But what is a Fourier series? From heat flow to drawing with circles | DE4

Fourier series, from the heat equation epicycles.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/de4thanks
12 minutes of pure Fourier series animations: • Pure Fourier series an...
Some viewers made apps that create circle animations for your own drawing. Check them out!
/ make_your_own_fourier_...
isaacvr.github.io/coding/four...
Thanks to Stuart@Biocinematics for the one-line sketch of Fourier via twitter. As it happens, he also has an educational KZbin channel:
/ @biocinematics
Small correction: at 9:33, all the exponents should have a pi^2 in them.
If you're looking for more Fourier Series content online, including code to play with to create this kind of animation yourself, check out these posts:
Mathologer
• Epicycles, complex Fou...
The Coding Train
• Coding Challenge 125: ...
Jezmoon
www.jezzamon.com/fourier/index...
For those of you into pure math looking to really dig into the analysis behind this topic, you might want to take a look at Stein Shakarchi's book "Fourier Analysis: An Introduction"
Timestamps:
0:00 - Drawing with circles
2:10 - The heat equation
6:25 - Interpreting infinite function sums
9:52 - Trig in the complex plane
14:11 - Summing complex exponentials
22:11 - Example: The step function
23:54 - Conclusion
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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These animations are largely made using a custom open-source python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
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Instagram: / 3blue1brown_animations
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Facebook: / 3blue1brown

[@pablotano352]

After listening to a lot of teachers, I have to say that you are simply the best explainer I know. Probably one of the greatest teachers of all time. By explaining complex and powerful ideas in such a beautiful and simple way, your work will surely have a big impact on the life of future scientists and engineers. Keep it going man, you are doing an amazing job!

[@ollerich32]

I have to subtract e to the i times pi from this comment

[@aryamanmishra154]

where did you do your PhD from?

[@Ludix147]

He truly is very good at what he does! You can tell that he has a good understanding of when his explanations are good, and you can tell he takes a lot of time to get to that point.

[@gaeb-hd4lf]

Completelly agree

[@wengeance8962]

I didn't do a PhD in physics and I think the same

[@DennisDavisEdu]

Grant, you have the unique trifecta: 1. Intelligence to understand these complex topics. 2. Ability (and willingness!) to explain them clearly. 3. Technical chops to animate and edit your explanations. Nothing you do is easy but it is all appreciated by a wide audience.

[@tanmayjaiswal5935]

Agreed. He is easily one of the best, if not THE ABSOLUTE BEST

[@randomdude9135]

Thanks for adding chops and trifecta to my vocabulary.

[@alex_evstyugov]

That is not a unique trifecta by any stretch of imagination. Many have it, and more to the point everyone can. *You* can have that trifecta. If you don't, that's your own choice.

[@whitewalker608]

A lot of Stanford/MIT/UCB etc Computer science graduates have all 3.

[@PrimerBlobs]

There's also a strong instinct for visual beauty, a tendency toward warm communication, and memory/empathy for what it's like to not know the concepts yet, which are what make 3b1b really stand out, IMO

[@legovd1018]

Somehow this wound up in my recommended, and I’ll be frank here. I’m in 10th grade and am nowhere NEAR the level of algebra and calculus comprehension required for this, but you explained this complex subject so well and fluently and thoroughly that even I managed to roughly understand some of the concepts! Excellent work, and keep it up!

[@foxphire0093]

I (a Junior Electrical Engineering Student) went through the same experience time and time again. But, remembering these after a time of learning is a great way to see how far you have come with understanding and knowledge; I did that as I was watching this because I’m going over Fourier Series right now in my Signals and Linear systems class, and I said to myself “wow, I’ve learned a lot, I actually understand where these are coming from!” when I watched this last year, I had no idea where things were coming from, but after learning about the fundamental concepts and applying them, the fog cleared and I actually recognized things. All of this to say, always go back to videos like this to see what you recognize to see what you now know - it’s the joy of learning.

[@andiehammettz4u265]

Agreed. How in the world did I just receive that when I'm mathematically illiterate yet artfully wired?

[@foxphire0093]

@@andiehammettz4u265 Because there is an art to mathematics, art comes from mathematics in the end as it describes everything in the universe

[@andiehammettz4u265]

@@foxphire0093 but when I draw I'm not thinking of a formula. The creativity is not pre formulated outside of how the parts relating to the whole. Form AND function.

[@will8070]

i’m a sophomore too and i’ve just been watching a bunch of his random videos and i watched this one like a couple weeks ago lol

[@mkoyama]

I am a Math Ph.D holder who's been both teaching and researching the Fourier related field and, HOLY CRAP this is one of the best visual explanation I've seen. I will definitely distribute a reference to this channel wherever I will go.

[@roncho]

mr Grant is a genius teacher!!!

[@user-ny3ke1iu1s]

The mathematics for Alfred Hitchcock, cartoon drawing, is about the same as a diamond Anvil.

[@AG-lm5uf]

I dropped out in 10th grade 25 years ago and your videos have inspired me to go back to school.

[@mjolnirforsworn]

It's never too late!

