But what is the Fourier Transform? A visual introduction.

An animated introduction to the Fourier Transform.
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/fourier-thanks
Learn more about Janestreet: janestreet.com/3b1b
Follow-on video about the uncertainty principle: • The more general uncer...
Interactive made by a viewer inspired by this video:
prajwalsouza.github.io/Experi...
Also, take a look at this Jupyter notebook implementing this idea in a way you can play with:
github.com/thatSaneKid/fourie...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Russian: xX-Masik-Xx
Vietnamese: @ngvutuan2811
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Animations largely made using manim, a scrappy open-source python library. github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.
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, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
Reddit: / 3blue1brown

[@halihammer]

It's crazy how Fourier was able to do this with just a piece of paper and his imagination, while I'm already struggling to follow this masterpiece of a visual explanation

[@seanriopel3132]

And super computers at your disposal. Most people have little comprehension and appreciation for how important and inspired the giants whose shoulders we all stand upon are. The dedication and sacrifice our ancestors dedicated their entire lives to, is without parallel. The one thing they all had in common is a desire for fundamental truths. They were outcasts whose very existence challenged the staus quo. Just because we avoid conflict just to keep things copesthetic doesn't mean it is the proper path forward.

[@ivoryas1696]

halihammer Yeah, for some people... "built different" is an _understatement..._

[@wfps488]

The guys bibliography is also insane. Born an orphan, became a mathematician, later traveled with napoleon and become a member of parliament. All while creating the groundwork for a lot of quantum physics. Guy lived one hell of a life.

[@123deserted]

@@wfps488 WHAT? damn, I feel bad about hating the guy when our professor was not even trying to explain the beauty behind fs and ft, I didn't understand them for shit back then, I wanna blame the prof but idk.

[@Zubzub343]

Oh man, I've been following your videos for a while and learnt everytime some new ideas but here you just touched my heart. I studied Fourier transforms a while ago, had good exam result and got a Master degreee in engineering. But still, I've always felt that I missed the correct intuition of Fourier transform. I did some research on my side after the class and got a way better understanding. But still, I think you just achieved what my professors and myself never manage to do, that is teaching/understanding correctly the underlying principle. I cannot thank you more for these video and this whole channel, and to all professors here trying to give some vague intuitions with bad drawing on the blackboard, please, redirect your students to this video. The next generation of engineers will thank you later.

[@3blue1brown]

Thanks for taking the time to write such a nice comment. Hopefully, you continue to learn more, there are quite a few great resources on the internet about Fourier Transforms and such (Better Explained has a pretty good one), and I think you'll find many more interesting perspectives and "aha" moments.

[@ibrahim_akalin]

and you just wrote down my feelings in this comment, I'm %100 with you! I've been following people on KZbin and Twitter that are able to learn and teach in deeper and way more intuitive ways, definitely a better path for lifelong learning. Thank you both!

[@daggawagga]

Your comment resonates so much within me. I had never even come close to getting an acceptable intuition about these frequency transforms.

[@VaderDarth512]

Your the reason i have an A in Calc AB. You make math so fun and interesting!

[@HonneyC]

Zubzub343 I was about to right approximately the same comment ! I’m pursuing mathematical studies but it’s the first time in 5 years that I feel I’m getting a good understanding of the way this formula was contrived ! Really had to see it in motion, thank you for that !

[@cusackprep]

You sir truly deserve an honorary doctorate - just for this video. Your impact to generations of confused engineering and math students will forever ripple through our society.

[@aaronlatapi2272]

I totally agree, as a very young engineer math and physics had been thought in only repetition but not in the actual application and manipulation we can do with them. How we can play and control math is a topic i find really interesting and in which im very new at it.

[@nosferato445]

@@aaronlatapi2272 I taught the Fourier transform 'wrong' for ten years.... until I saw his method. his ability to explain things is ungodly.

