the myth really took off.

And people started to claim these numbers were more than just numbers.

"Despite phi's seemingly mystical mathematical origins, "the truly mindboggling aspect of phi was its role "as a fundamental building block in nature.

"Plants, animals, and even human beings "all possessed dimensional properties "that adhered with eerie exactitude "to the ratio of phi to one.

Phi's ubiquity in nature clearly exceeds coincidence."

That is how one of the greatest writers in all of history put it.

And that's really the question, isn't it?

Does phi, the golden ratio, the divine proportion, whatever grand name you want to give it, really show up everywhere in nature?

Or is it our pattern-sensing brains making us think that we see it everywhere?

Like when you notice a license plate from another state and suddenly start seeing out-of-state license plates more than you used to or think you used to.

Well, let's look at some places people claim to see phi.

The human body.

It's claimed that the ideal ratio of a person's height to the distance from their naval to their feet is phi.

I probably don't need to tell you that beauty standards and different cultures vary a lot.

And people come in way too many shapes and sizes for that to be a rule.

The Great Pyramid of Giza, the Parthenon, Notre Dame Cathedral in Paris, the Taj Mahal-- a handful of ancient buildings that people claim were built with golden ratio dimensions.

The thing is, for an object that's fairly big, like a building, or complex, like a body, there are so many ways to measure it and so many measurements you can take that some are bound to be somewhere around a golden ratio apart from each other.

I mean, this video is a 16x9 aspect ratio.

That's pretty close to 1.6.

But it's not phi.

16x10 would be even closer, actually.

Lots of places that people claim to see phi in nature are just plain wrong.

Like the ratio of one turn of a DNA helix to its width.

Google that, and you'll see results that say it's 34 Angstroms high per turn and 21 Angstroms wide.

Both Fibonacci numbers.

Ooh, intriguing.

Unfortunately, that's wrong.

These are DNA's actual measurements.

Here's the key thing in any example that you find.

Phi isn't "approximately" 1.6, give or take.

It's exactly this.

If people go measuring things looking for the golden ratio, they often measure them in ways that ensure that they find the golden ratio.

Our brains love patterns.

And once we learn a pattern, like the usual arrangement of a mouth and a nose and two eyes to make a face, we see that pattern everywhere.

And that brings us to this, a nautilus shell.

A nautilus is a cool little mollusk-y thing that swims around with a spiral shell and a face full of spaghetti, and it's become basically the official mascot of the golden ratio in nature.

The claim is that if you trace the spiral of this shell, each ring is a golden ratio away from the next smallest ring, et cetera.

But people have gone out and actually measured loads and loads of nautilus shells, and they aren't golden spirals.

The ratios vary quite a bit.

Like snail shells or sheep horns, it's an example of what's called a logarithmic spiral.

I mean, it's really cool.

Each turn of the spiral grows by the same proportion because the nautilus grows at the same rate.

But that proportion isn't phi.

And that's too bad, because logarithmic spirals are cool, but no one pays attention to them because of this obsession with phi.

My point is, close is not enough.

If phi is a fundamental building block in nature, we should we able to show that it's more than a coincidence, that there's some reason behind it being there.

Which brings us to these.

Not every claim about seeing phi in nature is fake.

Phi does show up in nature in a really interesting way.

And if you've ever looked closely at a pineapple or a pinecone or a sunflower or an artichoke-- I don't know who closely studies artichokes, but maybe you have-- all of these plant parts show a special kind of spiral.

Let me show it to you.

Here's a pineapple, and if you notice, on a pineapple, there's these spirals going one direction like this, and then we can see a spiral going the other direction the other way around the pineapple.

Let me trace this out and make it a little easier for you to see.

So a little arts and crafts time here on "It's Okay to be Smart."

This is gonna be fun.

There's 13 spirals in that direction.

Well, now, let's count these spirals in the other direction.

So we've got 8 spirals going in one direction and 13 spirals going in the other direction.

And if those numbers sound familiar, that's because they're both Fibonacci numbers.

And you remember from before that within the Fibonacci Sequence, we find the golden ratio.

Okay, that's just one pineapple.

And maybe that's a big pineapple conspiracy coincidence.

So let's count something else.

I don't know if you've ever noticed this, but pinecones have these adorable little spirals too.

Let's see if there's any phi magic going on in there.

Eight spirals going that direction.

Whoa.

That's pretty spooky.

13 spirals going the other direction.

Those are Fibonacci numbers too.

Pinecone, pineapple.

Maybe these should have been called "phinecones" and "phineapples," but I digress.

We can also find Fibonacci numbers in these sunflowers, this rose, this cauliflower, this succulent thing.

This is fun.

Let's count the spirals on this artichoke too.

And five spirals going in that direction.

Eight spirals going in that direction.

Five and eight are Fibonacci numbers as well.

And there's this, the branch--ow.

Ah.

Hold on.