[@JimmyDuong97]

u just dont learn that in school but in uni. indeed, hes a great lecturer

[@ronaldjensen2948]

Amir, I'm about your age and just finishing a bachelor's degree... If I can, you can... And one of my favorite math teachers is also named A. Ghoreishi. :)

[@skydragon3857]

How do you go back anyways idk how it works

[@ronaldjensen2948]

@@skydragon3857 Look for a local college/university, visit their web-site and find the apply button... Also, look for non-traditional student services for help, advice and support.

[@spudzzy0373]

I’m imagining students watching this and furiously taking notes while I’m here at 2am thinking “oooh circles make shapes”

[@paulgoogol2652]

Most average students: "well it is 3am for me"

[@Wilker_uwu]

i'm bad at math ;-;

[@MrParry1976]

well it is 3 am right now here

[@abicol6010]

This video came out while I was taking my intro diff EQ class. Didn't see it till now at 3am. Would have been helpful at the time since my Prof gave zero context for how these equations work and how they can be applied.

[@ishworshrestha3559]

Ok

[@lol12313]

As I begin my journey to get a physics degree, I can't help but look back and thank you for all the content you've made. Truly inspiring and educational stuff :)

[@colummahony3364]

Such a generous guy, thanks for supporting this stuff on KZbin

[@spongbong0]

please give me tips as I also would like to have a degree in physics

[@lol12313]

@@spongbong0 im innately a curious person, so physics is a pathway for me to understand how things work. Im particularly interested in how things work on a fundamental level and look forward to learning about relativity and QM in depth. For me, choosing physics is like following a passion; for you, i dont know what physics might be, so take some classes and find out!?

[@spongbong0]

@@lol12313 I've expressed my feelings in that exact same way once. 🤣 I see we have similar feelings

[@Yomello69420]

​@@lol12313 I want to be an astrophysist please give me some tips on how to get good at physics I practice questions daily but I don't know what else to do

[@bleepin_bitch]

You are creating a generation of better engineers and researchers with your channel

[@Randomstuff-wx2ku]

Agreed

[@andtherefore8076]

True!

[@HotDogMarchant]

I’m in the process of writing my thesis for a PhD in theoretical physics and I’ll be completely honest, I cried at how elegant this explanation is. Thank you for helping someone who feels they’ve lost a lot of passion for maths and physics after years of hard work, realise that they still have the capacity to really care about these subjects. Truly thank you.

[@joynew1914]

I really admire those who study mathematics and theoretical physics.

[@austinconner2479]

What made you lose passion along the way? I just finished my thesis for a PhD in math and I'm still excited about the math itself. Worst parts of academia are still stuff outside the math (grant writing, etc..)

[@NovaWarrior77]

Oh man. Happy for you sir.

[@ashutoshsamantaray2572]

@@thesickbeat ya i am annoyed by these type of comments but then i feel bad abt promoting for some1 to not speak what they want

[@vutruongquang3501]

It's been 11 months, how are you going George ? I will you're all good and healthy ^^

[@lightyearahead]

Math is even more beautiful when someone teach like this.

[@chrissetyanwar]

Yes, i agree.

[@moujeshagrawal9393]

Teaches*

[@itsawonderfullife4802]

Yeah it almost becomes sexy.

[@Bjowolf2]

@@itsawonderfullife4802 Almost?!? 😉

[@randomdude9135]

I was your 1200th liker

[@skymaster4743]

The global academic ecosystem needs more people like 3Blue1Brown to teach Maths to students.

[@vmajed7103]

can't you just say "students" instead of all you wrote at the beginning?

[@helmanfrow]

It's amazing to see all these concepts described with animated drawings. I find it even more amazing that generations of people were able to learn and apply the concepts well before the advent of moving pictures. To really internalize these ideas and visualize these things in your mind must be an amazing experience.

[@eunhyoukshin7777]

Joseph Fourier must be so proud to have a Fourier-portrait of himself

[@gcewing]

And when it's built up from frequency components one at a time, it kind of looks like he's developing from an embryo!

[@alpineblob]

This is pure gold for an engineering student like myself. I hope you one day get the recognition you truly deserve for illustrating the ideas this beautifully and clean. Thank you.

[@FilippoSolari]

same for me... as an engineering student, i've never fully understood the math behind Fourier, limiting myself to apply it mechanically. I know what an FFT and the Fourier series are built for and their importance in signal processing, but the math behind them remains for me some sort of magic even after Math III exam. This, damn this is totally mind blowing.

[@ducanhdinh8574]

I've never known what even is a "differential equation", and I'm definitely never studying this kind of advanced maths in college. However, this is one of the best youtube series I've ever seen, and this super well-articulated, extremely elegant video has prompted me to express my utmost appreciation for your content, Mr. Sanderson. Truly the best Maths channel in the world.

[@skyclaw]

I’m sure I’m not the only fully trained mathematician who watches 3B1B purely out of appreciation for his teaching style.

[@MetapeterUndMetagreta]

Not only the style but this incredible calm and... special voice. If someone would exactly do what he did with a normal or "bad" voice: No one would listen to him/her. (crazy thesis)

[@jameshakai1662]

​​@@MetapeterUndMetagreta Personally, I never considered his voice. But what's important to me is his clear and easily understandable speech. KZbin is filled with videos explaining university level mathematics that I could never listen to for the sheer fact that I can't understand the accent. I really appreciate the effort, but a strong accent immediately makes me click away

[@Necron3145]

that list of immortal mathematicians is imcomplete it doesn't have you in it

[@48956l]

Link one research paper this guy has published..? Do you even know his name without looking it up lol

[@shivakumargujjari]

@@48956l so you really thought he genuinely wanted 3b1b's name in it? Really?