[@JtotheAKOB]

and physicists

[@howmathematicianscreatemat9226]

@@nosferato445 yes, he has both high IQ, really high empathy and the will to be listening towards others. Fantastic combination 😎

[@AK-xw2ie]

Fully agree A extremely gifted teacher

[@raphaelsbr1899]

Nobody is ever gonna read this, but i'm literaly mesmerized by the quality and accuracy of teaching this video posseses. It's nuts how well u explain such complicated things

[@PhilippeSaner]

I read that.

[@raphaelsbr1899]

@@PhilippeSaner i appreciate bro

[@simonhinterseer9974]

didn't read - just clicked thumbs up

[@raphaelsbr1899]

@@simonhinterseer9974 well done soldier

[@George-gq1ut]

Nope, we read it and we agree with you. I have 2 decades of constantly dealing with Fourier, Laplace, Smith chart, I came here for new perspectives. We live a revolution of teaching. Oh boy, I would have love to see this 20y ago.

[@adamcspanza5986]

The amount of clever someone has to be to discover this stuff is insane

[@theblinkingbrownie4654]

It's not just one person, it's hundreds of amazing people.

[@nischalada8108]

It’s not so much as to “discover” this stuff, but rather, using models and constructs to simplify the world around them in order to understand what was previously too hard to. These mathematicians were just so eager to see nature unravel in ways that unraveled cool secrets, that everything from calculus to Fourier transforms, etc. were built, rather, out of this curiosity.

[@AndersAlsDieAnderen9]

The amount of clever someone has to have in order to explain it in such a simple way! At university my (very good and passionate Professor) took more than 1,5 hours!

[@NovaWarrior77]

I am a newb in the regard of creating math, but I would assume it has more to do with seeing enough relevant constructs and making a small or big leap based on alot of background. By no means do you need to be excessively clever to come up with this stuff, not even have a killer amazing amount of background. I just have come to believe that that's the case.

[@aurelia8028]

I know rite? Especially in math, physics and chemistry, sometimes I'm just sitting there wondering how on earth people managed to find this or that formula, or realize that there was this connection to that, which makes solving whatever problem 100 times easier. Something I still really don't understand is how astronomers figured out the orbital period, velocity and distance to earth of the different planets just by looking with their _eyes_. I also wonder how on earth Newton actually tested his equations of gravity to find out they were correct.

[@Kevin-cy2dr]

If this lecture was delivered in a class you would surely get a standing ovation.

[@CaseyAtchison]

I'm showing it in my class next week.

[@akashchoudhary8162]

@@CaseyAtchison Well? What was their reaction?

[@CaseyAtchison]

@@akashchoudhary8162 More than one told me it's the most real science they've ever seen at their school.

[@MrAlRats]

@@CaseyAtchison So no standing ovation?

[@overtheworl]

@@MrAlRats funny haha

[@karmapa_TW]

After 20 years of graduation, I now realize what I was learning back then. Thank you for your production.

[@Kingd123asdf]

same here

[@zhenzhaotu8090]

As a math student I cannot believe there is a person can make math concept like this intuitive, amazing !

[@LordFennel]

I'm a first year physics student in the UK. Talking to friends in higher years, I've learnt to dread Fourier Transforms. They are spoken about in hushed tones like a mass genocide in the recent past. I realise this video probably only just scratched the surface of this topic, but I must say how I feel much better informed than I ever could have been by reading a Wikipedia article or even my textbook. Your videos are unique in the way they build up complex concepts from simple ideas in an intuitive, visual way. They are always a treat and have been a fantastic academic supplement in my first term at university. Thank you so much for all your content, 3B1B.

[@paulschmitz1275]

Just wanna echo that spirit. Has been a tremendous help to me to, even though my subject area is economics not pure maths.

[@NuxTheDragonSlayer]

Fourth year Physics/Mathematics student here and I've got to say first few times I did fourier transforms it was a nightmare and I really didn't know why I was doing it. But after just a few times I came to realise that they are much easier than people make out and so infinitely useful. So I wouldn't be put off of anything if it mentions that it has fourier transforms in it.