I came prepared for this thing.

And this, the branch of a monkey puzzle tree.

Yes, that is its real name.

This looks like some sort of medieval weapon for, like, a plant knight or something.

We can count the spirals on this thing too, very carefully.

I should be wearing safety goggles for this.

And eight spirals going in that very dangerous direction.

Come here, you.

13 spirals going the other direction.

Plants can't do math.

They can't count.

So why is there this connection?

Well, imagine that you're a plant.

You eat light.

So the more sun you can catch with your leaves, the better.

So as a stem or a branch grows up or out, where do you put your leaves?

Let's put one leaf right here.

That looks fine.

And then let's go, say, half a turn to where we'd be as far as possible from the first leaf that we laid down.

We could do our half turn rule again, and--well, wait.

Now we're on top of the first leaf, and if we continue this, well, we're not gonna be catching maximum rays, man.

Let's start over.

And let's pick a--let's pick a new fraction of a turn.

Why not a third?

So we put our first leaf down here.

We can turn a third of a turn right here.

We turn a third of a turn again for our next leaf right here.

Well, it's looking pretty good so far.

A third of a turn from our third leaf-- oh, now we're back over the top of our first leaf again.

So that's not gonna work either.

Maybe one over four.

So we can start here.

A quarter turn, a quarter turn again, another quarter turn.

There are plants that actually grow like this.

But we can quickly see as we continue this pattern, we overlap our leaves again.

This can't be the best strategy out there for catching maximum sun.

And it turns out that if we use any fraction of a circle with a whole number on the bottom, our leaves will eventually overlap.

A rational number isn't gonna work.

But what if we used an irrational number instead?

Remember that irrational numbers can't be expressed as simple integer ratios.

And it turns out that phi might be the most irrational number that there is.

So let's put our first leaf down, and then let's go a fraction of a circle one over phi turns around.

Now, remember that one over phi is equal to this.

So if we express that as a fraction of 360 degrees, we're taking a turn of about 137.5 degrees each time, which probably won't surprise you is called the golden angle.

I've made this little guide that's exactly that angle.

So let's fill in our leaves using our new phi guide.

All right, so we go 137.5 degrees from our first leaf and we lay another one down.

137 1/2 degrees from there and lay down our third leaf.

And we see our new leaves fill the gaps left from the leaves before.

We never overlap so far.

Let's keep going and see what happens.

One, two, three, four, five, six, seven, eight.

The Fibonacci number of spirals, they form all on their own, just from that golden angle turn rule.

Why do these spirals form?

You can actually do this yourself and play around with different sized leaves or petals or whatever, and you'll find that with the golden angle as your guide, you always put down a Fibonacci number of things before you get to these layers where things start to... almost overlap, but not quite.

And the spirals just sort of happen from there.

And there's a Fibonacci number of them.

This works for more than leaves catching sunlight too.

It's also a useful pattern for catching rain and funneling water down to your roots.

Or for packing more seeds and flower petals into a small space.

Looking at you here, sunflowers.

These are all things that can make you a better plant.

Quick side note... we didn't put a camera over there.

I'll just stay over here.

Quick front note: this explains a lot about why plants do this, but the how is a lot more complicated.

And scientists are still working out a lot of the details.

What we do know is that instead of having some internal leaf angle growth measurer thing or something, all these angles, well, they have to do with newly growing plant parts repelling other nearby plant parts.

Kind of like how the poles of magnets, if they're alike, they repel each other.

Only this is thanks to growth hormones and not magnets.

But the point is, there's actual biology and chemistry underneath all of this.

So, anyway, it's not like there's a gene in plants that's programmed to do math or something.

But plants have had gazillions of years of evolution to find the best way to do all the cool plant stuff that they want to do.

And it's not like every plant even does this.

Plenty of plants follow different rules, and it works just fine for them.

And that's all that really matters in evolution, is that something works well enough, not that something reaches some mathematical and irrational golden perfection.

But, I mean, you can't deny this is kind of beautiful.

And I think that's probably a big part of why we're attracted to this pattern more than other plant patterns, because, you know, ape brain like pretty pattern.

And that's the funny thing about beauty.

It can take so many forms.

We know that some artists, like Salvador Dalí or the architect Le Corbusier, occasionally use the golden ratio purposely in some of their work.

But, I mean, there's plenty of beautiful art that makes no use of the ratio too.

You remember when I asked you to pick the most beautiful rectangle at the beginning?

Lots of you probably picked one other than the golden rectangle.

Math follows very particular rules.

And things like beauty and life, they're a bit more messy because the world is a messy place.

And that's part of the beauty, isn't it?

That's okay because sometimes, in the middle of that mess, if we look hard enough, we can find some order after all.

Stay curious.

Okay, wow, this is-- this is way creepier than I thought it would be.

Can I be a person again?

I don't--I don't really like being a tropical fruit.

Anyone?

Hello?

Hello?