[@48956l]

Shiva Kumar ya and ur name belongs on the immortal dumbasses list

[@faridmammadov5989]

That, I guess, would be an overestimation but I have to admit that he knows what he is doing

[@anders5611]

Because he's not a mathematician, he's an educator.

[@leeschmalz6365]

"Taylor made polynomial" and then not acknowledging the pun, I love it

[@vinayakchaturvedi8006]

Hahaha I caught it too xD

[@connorconnor2421]

3:45

[@c4stmiranda902]

I dont get it. Can you plz explain.:(

[@Max-bz8ev]

@@c4stmiranda902basically, you can express many functions locally as an infinite polynomial or power series called a *Taylor* series.

[@markenangel1813]

@@c4stmiranda902 he has another video about Taylor series: kzbin.info/www/bejne/aZWZdaagfqetgJY

[@PBlague]

I watched this a few months back and although it was really cool I didn't understand enough... but now that I see how it is working and you were able to bring that image into the circle initial constants it was just soooooo satisfying... I've been watching your content for so long and you amaze me every time, keep on doing what you're doing.

[@miriammcfarlane6972]

You do such a good job of explaining this. It's been nearly 50 years since I learnt this kind of maths, so I'm a bit rusty on some of it, but I still get the general idea and could rewatch it...plus it's so calm and soothing to watch the drawings happen! 😊

[@blacklegend2304]

That must have been some hard-core editing Thanks for the great content

[@Schattengewaechs99]

Animating ≠ Editing

[@giacomo.delazzari]

Doing something like this with video editing software is almost impossible. Each of his videos is entirely produced by a Python program that uses a library he wrote (which is also open source IIRC). I guess that is the only sane way to create videos like this.

[@nuklearboysymbiote]

@@giacomo.delazzari still pretty insane

[@jl_woodworks]

Hard-core coding instead.

[@ollerich32]

@@giacomo.delazzari yep, it's online: github.com/3b1b/manim

[@ErikRyde]

After watching this for the 1000th time, I finally had the key intuition necessary to code my own Fourier Transform. As a musician, nothing makes me happier than having my own little Fourier transform now. I just want to thank you for these videos, they’ve completely resparked my joy for math.

[@dracovet777]

So you've coded your very own Melodyne?

[@ErikRyde]

@@dracovet777 Sort of lol. Melodyne is way more intense than my algorithm. Since I wrote this I've managed to optimize it using some linear algebra I just learned, but it's still just a basic fourier transform and not anywhere close to an fft

[@sidewinded1]

Could you tell me how you coded it? I would love to try!

[@ErikRyde]

@@sidewinded1 ok so I use python’s numpy library. Long story short I take the inner product of 2 vectors which are representative of the functions you get in the Fourier series. The fancy stuff is making it all compact so I can use it wherever I need to, I basically made that all into a lambda function. I know this isn’t the worlds best explanation but all I’m trying to say is it boils down to how much linear algebra you know.

[@punkt3260]

Nothing makes you happier? No wife?

[@starstrudel8417]

What an elegant and clear explanation. And the effectiveness of your visuals CANNOT be understated! Fantastic video

[@Lesics]

What an amazing explanation, this video deserves 10 million views!

[@pardeepgarg2640]

Not 10 million but 20 million

[@epicmoofish3726]

Aight, hold on lemme watch 6.5 million times

[@VineFynn]

Does everyone really need to know what a fourier series is

[@Shoya_Ishida_69]

No , it deserves 10000000000000 Trillion views and 10000000000000000000000000000000Trillion subscribers

[@Sciencedoneright]

@@jerryboy9652 even more... 10000 BILLION VIEWS

[@saurabhjoglekar]

Whoever first invents a time machine, please go back in time and show Fourier this video. Absolutely amazing!

[@dexter2392]

Fourier would be proud.

[@chupetaparabose1]

@@dexter2392 Or , he will create the fourires series because of the video

[@vinayakchaturvedi8006]

@@chupetaparabose1 Maybe he made this series because of this comment :3

[@FiasaPower]

Imagine how he visualised this inside his head back in 1800s.

[@veronicagorosito187]

@@FiasaPower I was wondering the same... What a genius could imagine this? Amazing minds.

[@bowtangey6830]

If I remember the history correctly, the Fourier series idea was first suggested by Daniel Bernoulli (son of Johann and nephew of Jakob. Euler's father and Johann had both roomed in Jakob's house in Basel when the father and Johann were both students at the University of Basel when Jakob taught. -- Daniel is the Bernoulli of "Bernoulli's Principle" in fluid flow). He was studying it in connection with not the Heat Equation, but the Wave Equation as a model for a plucked string (like a guitar string). He proposed in a letter to Euler that any realistic initial configuration could be written as a sum of sine waves. Euler replied both showing how the coefficients of the sine terms could be calculated, but also criticizing the concept. He noted that a simple initial configuration would be pulling the string out at the middle, giving the string (endpoints fixed) a v-shape. That shape is not differentiable at the vertex, but all sine functions used in the proposed sum are differentiable everywhere. So, he stated, the idea could not work in the generality that Daniel suggested. The idea was out. When Fourier first presented his results in a paper of heat propagation to a Paris committee in 1807, the paper was rejected. On of the committee members, Joseph Lagrange (a protégé of the now-dead Euler) specifically criticized the notion that any function could be represented by Fourier's trigonometric series.

