Early view of Fourier series!

The frequencies are all integers, in this case between from -124 up to 125. As to the amplitudes and starting angles, that's exactly what the video describes. Can you isolate which part was unclear?

3blue1brown

Hi Grant ! Great job as always. I was wondering how did you find the 250 frequencies and amplitudes to draw a given shape? (The Fourrier drawing in this case)

I think of this as just spelling out what the L2 inner product is saying in this case. It would be nice to properly make the connection to linear algebra here, discussing how computing a Fourier series is a kind of diagonalization. But it's probably best to have a little background in the relevance of eigenvectors to ODEs before any of that.

3blue1brown

The way you derive the coefficients is beautiful. And deriving them using the L² inner product is also beautiful but in a very different way. I wonder if it’s possible to reconcile these approaches in a similarly beautiful manner.

Roman Odaisky

Also, hello Tim 😉

I like how you managed to pose technical details as questions for real analysis, thereby encouraging curious viewers to go look them up.

I'm glad you enjoyed it, thanks!

3blue1brown

I expected none of this to be new to me, I have built an intuition for why transforms work and how coefficient selection works, but this made even more sense and managed to fill in a few gaps I wasn't even aware of, it's incredible... thank you

Joseph Fourier must have been so happy to find his cartoon drawn using his invention.

every time a new video comes out i think, man i can't thank this guy enough.... one thing i notice in a lot of videos though (not just yours grant) is that when there are caveats/disclaimers (such as at ~18 min in this video), it can be hard to read because of how when you pause youtube, the playback bar at the bottom and the title bar at the top become permanent and you can't truly see the full screen. i end up having to pause/unpause/rewind/repeat to read some of these quick bursts of text, which are often my favourite or just really drive the point home

And not sure if it was on purpose but the subscript font sizes for the coefficients looked a bit odd starting at 15:42 - especially for C1 where the '1' looks like a larger font size than the rest. Later, the font sizes changed a bit in the subscripts but that might have been a stylistic choice.

That was brilliant, Grant! I'd never thought of that intuitive motivation for the FT as freezing out each coefficient and letting all the others disappear in the average for each integer frequency. Very cool.

Well, at least as I've always seen things, "square wave" will refer to a periodic function, whereas at that timestamp the discussion and application are just over a finite interval.

3blue1brown

If you focus on any one particular input, even if at one point it's part of a "spike", at some point there-after that specific input will approach its limit point. If your unfamiliar, you may be curious to look up "uniform convergence" to get a sense of how mathematicians formalize the different ways that functions can approach each other.

3blue1brown

Hmm, maybe. Do you think that would actually make clearer why the computation works? I'd be a bit worried about distracting with an intriguing visual if there's enough cognitive load from the computation (which is the primary goal).

3blue1brown

Ah, thanks. That's actually where it usually is to make room for the end screen elements (not included in this draft video on the hidden channel). But your comment does make me think it might look nicer, in either case, to keep things properly symmetric, so I think I will change that.

3blue1brown

Hi Bob, thanks. This one was a little tricky to make because I have two different viewer types in mind. One is the core follower, who watches everything, and has come from the previous two chapters. The other is someone who hasn't seen those but is curious to learn about what Fourier series are. I did my best to include something which (hopefully) felt like a concrete example finishing off the heat equation discussion, leaving only the need to understand where these special constants come from (ending around 9:45), and then to have a discussion which concludes by understanding where those constants come from (in full detail if one engages with the exercise). I'm worried that if the final thrust of the video is all about the heat equation, the second viewer type might be left more unsatisfied than they need to be. What do you think?

3blue1brown

Hey Johannes, thanks for the feedback! I didn't end up plugging the Fourier Transform video because I thought the call-to-action might be better spent pointing to some other creators resources on Fourier series. E.g., if a viewer hops over to the Coding Train video, they'd get the chance to dig more into an example of code implementing this kind of thing. I tried to keep this as related to the heat equation progression as possible, while at the same time acknowledging that almost certainly people will be landing on this not having come from the previous two, but just wanting to learn about what Fourier Series are. Maybe that's a needless balance, but I did my best to answer the question which remained after the last chapter, while still making it feel conclusive to someone coming in from a different context.

3blue1brown

Not an error, but I think it would be really good to write e^(n×2pi×i×t) as e^(i×n×2pi×t) which sounds incredibly pedantic, but it better highlights that it's a purely imaginary number (so no growth), and that it's the n'th harmonic of the angle 2pi×t. Otherwise I love it!

Zac Harrold

(@David Andrew Kenny) The Gibbs phenomena spikes at the discontinuities tend to 0 width as the number of harmonics tend to infinity.

I saw the square wave over shooting which I first discovered was not a error on my spreadsheet. Its called Gibbs phenomenon and I'm really hopping you will do your magice with it one day. How can it trend to a square wave at infinity if the spike never gets smaller?

Hey, around the 20:00 mark, theres an opportunity for a bit more of a visual explanation for why it makes sense to "multiply by e^(-k2pit)". You could show an animation of the Fourier series rendering (of Fourier's face, say), but where the entire picture is rotating so that only that one vector is seen as constant, as the whole picture rotates around it. Not sure if it adds value to the discussion but would look neat.

Jacob Mirra

Really nice work, and a nice start toward the Laplace transform (for those decaying harmonics!). I have tried in the past, without success, but I'm still hopeful of finding some way to communicate to you a complete 3D visualization of W=e^S where S and W are complex so you might consider using it for explaining the Laplace transform and others. All you need to add to what you have already done in the past is an e^x horn inside the S plane cylinder, that is cut by the flat W plane, (I think its new but nevertheless very obvious once you see it!). Thanks again for all the outstanding work and videos.

Just FYI, the final "special thanks" scene is shifted to the right. Probably a glitch in the edit!?

Great video Grant! The only feedback I'd give echoes Johannes' above: perhaps include a quick summary at the end going over how Fourier series could be used to solve the heat equation (maybe using a specific example, like those you talked about at the start. As is, I felt more like I understood we could use the Fourier series to generate these images and generally express curves as a sum of sinusoids than how I could apply it to the heat equation (which isn't a great leap, but perhaps a concrete example/summary could clear it up for someone new to the subject). Then again, it's been a while since my PDE's class and maybe it's just me being slow. Great work as always! :)

Looks like magic!

Daniel Armesto

Hey I really loved the video. However I've got some feedback: 1. Around 11:50 you talk about how 'unusual' it is to think about a sine wave that way. However it's actually not (I think) since most people should be familiar with the sine circle diagram, where cos x is X and sin x is Y. Also why don't you plug your old video (series) on the Fourier transform? Furthermore it would be nice if - since this is part of a series about the heat equation - we could have an ending to that series (assuming it's over) where we look back over the process and summarize the general solution in 3 easy steps (compute fourier, use the cos part, ...). Also your subreddit recently had a question/discussion on how and why complex numbers and vectors are isomorph(ous).

Not as much an "error" but more of a semantics question: At timecode 8:30, wouldn't that be a square wave rather than a step function (since we're defining the value of the function to be 0 at the discontinuity)?

RedAgent14

Got here so early 360p was the only available. I'll let it sit a bit before I watch, but I really can't wait to see this since these circle drawings are some of my favorite things to watch.

Bpendragon