# 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

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really believe in Kiwi Co. 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.

Holy cow, Destin how are you today?

Wow! I failed Fourier Transforms at third year electrical engineering. This would have got me through. Great work, now i'll look up Laplace Transforms. Failed that too.

Infinite thank you…………..

Could you generate piano sound with this method? Like additive synthesis.

It is a delight to study these things indeed…

IMPORtant NOT IM-POR-DENT

Georgia Tech ♥️

Absolutely top video, thank you very much. I’ve learned a lot.

It's gif, not jif. Do you call it a jraphics interchange format?

Nice visualizations! So far, it shows the terms of the Fourier series and how they sum to approximate the target function. To take it a step further, I wonder if there is a way to visualize the systematic procedure by which the terms of the Fourier series are obtained from the original function?

So anything could be built from rotating circles?

Mind blowing 💥💥

1:30 bilim ne güzel lan 😂😂

Destin you are an upstanding person. Great video!

Amazing

Who is watching it from India…hit like button

Sir, Also plz make video on practical implications of Laplace transform and fourier transform..

whU_Ut whaaaaaaaaaaaaaaaaaauw, My question is how many of those whipers do you need to draw any kind of line!?

cooool bro!

Wow just subscribed. Really cool video. Kudos

Incredible work !!

So, literally the one and only thing I understood was the GIF reference. I'm pretty regular.

I also use Mathematica but I can not create such type of thing but all I understood from this is that we to use Curve Fitting in Mathematica

I need access to this program.

Hi

At 2:45 it looks like Ptolemy's geometric constructions to expalin planetary motion i.e. deferents and epicycles. Would you agree?

The fun bit is understanding how the Fourier integration works – how when you sum over all time the product between a sine wave and your function you get a magnitude which is your Fourier coefficient at that frequency. If your input is a sine wave and you graph the Fourier transform you get an impulse at that frequency. At the given frequency the result is infinite. As you wonder slightly to the side then over infinite time the errors add up and the result is zero. So the impulse is infinitely narrow and infinitely high and mathematically has an area equal to the magnitude of your sine wave.

So I’m basically a Fourier series. Got it.

Amazing !

Hé! C'est la même chose quand on joue de la musique. De simples notes bien ordonnées font des choses extraordinaires! : )

OMG!!!

Amazing helpful visual demonstration. Thank you!

you say "I play jolf" or "I play golf" ? it's a G man 🙂

Anyways, these math visualizations are amazing for students to help them understand!

Doga will say "gif" as in girl for "jif" if he is a egyptian. But for "Egypt" they say "Ejipt" only..!!!

Do all the wipers have the same angular velocity? What happens with differentiating angular velocities?

bilim ne güzel lan! greetings from Turkey.

Bruh we only speak fourier series at georgia tech… For some reason I expected him to go to tech…

I like vlog about math

When the G in GIF stands for Graphic, how is it still JIF? It’s gif as in gift, NOT JIF as in Jrafic.

Retired after a long career and still feel like a college student. Wow

Great as all the other videos. thanks!

Not really impressed with a higher octave's

Wow

No, that is not illuminating. The addition of waves is an addition of waves and nothing else. BTY he pronounced giv correctly it is Giga that is pronounce with a J. You have the swagger but not the punch.

Thumb up if you also store your old university notebooks 🙂

Dude you're awesome

Thats simply mind-blowing facts you've just shown….

I have expected thousands of Turkish comments

Congratulations doğa from Turkey

Congratulations doğa from Turkey

Wow! I have a degree in acoustics, and it still blows my mind to see it presented like this.

Sorry, but it's pronounced as gif not gif

Wonderful video

Beatiful visualisations animations graphics.. Wish I had them when i was a kid. Fourier derived formula for getting the amplitude and phase of the oscillations which build up the resulting function you want. However it has certain limits on discontinuous functions etc and we see this very nicely as including more and more components gets better BUT around 2:58 and 3:33 there we see an overshoot which doesnt seem to be going away anytime and in fact was all predicted hundreds of years ago by these great mathematicians who kind of saw these graphics in their heads apparently.

did you study EE?