[@michaelimbesi2314]

You have mastered the art of simultaneously producing content that is sufficiently interesting that I will watch all of it and the exact level of verbosity such that increases watch time to the maximum amount possible without enabling people to skip the unnecessary portions

[@biubiuty]

Your visualization skills alone deserve a Nobel prize :O

[@hellofromc-1374]

*field

[@abcdefghijklmnopqrstuvwxyz1062]

*field

[@gdash6925]

Nobel prize for optics?

[@DavidLindes]

@@hellofromc-1374: as in Fields Medal? Indeed.

[@Funkaar_Studio]

Agreee👍👍

[@nandospm]

I am an electronic engineer. I took a dozen of courses about Fourier series and transform. I work with spectral analysis on daily basis and this video blew my mind

[@0xFDFA]

Same here, though I only apply dfft occasionally in my professional life.

[@1Chitus]

I'm taking a masters course on Fourier Analysis right now, and rewatching this video just now shows me that the level of (even technical!) essence he manages to address without introducing new jargon is astonishing. Especially considering how long-winded definitions of things like even Cauchy sequences and limits actually are! Around 9' he's actually talking about Cesàro convergence, for instance, something I never saw in a bachelors course. And yet he doesn't miss it! And still he acknowledges that he swept some things under the rug! This is a great man.

[@antoniojpan]

The work and genius that goes into making this video make it a true piece of art.

[@aforslow]

I've finished a 5-year degree of applied physics, but I gotta say that I never quite fully understood why the Fourier series works. A big reason for that is that there's so many things to learn that you mostly learn the 'how' instead of the 'why'. It makes for quick gains of knowledge, but it simultaneously makes for quick losses of knowledge, as it really is the 'why' that makes for the building blocks of knowledge in the long run. Your channel, and especially this video, masterfully compresses the 'why' into its central components and makes for quick and stable gains of knowledge. In the long run, you will have helped creating a much more efficient, robust and accessible education for everyone. Great job, and keep going!

[@aforslow]

conacal rubdur it’s a very broad engineering degree (I should probably have written Engineering physics, as that’s its real name), so you can get into most engineering things (except for chemistry, I guess) if you choose the right specialization (i.e. Master’s degree). I myself am working with software development and machine learning!

[@aforslow]

conacal rubdur Yes, it is computer science! I’d say that more and more engineers go into this field nowadays, despite coming from quite different backgrounds in their bachelor’s. As you might realize, my Engineering physics degree was quite different from what I’m working with now. Basically, my first 3 years (i.e. bachelor’s) consisted of lots of heavy maths and physics, which has given me a solid mathematical foundation to understand most research papers I read on AI and other subjects. In your case, I’d say that most engineering degrees give a good mathematical foundation, though (except maybe for chemical engineering; I don’t know how complicated maths needs to be there), so it’s probably more important to choose a bachelor’s that you could feel motivated for, rather than choosing something that could boost your career. Anyways, owing to your list of subjects, I’d say that mechanical or aerospace engineering would probably be the best bet! I think bachelor’s = undergrad, so you have to decide on your bachelor’s before you get into college. Don’t worry though; the math you learn in one field is often used in the next (such as fourier series), so choosing one specialization doesn’t mean you can’t change your mind in the future! :)

[@aforslow]

conacal rubdur Yeah, I’d go for mechanical engineering in that case, since it’s broader and, who knows, maybe you’ll find something else that’s more interesting there, and then it’s nice to know that your broad education gives you better opportunities to specialize in that if you want! I have no idea what the chances of landing the jobs you’re listing are, and I’d definitely say that it depends on where you are. Engineers are pretty popular on the job market though, so finding a job shouldn’t be that hard. Regarding the major, I’d say that it also depends on where you are. I’ve heard that in U.S. it’s pretty common to do a bachelor’s work for a while, and then get back to a master’s. Many here hope that the company offers a paid master’s, since doing it on your own very likely costs more than you get back from a future career. In Europe, however, the situation is very different, as we don’t have to pay for each semester. Therefore, we often do a bachelor’s and master’s consecutively, without a break, in order to actually finish our degree. I myself would reason that if I started working after my bachelor’s, I wouldn’t wanna go back to school once I’ve settled into this new lifestyle of getting paid and not having a constant feeling that I have to study. People are of course different, but if you’re in Europe, I’d recommend bachelor’s + master’s in one go, since you then finish everything in one go, and it feels like you continuously improve your lifestyle!

[@aforslow]

conacal rubdur well, location matters in the sense that your society works differently than in Europe. For you, it costs a whole lot more for each additional semester you study; whilst in Europe, you don’t have that much to lose to study a couple more years. Regarding your question about applying for jobs all over US, I don’t know what’s best, since I don’t live there and don’t know anything about the costs of travelling or how valuable it is to move for jobs there. If it had been me, though, I would have started out looking for jobs nearby, since it’s simpler!

[@aforslow]

conacal rubdur Yes, that sounds good! Those unemployment rates are very low, so you shouldn’t worry about that. Additionally, if I were to worry about unemployment after graduation, I would take into consideration how easy my job is to automate, since that is a trend that will accelerate. For most engineers, though, I’d say they’re pretty safe!