[@TAP7a]

Yo dude, I'm in my 4th year at Manchester, hit me up with all the questions you want about undergrad physics. I can certainly remember how intimidating the whole thing is, so I would be honoured to do anything I can to help you along the path (I've just taken exams in GR, QFT [which is *all* Fourier Transforms basically] and Statistical Mechanics [also contains a massive amount of FTs], so hopefully I've picked up a few things you can make use of)

[@spiritofnux9105]

Came here to say the same: I studied in France and I too hated Fourier transforms... until I watched this video; I had no idea it was this elegant! Many thanks Mister 3B1B 🙂

[@jackkingers]

I would say if you are confident on pure algebra and calculus, don’t worry about it. If you struggle with the concepts of what spaces you are transforming into, just don’t worry about it. That will come with time and experience. Just crack on with the maths. Worked for me, and I’m doing my PhD in spectral decomposition electromagnetism!

[@miiortbiiort4610]

Funny how we pay loads of money to learn this stuff at university, yet we all still come running to free online video lectures because no professor can match them in terms of quality.

[@vapourmile]

I was just thinking about how the reason I dislike this kind of KZbin channel is for the same reason I dislike other similar channels, such as Veritasium, Numberphile, SmarterEveryDay and VSauce. The trouble is, you aren't really learning anything. In University it usually takes a long time and you have to start from the foundations and work your way up through the rungs one by one until you arrive at the point where you can follow what you've learned into the workplace and do it professionally. This kind of channel uses the "wizzbang" or "wow" approach of endlessly cherry picking isolated curios and well known and popular but poorly understood ideas. In University after this you would be tested by having to produce maths or write programs which use what you've learned to show you can. You don't get that here and it's my guess almost nobody who watches this video goes on to actually sit down and write their own FFT. In the end, you just add it to your store of useless information.

[@miiortbiiort4610]

@@vapourmile Actually, I got to disagree with you on that one. I think you misunderstand the goal these types of channels have in mind. They aren’t academically ‘rigorous’, because such long-winded video’s would only be watched by those already schooled in physics/mathematics. The aim is to help conceptualize idea’s and inspire people to investigate for themselves. The importance of communicating science to the general public is something that’s often overlooked. As for 3B1B, I actually did learn something from this video. My university did nothing to provide me with a good concept behind Fourier transforms. Basically, I was handed a 10 page document with all sorts of mathematical wizardry going on, so that by next lecture they could use Fourier theory. I might have kinda understood the derivations, but I missed any visual intuition as to why Fourier transforms worked. After watching this video I went back to that document and it made a lot more sense. I believe many people just end up memorising the formula’s, without much understanding. It’s a lot harder to find new applications for a theory if you don’t really understand where it comes from and what makes it work.

[@petermarsh4578]

@@vapourmile I mostly agree with you here. The aim of the videos isn't to give a deep understanding, but to give the viewer a neat intuative sense of what's involved in the topic. However, I love these channels! It makes these difficult topics accessible for, well, anyone. And in my case, I often come to these sorts of channels to supplement my academic learning. I find lectures to be quite dry and difficult, but when I combine that rigorous understanding with this intuative understanding, that's when I'm most easily able to work with and exercise these ideas!

[@henokhsihotang9291]

​@@petermarsh4578 Exactly, people often think that this is all formal academic course teaches you and say "school and college bad!!!". Videos like these are meant to explain the bigger picture of a subject, not to replace rigorous academic study of it. At best, these are meant so that the general public understands more (not all) of the topic, and as a "companion" for those studying it in an academic institution.

[@Happyduderawr]

i had some excellent professors, however it doesn't really matter too much if the professor is bad or not since learning math takes a ton of self study anyway. The people complaing about bad proffesors may have valid criticisms, but when you do research you're basically on your own and it doesn't really have much of an effect on your education, though all your colleagues will be massive math nerds.