@UC6107grRI4m0o2-emgoDnAA why n=1, or 3 or 5 …in short why only ODD numbers are used, when the fact of the matter is that there are two wipers used for n=3 and 3 used for n=5? Please respond.

@UC6107grRI4m0o2-emgoDnAA why n=1, or 3 or 5 …in short why only ODD numbers are used, when the fact of the matter is that there are two wipers used for n=3 and 3 used for n=5? Please respond.

Fourier transforms, fourier series, cosine transforms, k-l transforms, etc is what i've been learnin these years.

I am now drunk

First time I wished I could like a video more than once

Interesting thing in this video is that you are wearing the same shirt I had in my 11th or 12th grade. LOL

I hated Fourier when I did engineering Maths, back in Noahs class. I think, for the first time, I now understand why. You kids, you thing your so smart…. 😉

Süpeerrrr.Türk birini görmek ne güzel.

Supeerrrr.Very nice to see a turkish guy.

cool. I did this in my Engineering mathematics but never had the chance to visualize the behaviour of the equations this way

Finally, thanks to you, Destin, and Doga, I can visualize additive functions with a Fourier series! Thank you!

impressive!

Thats elegant. Im a caveman so what do I know but I'll be damned if that is not elegant.

Smarter you may be Gustin, but you don't dis your guests in some perverted form of humor. Especially if you are wrong! It's not Jif as in peanut butter. Gif as in Humility.

Ok. I'm out. Over my head.

Derpy?

I don't believe that math is the language which describes the physical world. It's more so a translation of observation. Approximating the visual observation to a language that can be recorded, studied and shared. Like, a relative map for future studies.

Gifs are not pronounced Jifs. It's not peanut butter, or somebody's name. You can add to a language, but you cannot start changing pronunciations… otherwise pretty soon everyone would be speaking "jibberish". I've been around long enough, to study beginning techniques of gifs. Get with the program dude.

You don't need Mathematica to do this. Unless you have a Raspberry Pi, it'll set you back a couple hundred bucks. I've done fourier series in Geogebra, which has a standalone or web based version. Or Desmos, which is a bit easier to learn and runs in a browser. I've also made simulations of electronic circuits which can approximate fourier series.

Wow that was absolutely amazing! I haven't seen something on YouTube that caught my interest so well in a long time. I subscribed and I hit the Bell.

Amazing work guys keep on going

Is "The Roots of Unity" concept considered a Fourier Transform?

I am curious why he was showing the fractal at 4:27, what was that? does it have a relation to the fourier series?

Destin : you know what are gifs ?

Doga : am i a joke to you?

love it!!

chaos pendel!!

That circle is very wonky

What is different between this method vs finite fourier series method taught in college math courses. Is this just a way to visualize what’s happening when sine and cosine terms are added via the Fourier series or is it a different technique.

Can we use this in a 3d printerss i wonderr

3b1b

Im in an intro to quantum class and this has honestly been pretty helpful considering how mentally in debt I fell towards the class. Cheers Mate 🙂

Excellent

Please do it for Fourier Transformation

How did he converted the brain-drawing into functions ?

please share Matlab code for drawing any shape with Fourier transform

As bayraklari As 🇹🇷🇹🇷🇹🇷

its time

This is the best posdible explanation for fourier series.

Thanks a million.

Reminds me of something that would be used by the mechanics of the automata

nice point of view

Doha seems like a really d1k…

I want to see the list of sine waves that makes up the neon sign.

it’s not pronounced “gif”, it’s pronounced “gif”

Perfectly said order out of chaos that came to my mind when I first watch this on 3Blue1Brown channel on Fourier Series/DE, I recommend all to watch that.

What tie knot is that?

Great addition to this video is 3blue1brown's video on it. Both give different ways to understand the topic and in conjunction they give a great intuition.

Go to a top 10 university and 0 of my professors have been able to explain the underlying concepts of Fourier series this well.