[@3blue1brown]

To those asking about the software behind these animations, take a look at 3b1b.co/faq If you want to play with these animations, I might actually recommend looking at the video by The Coding Train, since the code he shares is probably easier to get started with, and the video itself involves walking directly through his implementation: kzbin.info/www/bejne/g56Vloycn5WGnaM

[@gabriellord9214]

Please could you explain FFT (fast Fourier Transform?) It's used in audio manipulation and so on, which I'm super interested in... I don't know whether this video covers it (only just started watching haha 👍😊) Cheers 😊

[@randomdude9135]

Thankyou for your contributions and efforts. 😊

[@gabriellord9214]

Ahah! ~5:50 makes a whole lot of sense! It's possible to think of sounds as sums of various sine waves! So it all fits! Enjoying the video! Thanks 3B1B 😊

[@TheCodingTrain]

Thanks for the shout out! The video you reference just shows the basic fourier series for a square wave. I also have these three follow-ups which show how to draw a path with epicycles. The demo is here: editor.p5js.org/full/ldBlISrsQ/ Code: editor.p5js.org/codingtrain/sketches/ldBlISrsQ And these three videos show how to write the code! thecodingtrain.com/CodingChallenges/130.3-fourier-transform-drawing.html thecodingtrain.com/CodingChallenges/130.2-fourier-transform-drawing.html thecodingtrain.com/CodingChallenges/130.1-fourier-transform-drawing.html

[@thefantasicm_2407]

Your work is awesome man :-)

[@meatlover6916]

18:50 this kind of gave me a deeper intuition on the fourier transform; thank you.

[@gabemathews1216]

You are seriously the missing link in my education. The way you explain mathematical concepts is unmatched and it gives me so much more perspective and appreciation for whatever I'm studying at the time. Thank you!

[@jiemin3065]

10:08, a beautifully wordless proof of e^-ix + e^ix = 2 cos(x). Thank you for this!

[@squibble311]

i actually didnt think about that, ur a smart guy

[@alberto3071]

If you're interested, you can prove 2isin(x)=e^ix-e^-ix just by looking at the difference vector of the ones shown. It's modulus is 2sin(x) and is vertical thus the vector is 2isin(x).

[@abdulhamidwalidabdulhamid2353]

I have an exam tomorrow on Fourier Series and I can tell that no Instructor in my college managed to make me understand it this way. Despite the fact that they all have PhDs but its always the way of explanation that matters. Your teaching skills are on another level

[@toxickid2456]

Bruh ok, did i ask?

[@osteoporosiscronica4637]

@@toxickid2456 yep, your username fits

[@toxickid2456]

@@osteoporosiscronica4637 ik, and urs not

[@osteoporosiscronica4637]

@@toxickid2456 yep

[@eigenrauflinog9069]

@@toxickid2456 no one cares what you think

[@francescolapietra930]

I watched this video along with the one on the Fourier Transform a while ago while studying for a class on signal theory, and recently in another class my professor suggested this to us and i remembered this gem from your channel

[@redstart67]

This is a truly amazing visualisation of some beautiful mathematics & can't be praised highly enough on so many levels. I'd known a little about Fourier transforms & their applications ever since an encounter during a molecular spectroscopy practical class at university where a previous student had written some basic code on an old BBC micro that analysed a spectrum of mixed fundamental frequencies using a reverse Fourier transform. Ever since that first glimpse, I've wanted to learn a bit more about the maths involved but in many decades of casual interest had never encountered such a brilliant concise & understandable explanation. The astonishing graphics both capture the imagination with their beauty as well as perfectly illustrate your introductory point about the extent of applications for this elegant maths. Absolutely brilliant video which I sincerely hope many eager young mathematicians will get the chance to view & like me, watch in awe & amazement. I only wish that Fourier himself could watch this on Heavenly KZbin.

[@charleshoots4720]

I'm a seismologist and even I don't understand Fourier theory at this level of ease and intuition. You took one of the most complicated concepts in science and made it so beautiful and intuitive.

[@raquibkhaneee]

I took courses on signal & system and digital signal processing before KZbin was founded. I wish I was born later so I could take advantage of the best explainers like you and the best of visual presentations like this. The timings of items you throw on the screen, the sequence, the connections, and the visual ques you use is beyond precision, beyond exact. This is a God-mode representation of Fourier series in terms of rotating circles! Take a bow!

[@foxphire0093]

I’m in Signals right now and watching this definitely helps my overall understanding of why things happen as opposed to just accepting it and moving on

[@david203]

This is perfection in explaining mathematics. Almost all of the insights that I gained when I worked with Fourier Series on computer as a much younger man, years ago, are revealed here through clear audio descriptions combined with a variety of related simulation-demonstrations. Only lesser-related but very important implications are missing: showing the precision tradeoff between measurements in the amplitude and frequency domains (leading to Heisenberg's uncertainty principle), showing how eigenvalues happen, discussing pure and mixed states, and the Pauli functions/transformations revealed by the quantum mechanics of photon polarization.

