What is a Fourier Series? (Explained by drawing circles) - Smarter Every Day 205
- What up? 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. I reached out to Kiwi Co and asked them to sponsor Smarter Every Day a long time ago because this can change the world. They send a box to your house for a kid to open and build a project with their hands.
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The kit comes to your house, there's really good instructions in there. The kid gets to work on a project themselves, and at the end of the project they have something they built with their own hands. Ultimately, I just want you to do this for your children. Or a child you love. And I want more of this in the world.
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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.