Today we're gonna talk about waves.

This is a circle, you probably knew that.

If we were to turn this circle on

and watch it go up and down and up and down

and trace that motion out,

you get what's called a sine wave, which you know

to be important in things like pendulum motion,

particle physics, things of that nature.

Sine waves are important but for my money,

the coolest thing about 'em is you can add them together

to do other things, which sounds simple until you realize

this is how the 2018 Nobel Prize in physics was won.

My buddy, Brady Haron, has a really good video

about that overall on Sixty Symbols.

There's some fancy math I learned

at the university called the Fourier Series.

These are my old notebooks and check this out.

The teacher challenged us to create this graph

by doing nothing but adding together curves.

And I found where I did it, it's right here.

And it took me, it looks like four or five pages, yeah.

It took a lot of pages and I ended up with this.

I was able to make the graph

by adding together a bunch of waves

and to demonstrate that, I created this.

I had to get a tripod, here's my flip book.

So it starts with one sine wave

and then we add another one and you can see,

the more waves you add together,

the closer the function gets

to what you're supposed to make,

because you can see that and that look very similar.

That's 50 waves added together.

So it's cool and it's one thing to

know how to do the Fourier Series by hand,

it's quite another to understand how it works.

And I didn't really have that moment

of it clicking in my brain until I saw

this awesome blog by a guy named Doga

from Turkey, he's a student at Georgia Tech.

I want to show you this,

this made it click in my mind unlike anything,

this transcends language.

So let's go check out Doga

and let him teach you how a Fourier Series works.

I'm in Georgia Tech, this is Doga.

- Hello.

- You have visualized, via animation, a Fourier Series

in the most beautiful way I have ever seen in my life.

- Thank you.

- Sine waves are probably the simplest kind of wave, right?

The second most simple kind of wave is a square wave.

But the difference is you have sharp edges in a square wave.

The first thing Doga did to impress me

is he used curvy waves to make sharp-edged square waves.

We have to add up different oscillations or simple

harmonic motion here. - Harmonic, harmonics, yes.

- [Destin] Yeah, and so, the first harmonic, n=1,

gives you this. - Yes.

- [Destin] Which looks nothing like it.

- Not to me interesting, just boring sine wave

and I add one more, it's actually like it.

I'm adding one harmonic and another one,

well one third of that harmonic.

- So you're adding a basic

well what are we going to call these, wipers?

- Yeah let's call them wipers.

- Okay so we're going to add a wiper on a wiper

and by doing that and we graft the function.

- [Doga] And then follow the tip of these wipers.

- [Destin] Yeah?

- [Doga] And then draw that with respect to time.

- That's awesome man!

Like this is really really beautiful

and really really simple.

- [Doga] So, I can add more wipers.

Making us more harmonics.

And I add.

Fifteen harmonics is something really cool.

- [Destin] Oh wow that looks like a whip.

- [Doga] Yes.

- So you're saying

so basically, here's the up-shot

a Fourier series you can create any function

as a function, or an addition

of multiple simple harmonic motion components, right?

- Yes.

- All Doga is doing is he's taking these sine waves

that we explained earlier and

he's stacking one on another sine wave.

He's stacking the circles, to add together these waves

to create a Fourier series.

These visualization techniques that Doga developed

worked on any version of any function.

For example on a sawtooth wave,

you can see at n=8,

how the Fourier series starts to play out.

It looks really cool.

How did you do this?

Like what program did you use to visualize this?

- [Doga] I used Mathematica.

- [Destin] Mathematica? - [Doga] Mathematica, yes.

- [Destin] Really? - [Doga] Yes.

- [Destin] So if I give you any function can you create this

but you had to flip it into video format somehow,

how did you do that?

- I exported in like, gif.

I created a table of the different times

of this animation.

And then I just exported those tables into gif.

That's all that I did.

- Okay, here's an interesting question, are people

It's actually "jif" I don't know if you know that.

(laughing)

So if I were to give you a function,

like if I were to give you

a super, super complicated function.

Like a really weird curve,

you could make a graphic like this?

- I can, yes.

- [Destin] So I can challenge you?

- Yep

- Let me explain what's happening here,

amongst academics there's this

thing that I just now made up,

called "mathswagger" and basically,

it's when a person is good at math

they like think they can do anything with it.

It's not like a prideful thing,

I mean Doga is a very humble person.

But you could tell he was very confident

in what his abilities with math were.

So I can challenge you? - Yep.

- Which is why I'm challenging him

to draw this with the Fourier series.

It is that Smarter Every Day thing

that you see all over the internet.

I totally am geeking out right now, I love this.

It's a hard image to draw using math,

it's got like curves right.

It's got little sharp points and switch backs.

It's self-serving for me,

so this is an appropriate challenge

for somebody that's demonstrating "mathswagger".

The problem is, he actually can do it.

He can model this using nothing but

circles and the Fourier series.

Which is completely impressive.

Check this out.

The first thing that he has to do

in order to draw this image

is to extract the x and y positions that he would need

to make functions for in order to make this thing work.

He then needs to create a Fourier series

for each one of those functions

so that he can add them together.

And as you can see, these first few were not winners.

I mean like no stretch of the imagination

could make your brain think this looks like

the side profile of a human head.

Everything's a bit derpy.

But as he starts to refine it,

and he adds more and more waves to the functions,

things start to hone-in and it starts to look really good.

At about 40 circles in this whole function,

things start to look really good,

and your brain would totally think

that you're looking at a drawn image

instead of a mathematically drawn function.

If you look closer at just one of these arms,

you would think that it's chaos.

But it's not, it's complete order

backed up by a mathematical function.

In fact, this is why I love math,

it's the language that describes the entire physical world.

We can approximate anything,

as long as you have enough terms.

This is the beauty of the Fourier series,

you take simple things you understand

like oscillators, sine waves, circles,

and you can add them together

to do something much more complex.

And if you think about it,

that's all of science and technology.

You take these simple things, and you build upon them,

and you can make a complex system,

that can do incredible things.

A simple thing can lead to something incredibly powerful.

Speaking of the power of simple things,

I want to say thanks to the sponsor, Kiwi Co.

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Kiwico.com/smarter, thank you very much

for supporting Smarter Every Day.

- I appreciate your work

and I just wanted to say that. - Thank you, thank you.

- That's why I came to Georgia Tech.

Thank you very much.

That's it, I'm Destin, you're getting smarter every day.

I'll leave links to his website below.

Have a good one - Thank you

have a nice day. - That cool?

If you want to subscribe to Smarter Every Day

felt like this video earned it

you can click that, that's pretty cool.

Whatever.

You're cool you can figure out what you want to do.

I'm Destin, have a good one, bye.