[@whiteComett]

Brilliant! I almost cried man...Throughout engineering, god know how many times I had to use Fourier Series/transforms and heat equations, not to mention the never PDEs, but not once had I gotten the chance to understand where it comes from, the beauty of its complexity and capability, which I something I would always strive to understand. But just dedicating 20 mins to this video, gave me so much closure. Thank you!!

[@maxlopalegz5500]

This video is being recommended to me by my university maths department! All the best from Berlin! Awesome work

[@Jagdpilot]

Digitalisierung be like

[@AR-vb4xy]

Wie geht es deutschen Hochschulen?

[@halbeard2996]

Every youtube science channel recently: draws these fourier shapes with a couple of arrows on screen 3Blue1Brown: hold my beer

[@eunhyoukshin7777]

When a mathematical genius is also a programming genius:

[@user-uu1nw1bl9j]

@@eunhyoukshin7777 This dude has a greater mathematical insight than I ever dream to have, but even though I don't know what he does outside these videos, to call someone a genius for explaining subjects you already encounter in your first year of studying physics or mathematics is a little bit too much.

[@thegreatwebstar]

This is tremendous, well done ! It Helps to rotate axis to visualize. Excellent work. Imagine the things we will be able to create tomorrow, imagined

[@WildcatRA]

Just spent a week learning about the diffusion and heat equation along with Fourier series. Well done!

[@alexandre.hsdias]

In all my years at college, I've never seen such a stunning presentation about Fourier series. I can't help but say thank you . This is in a total different level of explanation...

[@DonArmadillo]

3b1b Has allowed me to find beauty (and even a little love for maths) a subject I previously (and kind of still hate/dislike). I've went from barely passing to acing my college math exams, much thanks to the conceptually and visually (yet incredibly accurate) descriptions Grant has given. The overview stipped of the detailed theoretical information is priceless. So much respect to you! Please join his patreon!

[@denelson83]

I'd say he Grant-ed you the insight you needed.

[@sadkritx6200]

@@denelson83 Who Grant-ed you the permission to make such puns?

[@-guitarhero]

@@sadkritx6200 joe

[@ishworshrestha3559]

Kk

[@-guitarhero]

@Aaditya Sabharwal you know

[@jezza10181]

Your explanations beat anything by any other content creators... You are literally god level.

[@blacksparrow4635]

you were the first man who teach me the real patterns which connect physical entities to mathematics and as a fact it is helping in me in programming domain , thanks for that.

[@ArThePig]

What a genius you are! You open up what i could not visualize during my teenage years in university. You make math such a wonderful and enjoyable subject, even fun to keep sharpening my brain watching your vdo’s.

[@elkalajeer710]

Guys I think I got it Those at the start are circles

[@vampyricon7026]

Hugh Dennis answering Picture of the Week on Mock the Week: "That's Boris Johnson."

[@Diego-ji6nl]

Actually they are circumferences

[@drone_better7757]

@@vampyricon7026 Oh, no. I'm not brave enough for politics.

[@spacejunk2186]

Whats a circle?

[@jonperkins4460]

👏👏👏👏👏👏👏👏👏👏👏

[@krishnapandit8190]

Okay. Grant Sanderson is the best math youtuber hands down. It's surreal that you can relate to so many concepts with just the Fourier series.

[@HosRo4161]

Beautiful. Minute 10 helped me see the connection between cos() and cosh() via (e^x) versus (e^ix)... Thank you!

[@ltsgobrando]

11:35 This is the time where I figured out exactly where you were going! You were 100% correct, this was intuitive. Also wanted to say that I appreciate you so much ❤

[@volvocineseries5839]

"ah shit, here we go again" -me about to spend the next indefinite amount of time pondering maths

[@saeidakbari4788]

LOL That's me exactly! :D

[@PauloConstantino167]

wow. congrats on your world changing comment

[@nolan412]

Begins grinding gears.

[@planktonfun1]

there goes sleeping

[@veljko100able]

Please make video about Laplace transform and (or) Z transform, I am sure there a lot of others confused by that topic. And I haven't seen better person to explain it to us

[@Schattengewaechs99]

Yes, that would be a very interesting topic!

[@thomas.02]

i believe he mentioned in an earlier video that he will explain them in a later video of this series

[@diarya5573]

Yes! I have so much trouble visualising those

[@veljko100able]

@@thomas.02 oh that's nice, I am looking forward to it!

[@porschepanamera92]

Laplace transforms are for example beautifully used in structural dynamics. It would be nice to see something like that visualised indeed. Not an easy topic to study from text books only lol

[@kevinxin1545]

Grant is officially my hero and I want to be just like this guy. But Grant you have set the bar so high that watching you pull these off is just so... I dunno it just hits me that I might just never get anywhere near the level you are. You are insane.

[@DylanUPSB]

I really wish that every intro to Fourier analysis courses/videos/books gave the heat equation/limiting series motivation, and your explanation of transforming the series so that each term is a constant around 19:00 - right at the start of the course. Those two things made the concept feel so much more motivated and transparent. Guess they should all just watch this video!

[@randomdude9135]

That moment when Rotating vectors draw better than you 😑

[@shizotypical]

Such is the power of math

[@kenzostaelens1688]

i feel your pain

[@halbeard2996]

Maybe you're not rotating enough when drawing something

[@DarthZackTheFirstI]

thats not true. he used svg data which was taken from a picture drawn before. he just followed the tracks. so the artist was first and then the program ;P

[@ariat3381]

@@DarthZackTheFirstI you killed this poor little joke, how dare you!