[@evil-wombat]

I've been an engineer for over ten years (my roommate was the signals junkie and I was the embedded systems dude) and you are the first person to actually make me understand the derivation behind this thing.

[@eccentricOrange]

> signals junkie I'm stealing this phrase

[@chrisengland5523]

I remember struggling through the maths of Fourier transforms at university, many decades ago and I've long since forgotten it all, but this video is a very intuitive explanation. I just wish we had had such graphical illustrations when I was trying to understand it.

[@ManjunathMarkal]

After 13 years completing engineering , i understood use of Fourier Transforms. thank you sir.

[@theowleyes07]

Correct

[@AdityaX2703]

bas formulae ratt ke aoge toh yahi hoga

[@BharCode09]

@@AdityaX2703 What are you doing here? You better stick to Bhuvan Bam! Looks like typo in searching. Do you speak to your mom in that mouth? Yakk!

[@AdityaX2703]

@@BharCode09 and i hate bb and indian youtubers

[@jaroslavzaruba2765]

Thanks for saying that. Now I don't feel that much as such an idiot. Cheers.

[@MrPlasticsnowman]

I want to say one thing: Your skill in not only understanding and vocally explaining these concepts is so perfectly complemented by the animations you create that it blows all other resources on this topic out of the water. Even my college professors recommend this video specifically because, and I quote, "I could never create something so masterful that so aptly explains what is going on inside my head." Your skills are a gift to us all! I hope you always find as much passion in creating these videos as I do watching them.

[@benjaminknudson5997]

It's incredible honestly, 3blue1brown and Sal Khan have made such massive impact; I might need to learn some chaos theory just to try to measure how we have profited as a species from just those 2. I would like to recognize educators in general for their hard work and passion, although, I think we can assess that a few individuals can be attributed a great deal of impact, similarly to say physics. There is a subset of a subset of individuals we can point at who we say have made great contributions, although if you point at them they almost certainly just point somewhere else; a giant standing on giant shoulders perhaps.

[@alien9000555]

He is doing history

[@aussiedog5221]

Become a supporter of Grant. I did....after watching his videos for a few years. His math videos are worth a few bucks a month.

[@yt.abhibhav]

@@benjaminknudson5997 You should include "StatQuest" as well in your list sir

[@kooljirl161]

why is it that I spent tens of thousands of dollars on a engineering degree, and yet these videos do a much better job at teaching me what the math actually IS than any of my professors... Keep up the amazing work!

[@ronitganguly3318]

university degrees are basically a business

[@ttt69420]

It's because even someone who has never taken pre-calc can see immediate applications to, say fluid dynamics, with inverse fourier transforms if fluid dynamics is something he/she is into. Those are the people that are likely going to use them in research roles or graduate school. Whereas vast majority won't ever touch them in a typical engineering job. It's like throwing out bait to hook one or two students a year.

[@SakiDG]

Yeah, my main problem with my degree so far is they teach the pure math... not what it's for. Showing me equations upon equations does nothing for me without a base explanation of the usage. People say these videos don't touch on a proper university education, but without this video I had zero clue. Now I can actually understand the lectures and the math 😅

[@asciizero1099]

This is by far the best, simplest, clearest and nicest explanation on the subject. Thank you so much! You're a hero

[@TheCodingTrain]

This video is incredible, thank you!

[@3blue1brown]

Thanks Dan!

[@shreyanshdarshan3199]

The Coding Train good to see you here!

[@pierreandferb]

You're my hero!

[@evanbecker6743]

It's Dan!

[@ianprado1488]

You are a good guy

[@SomeoneCommenting]

Remember kids: Fourier (1768-1830) didn't have computers nor nice animations to come up with this. He did it purely with paper and ink, using his brain.

