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
how they look at things. You might like to work on
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
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. Go to kiwico.com/smarter and select whatever kit makes the most
sense for the kid in your life. Get the first kit for free,
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.

100 thoughts on “What is a Fourier Series? (Explained by drawing circles) – Smarter Every Day 205”

  1. Wow! You are keen to pick up fascinating things! I'm a fledging phenomenologist. I love the p philosophical study of conciousness, reality, and experience. I love your channel! I'm working on some philosophical issues and wondered if you could use this application to show why light waves slow down and change direction through different mediums. I think they slow because they stack together to get a slower wave speed. How would stacking circle makers slow the speed of the wave? I'm working on areas of complexity and this geeks me out! Thanks Brother! If you really want to make me happy, see if bell's inequality can relate to how the fourier circles pass through Quantum filters. I have large questions! Thank you!

  2. I'm no mathematician or coder (MD by trade) but managed to get a square wave going – much like the one above with all the circles whipping around – using Javascript & p5 library. It was actually easy . The series is basically sin(wt) + sin(3wt)/3 + sin(5wt)/5 ……………. t is your time step , w = freq . The more terms the squarer. Getting the graphics looking good & moving was the tricky bit.

  3. Hey Destin, I'm not an academic but I am a science enthusiast, I've tried learning many things and look at stuff through the eyes of a philosopher…
    I've followed you and others like Veritasium, PhysicsGirl, PsiShow, FermiLabs, Etc…
    This Fourier series has peaked my interest in encapsulating a GUT or TOE …
    What if you applied that series to string theory…
    Where as matter is a simple wave function of a virtual uncertainty particle/wave, that simply builds on itself…

  4. 2:00  You have significant overshoot on your oscilloscope square wave pattern.  You need to adjust the capacitance of your oscilloscope probe to correct this.  Do you know what Gibb's Phenomenon and the Lanczos Sigma Factor are?

  5. Please build that out of Spirograph™ pieces! What an amazing "Retro-machine" illustration that would be.

  6. 6:19 I really love what u said there… "u may think it's a complete chaos, but it's not, it's complete ORDER" makes me think of how the universe is like….
    people say particle in this huge boiling kettle we are livining is in complete chaos… uncertainity of where the particles are and will be… uncertainty & probability, but people don't realize that this uncertainty… emerge from the idea of how we are limited in observing these "uncertain" particles with a clear eye… but deep instead of how they particle behave there might be an order. , we just don't notice it or know about it. After all they probabilities them is government by mathematical functions which determines the particles con-straits and abilities?

  7. You can presumably do this with any two dimensional line art, right? So with enough functions and such you could make say, the map of the continental USA (the outline)?

  8. At 77 years old I have experienced many times that I don't know everything. In addition though I majored in Physics and minored in math nearly 60 years ago (npn junctions were the thing along with having a bamboo slide rule) and more that I have not used for 50 years – I became a Federal Agent instead. I enjoyed so much your Fourier Series presentation I am tinkering with the myriad of apps out there that make math fun – super way to keep the brain growing while I get smarter every day. Thank you.

  9. To be fair oldschool oscilloscopes allow you to manually change the status of signals, if you use a function generator, you need to do some math. And if you want more sine waves, well you can use programs on newer oscilloscopes or additional generators and that way get similar outcomes.

    Guess what I want to say by this. is that yes there are many ways to do things with math, but honestly to me this isn't at all that impressive, since most of the electronics class students were just messing with sine waves and none of us ever got famous or anything for doing that. 😛

    Oh well, guess it's good that the world know what is being done.

  10. Destin, whatever you have to do keep that man in this country! He is brilliant and we need brilliant people. Point him at cancer or AI or ??? I noticed in the comments that he’s Dr Doga, not surprised.

  11. I remember how brutal that math was back in engineering school. This video somewhat takes the pain away from my memories of battles with the Fourier series.

  12. Hey. There's a problem with doing that. Please look into "Gibbs phenomenon". You'll always have an error.. and it'll be higher in the edges (at points of more impulsive jumps or jump discontinuities). A square wave has jump discontinuities and sines are continuous functions. There's no way you can simulate square wave exactly using linear combinations of sine functions. You can only get an approximation.

  13. Cool. But I already knew this insight during my bachelor physics at Ghent University. All the amplitudes of your fourier series are all the diameters of your circles that are chained on eachother and the frequencies are linked to the different rotation speeds of the circles. It is pretty simple actually.

  14. He is proud of the GIF, but remains annoyed that there is still any debate over the pronunciation of the format.

    “The Oxford English Dictionary accepts both pronunciations,” Mr. Wilhite said. “They are wrong. It is a soft ‘G,’ pronounced ‘jif.’ End of story.”

  15. One problem with is the ringing at the edges of the square wave is caused by the lack of band with gain of the device amplifying the signal. As a square wave has an infinite band width. This was demonstrated by Tektronix in the late 1960 when they came out with a super duper Sync Generator For Broadcasters. Thus came the rise time standard in broadcasting. The signal was too perfect. Check it out

  16. Well : now we know that between Leonardo Da Vinci ‘s flying machine sketches and the actual method of drawing them is nothing but pure fractal

  17. Woah I wish I had this channel during my IIT JEE , today's generation is lucky to take leverage of technology , this could compensate for bad teachers we had.

  18. I am interested to obtain the mathematical formula for the closed curve on the flat surface of the cross section of the coil -perpendicularly to the long axis, the coil is bent from the pipe of the diameter d on the coil radius D and wirh the pitch d. I think it is possible to do it by the method showed in this film. It will be the some kind of the ellipse, bent on both ends in one direction.

  19. I wrote a Fortran IV-E and DOS Assembler IBM 360/44 computer program in 1969 for Fast Fourier Transform for oil/gas exploration for fast digital filtering and fast deconvolution. I don't remember any of it. I just relax now I'm old. I think I'll switch over to the New Housewives Of Real Jersey after this.

  20. My interest is forecasting stock market prices. Could this be done by turning say the last 5 or so candlesticks on a chart into fourier waves ?

  21. 6:53 – My three girls had only dolls and blocks and paper for Origami, and other non-electronic toys growing up, and now my first is 5th year "Civil Engineering", by second was a child prodigy on the guitar, now second year in music, and my third was 3rd out of 154 aspirants for "Industrial Design". The kicker is that they were all Valedictorians in their prep schools.

    You're a GREAT DAD! to have her play with cool toys that challenge all the senses. I can vouch for it! I just sub'd!

    I noticed LOTS of cycloids as he added circles. Cycloids rock, and because of them, my hero, John Harrison, was able to invent THE most accurate clock without pisoelectrics or any other electronic means – pure mechanics with no power tools – the world has yet to see. I think he finally got into the World Record Book for having a clock after 100 days that lost about a second. My Japanese quarts movt. cannae do that!

    Great stuff! Looking forward to more!

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