[@Qsdd0]

3:45 Fourier-made, not Taylor-made. :P

[@NightOmegaX]

I'm a little upset that I got beaten to this

[@AlienValkyrie]

Damn you, I already made that joke on the patreon draft. Don't you go stealing my puns!

[@lukasmiller8531]

I usually hate puns, but this one...

[@Mayank-mf7xr]

if you look at it , taylor series is a similar idea of breaking down a function ( must satisfy some conditions first ) to an infinite sum of its derivatives , each amped up a certain amount .

[@VikeingBlade]

I thought that was an actual mistake while watching it, then realized tailor-made

[@AlexandreFerreira-jb2jl]

Beautifull. And i understood part of the math behind it. Its impressive how complex numbers gave fourier series a new context.

[@rudrajain5563]

These videos makes me love maths more and more, that I will be able to apply them in my programming knowledge and build something beautiful out of it.

[@japanlovescali]

I feel privileged to discover this channel even before my college begins

[@santiagos4290]

Just luck + time

[@aravndhanr7241]

Yes you are. Make the most of it. Enjoy learning. Very happy for you

[@akashsudhanshu5420]

Mann, salute to you. I have to give a seminar on fourier series 2 days from now. And I have studied the theory to Write fourier series, prove convergence, etc. But still I find this video has new information, showing how deeply to have understanding of the smallest bits . Its inspiring me that I interpret maths in my own words, in my undrstanding, taking words from real life. Thankyou, Brilliant work, brilliant brilliant brilliant ..

[@geogi_bodies]

I came across this video by chance. I don't know what's going on but seeing the rotating vectors drawing out musical notes is fascinating!

[@0xFDFA]

I'm an electronic engineer amoung other degrees, and I've mastered the math back in the day and used dfft on quite a few occasions proffesionally but this was a wonder to watch and made me look at it from a new fresh angle. This video should be part of every curiculum that includes fourier transform. The part about how the Fourier got the idea in the first place was especially worthwhile for me. Thank you

[@JohnRobenault]

Exactly. It's like I've been doing the math blindly, but this gives me an opportunity to actually see what I've been doing.

[@absoultethings4213]

Uii

[@Mayank-mf7xr]

i earlier told how your essence series are in sync with my courses in college ... i studied calculus in 1st sem ... just as your series rolled in . i studied linear algebra in 2nd one ... longside your your series on it . i have to study differential equation for 3rd sem and guess what ... you put out what i exactly need . my core course contains ' fourier series and special function ( gamma , bessel and so on )' and you just bring this gem . i am so lucky .

[@Abdega]

Wish I was, I always just miss them :/

[@crziekidify]

I loved ur channel man, it made me loved math again. Been practicing it ever since ive watch your series on int and diff. Thanks man

[@danish07delhi]

Simply an awesome collection of visualizations explaining sophisticated mathematics, thanks a lot.

[@leopoldsimmons4804]

friend: how well can you draw? me: how many arrows you got? friend has left the chat

[@zionj104]

100th like

[@thawng9346]

jemma??

[@NovaWarrior77]

Thi is why you and I have no friends. 😢

[@felixalexander8986]

lol

[@evariste1686]

beautiful! But how do you add colour and shade to the last arrow? In that sense the arrows don't draw as the points are invisible until....

[@johncao6516]

0:25 That's the Nail and Gear of HELLO INTERNET!

[@changyang1230]

I was pleasantly surprised when I saw it too! Great Easter egg!

[@changyang1230]

And hello Tims!

[@coryman125]

Probably my favourite thing about "this side of KZbin" is how often people will reference each other's channels in odd ways like this. Came looking for a comment about it as soon as I saw it :P

[@doktorcool3740]

Yes - this Tim noticed it too and was delighted! :-)

[@greghmn]

Crazy how we can sometimes detect symbolic styles subconsciously. I wasn’t sure what that logo was at first, thinking it was either for CGP Grey or one of Brady’s channels. Guess I was right on both. Maybe the gear was a giveaway.

[@frrosp]

Really appreciate the quality ambient music in these videos! Vincent did one hell of a job!

[@maxwang2537]

Stunned to say the least. This is probably the most beautiful visualisation of one of the most beautiful mathematical theories I’ve ever seen so far. I’m extremely keen to know more about how a video like this, particularly this one, was ‘developed’, which would obviously include scripting, coding, animation, as well as all the tools and software involved. Just watched your another vid and learned that Minim is probably one of the animation software you are using. It now on my to learn list.

[@renedekker9806]

It is the ultimate high praise to your teaching skills that a video on a complex mathematical subject like this, gets 2.6 million views. Absolutely brilliant.

[@midouar4859]

10 million views 😅

[@ammyvl1]

4:19 the people are (in order) Pythagoras, Euclid, Archimedes, Fermat, Newton, Leibniz, Bernoulli(?), Euler, Fourier, Gauss, Riemann, Cantor(?), Noether(?), Ramanujan, Gödel, Turing. Just for reference, I put a ? besides the ones that may be wrong.

[@hwg5039]

No Taylor, Lagrange, Descartes and Pascal? Damn...

[@dimitrisdimitriadis4913]

Shamefully missing Thales among the immortals as well. Video dropped from 10/10 to a 9.5/10 just for that.

