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. They’re not on a phone,
they’re not on a tablet, they’re building something
with their hands, and that’s going to change
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the kit with your child, or it might be important to
have a hands-off approach and let them build something on their own and see it through to completion. The kit comes to your house, there’s really good instructions in there. The kid gets to work on
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built with their own hands. Ultimately, I just want you
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you just pay shipping, you can cancel the
subscription at any time. It makes a great gift, I
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.

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100 Responses

  1. Ben Schofield says:

    Holy cow, Destin how are you today?

  2. Doug McClelland says:

    Now that we have discovered the sign wave and function, lets see where this function is applied in the physical world. If your display on an oscilloscope a wave of electrical energy as seen on a antenna cable, often called standing waves or the pulse wave you would see in a TDR (Time Domain Reflectometer ) instrument, you will notice that you can find the same sine function based energy transfer in a wave of water that ripples across a still pond. If you were to put the water in a long square plexiglass tank and start a water in the tank, you could look at the water wave as it rippled along the tank wall and your would see a sine wave of water.
    The same laws of physic and sine functions apply to both the electrical pulse and the wave of water. The only difference is the physical matter is different. In one case the physical matter is the mass of the water molecules and on the wire it is the mass of electrons.

  3. pjb98422 says:

    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.

  4. RAKESH KUMAR says:

    Infinite thank you…………..

  5. David Hemingway says:

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

  6. Ger peter says:

    It is a delight to study these things indeed…

  7. Ralph says:


  8. Scientific_Mind says:

    Georgia Tech ♥️

  9. Audiophile Vintage says:

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

  10. Darin Schlemmer says:

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

  11. Poly Hexamethyl says:

    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?

  12. RoastHardy says:

    So anything could be built from rotating circles?

  13. Prathap says:

    Mind blowing ??

  14. HÜSNÜ CANBOLAT says:

    1:30 bilim ne güzel lan ??

  15. CarbonCore Multicopter says:

    Destin you are an upstanding person. Great video!

  16. John Rambo says:


  17. RONNY BHAIYA says:

    Who is watching it from India…hit like button

  18. amitkumar siddaraddi says:

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

  19. Jona OConnor says:

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

  20. Kazi Sati says:

    cooool bro!

  21. Marcos Alexandre P. de Souza says:

    Wow just subscribed. Really cool video. Kudos

  22. James Alexander says:

    Incredible work !!

  23. shad covert says:

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

  24. Abhishek Tiwari says:

    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

  25. Joseph Gassner says:

    I need access to this program.

  26. Max Mulhern says:

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

  27. Richard Corfield says:

    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.

  28. manju4ever says:

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

  29. Andronikos Kapsalis says:

    Amazing !

  30. Eric D'Amours says:

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

  31. Eidie Salim says:


  32. Daniel H says:

    Amazing helpful visual demonstration. Thank you!

  33. Michel Engelen says:

    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!

  34. Ramesh Chandra says:

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

  35. Jacques de Wet says:

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

  36. metal fingerz says:

    bilim ne güzel lan! greetings from Turkey.

  37. Nikhil Malani says:

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

  38. JOKER Company says:

    I like vlog about math

  39. kcirtapw says:

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

  40. Akram Rabah says:

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

  41. Francesco Gagliardi says:

    Great as all the other videos. thanks!

  42. Joe Blanco says:

    Not really impressed with a higher octave's

  43. 7thSon Mizo says:


  44. David Hunt says:

    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.

  45. Leo3ABP gaming says:

    Thumb up if you also store your old university notebooks 🙂

  46. Venkatesh Iyer says:

    Dude you're awesome


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

  48. Aristo Teles says:

    I have expected thousands of Turkish comments

  49. Aristo Teles says:

    Congratulations doğa from Turkey

  50. Aristo Teles says:

    Congratulations doğa from Turkey

  51. chaos says:

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

  52. Jim Moriarty says:

    Sorry, but it's pronounced as gif not gif

  53. Shahzad Hashmi says:

    Wonderful video

  54. blanca roca says:

    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.

  55. Giray YALÇIN says:

    did you study EE?

  56. TPHBLIB says:

    @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.

  57. TPHBLIB says:

    @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.

  58. Loris Lewandoski says:

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

  59. Manie de Bruin says:

    I am now drunk

  60. jak frost says:

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

  61. Nishant Celestine Tirkey says:

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

  62. Nico Brits says:

    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…. 😉

  63. Seyfullah Zahid says:

    Süpeerrrr.Türk birini görmek ne güzel.
    Supeerrrr.Very nice to see a turkish guy.

  64. Anderson Pyaban says:

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

  65. Marc Paul says:

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

  66. Sherwin Imperial says:


  67. Samwisegamgee The Brave says:

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

  68. Papa Rob says:

    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.

  69. stevecytfme says:

    Ok. I'm out. Over my head.

  70. stevecytfme says:


  71. daker macdonald says:

    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.

  72. VPXM says:

    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.

  73. Nothing says:

    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.

  74. James Bonanno says:

    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.

  75. Kanishka Wijayananda says:

    Amazing work guys keep on going

  76. gbob99 says:

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

  77. Bart Burkhardt says:

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

  78. Ruchir Rawat says:

    Destin : you know what are gifs ?
    Doga : am i a joke to you?

  79. Udo Matthias drums says:

    love it!!

  80. Udo Matthias drums says:

    chaos pendel!!

  81. C R says:

    That circle is very wonky

  82. Joseph Madratz says:

    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.

  83. 0n. GUVEN says:

    Can we use this in a 3d printerss i wonderr

  84. Girish Garg says:


  85. LYNDONisHERE says:

    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 🙂

  86. Subhash Chowdhury says:


  87. Subhash Chowdhury says:

    Please do it for Fourier Transformation

  88. Constant Rohmer says:

    How did he converted the brain-drawing into functions ?

  89. Amit Shende says:

    please share Matlab code for drawing any shape with Fourier transform

  90. Mu's says:

    As bayraklari As ??????

  91. norm colbert says:

    its time

  92. sankar r says:

    This is the best posdible explanation for fourier series.
    Thanks a million.

  93. Kevin Gilkey-Graham says:

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

  94. Alan Lenisa says:

    nice point of view

  95. marlontrujillo1080 says:

    Doha seems like a really d1k…

  96. hareti says:

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

  97. Eufalesio says:

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

  98. roladun says:

    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.

  99. Sergiu Muresan says:

    What tie knot is that?

  100. Arno Claassens says:

    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.

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