[@MarioMicheli]

Yes, that is Emmy Noether

[@-guitarhero]

I thought that fat dude was JS Bach

[@ishworshrestha3559]

Ok

[@BeamMonsterZeus]

This adequately touches on a hunch I had growing up, cementing my curiosity in the clockwork nature of the universe.

[@sdsa007]

Everyone should learn this.... this is the best explanation! I never thought I would understand it until now. Thank you! This is so crucial to the language of math!

[@KianaMaillet]

Physics student here!! Thank you so much for everything you do, your content is SO valuable to me and many other students, I've even had 2 professors send us your videos for better understanding. You are a blessing!:)

[@trevorjohn3632]

I have been using Fourier analysis in my work for over 50 years and I was taught the old way in terms of transforms between frequency and time domains. I guess I understand the principles, techniques and math really well, or at least I thought I did until I saw this. This visualisation is truly mind blowing. What a fantastic video and all done in less than half an hour!

[@dajimaak7681]

I made this back in my high school day. Some of the coolest math project I've ever done

[@andrewsheehy2441]

This is really excellent! I imagine that this video would hve taken WEEKS to create. Well done 3Blue2Brown!

[@sergiucimpoi8497]

The first time I visited this channel, I was: "How on Earth did he make 3.5mil followers?" Now: "How on Earth does he only have 3.5mil followers?" I guess, even for a non math head, this would be a pleasure to watch. Keep them flowing!

[@sethdon1100]

He now have more subscribers (followers) than Numberphile.

[@breezyashell]

being a visual learner I never really understood Fourier series. This is amazingly helpful!

[@steve-o6413]

Your not alone...

[@Cuasimodo2372]

Everybody is a visual learner, you just gotta think hard to visualize what formulas mean

[@steve-o6413]

Abstract thinkers (3D) use more visual than Straight thinkers (2D)...

[@Carofdoom1126]

Seriously 3B1B had reached math education immortality

[@steve-o6413]

Taking 2D Concepts and turning them into 3D Concepts is what this Video is about at least in my mind. Left Brain Right Brain...

[@raghaven92]

Wow!! I liked the animation of vectors!! Different perspective for complex Fourier series

[@transient_moonlight]

Man, this video made me understand Fourier series better than my textbooks. Thank you so much!

[@shivamsoni5894]

You make me love math.After watching one video I go out to relate everything in nature and i almost never fail to find a relation . Math to me now feel like a suject of great wisdom and it is more of a skill when practised for enough times can improve your understanding in it thank you man !

[@williamdeng5966]

This is sick! As usual, painfully clear explanations, and the exercises at the end really helped me to solidify my understanding and to get a feel for the beauty of Fourier series! I couldn't even believe I was actually expressing a discontinuous step function as an infinite sum of trig functions! Thanks for all your hard work Grant, and I'm looking forward to the lecture on the Laplace transform!

[@ramstube08]

Once upon a time, I would have been able to understand all of this, now I can't even make it till the end... I feel jealous of you guys.... Don't give up on science.... it's the most beautiful thing!

[@coolcat23]

The way you drill down to an intuitive understanding of what are essentially magical concepts is nothing short of brilliant. Most teachers are satisfied to provide some explanation which is correct but not illuminating. You, instead, drill down to the most insightful perspectives. Brilliant. N.B. I don't use "magical" in the sense of "supernatural" but in the sense of "mind-boggling" (in the Arthur C. Clarke sense).

[@user-fn6lz8pp6j]

Just want to say you have no idea how inspiring and influential you are to every person that is interested math. Definitely not being the only one, I am greatly delighted, encouraged, and motivated by you on the journal of learning math. Thank you for your contribution, thank you, thank you, thank you.

[@dancingbubbles1126]

The calmness of your voice, the music in the background, and the fluidity of the animation make watching this really soothing. It's also beautiful.

[@poonkodisrikumar]

Awesome explanation delivered by expert , got indepth and clear understanding.

[@parlancex]

The fact that a video on this subject has 15 __million__ views is nothing short of amazing. We love you Grant! ❤

[@FerroNeoBoron]

You were expecting trigonometric functions but it was I, complex exponential!

[@randomaccessfemale]

Euler's equation is just so fundamentally beautiful. It is superior.

[@alexeie.6187]

@@randomaccessfemale superior to what?

[@vinaytanwar7840]

*Evil sound plays in background*

[@xtdycxtfuv9353]

trigonometric functions’ bizarre adventure

[@johnjackson9767]

Twas i!

[@Mau365PP]

Explanations like this 20:09 are the reason I love this channel

[@benjaminglass2694]

My jaw dropped when I got there. I've been looking for a way to understand that concept for years.

[@eunhyoukshin7777]

Yeah its much better than just stating orthogonality of sinusoidals

[@janv.8538]

True

[@Mayyouknow]

same here!

[@blazegor9612]

This video is so great, that my teacher in Chile showed us this video so we can understand the idea of the Fourier series. Amazing job!!

[@jaimecaballer1625]

Huge congrats. Thanks for doing this. Top quality pedagogical work.

[@vanster]

I rarely write comments on KZbin, but right now I am so overwhelmed by your visualization and explanation that I cannot remain silent. I think you are a genius at explaining things like this. Thank you very much for what you are doing.