[@Joel-oe7ud]

EXACTLY

[@Evanhutton19]

yeah you try doing it and tell me how easy it is

[@ImXyper]

@Deep Moticons it was a massive feat for one person to do with just thinking and writing

[@sanghyunlee4703]

Damn

[@corporalcookie5299]

uhhhh so what?

[@mahxylim7983]

Words cannot convey my gratitude to you with making these videos, you are helping so many people all around the world to see the beauty of math, for what it really is. Thank you. You taught me more than any of my Math teacher ever did.

[@p337maB]

This is not only helping many people to see the beauty of math. People who understand these concepts more intuitively become better engineers, physicists, software developers, ... Better teaching creates a better world. I wish there was a 3Blue1Brown for every field

[@mahxylim7983]

@@p337maB Let's contribute to the society in ways we are capable, and eventually more 3B1B will appear. :)

[@cagedgandalf3472]

@@mahxylim7983 That is true

[@thehumbleworldtrotter691]

If I had had a chance to look at your vdo 30 years back during my electrical engineering classes, I would have fully grasped what I actually completely lost back then. My goodness, your explanation and graphich presentations are unmatched by anyone on this planet.

[@macronencer]

I think you might be the best communicator on KZbin. This is a flawlessly clear and concise presentation. I'm so glad you are planning to make a sequel too, and get into some other concepts!

[@47Yeoman]

Agree. Agree. Agree.

[@Duerschaedel]

totally agree!!

[@sanket7402]

Yeah! Agree..

[@46pi26]

This should be a template for all comments on KZbin. It's perfect.

[@brewbrewbrewthedeck4138]

Now now, let's not exaggerate.

[@Capitalust]

This channel is literally one of the best things that has happened for the mathematics community. Such valuable content.

[@adityasanthosh702]

@Piyush Satti Haven't you been seeing other comments which are recent? And 3B1B channel is growing day by day. I just found this channel a few days ago for which i am very thankful

[@SuperReddevil23]

True, so true, my conceptual understanding of fourier transform has grown exponentially

[@LarryMcLarren]

I would go as far to say for me, it is the best channel on yt

[@rohan1002]

I would also recommend mathologer

[@Pabloghghgh]

I'm struggling a bit with FT-NMR concepts and this has been INCREDIBLY helpful. Can't thank you enough.

[@ulasozbent9248]

It is unbelievable that there is such a simple explanation for the "thing" that the teacher has been teaching for months but left undefined! Congrats sir!

[@IceMetalPunk]

I somehow managed to get a computer science degree without once learning about Fourier transforms (though I narrowly avoided them several times). I'm not sure how I did that. But watching this video, they make complete sense! So often, "complicated" math concepts are taught purely symbolically, but understanding *why* they work, from the ground up, is extremely important -- and, for me, it's the thing that makes math fun! I guarantee I won't remember the details of this in a few months unless I watch multiple times, but if I ever run into a need for Fourier transforms, I also guarantee I'll be thinking of this video to figure out what I need to do :) So thank you :)

[@dp0813]

IceMetalPunk Explain it to yourself a few times after watching the video a few times, then explain it to others so they also see the big picture. That is one of the best ways to learn it. 👍😊

[@umair5602]

7:07 I just realized the reason it's giving a big spike at the zero frequency. When you move the frequency graph up, you're basically adding in another wave with a frequency of 0. The fourier transform still works

[@mahxylim7983]

You're right!!

[@Zylarlander]

Woah, great insight

[@victorwilburn8588]

Yep, good point, it's the "DC coefficient", as we would call it in the context of the Discrete Cosine Transform (similar in concept to the Fourier transform) that is at the heart of MPEG video compression (though I'm sure that term is used in other contexts as well). As opposed to the "AC coefficients". Borrowing terms from electronics.

[@Vancuum]

Look into the Dirac delta function. It turns out that the fourier transform of a constant is the aforementioned function. Your realization is more profound than you realize!

[@mjolnirswrath23]

@@victorwilburn8588 correct you want a pure DC waveform molecular Attenuation Field Density , AC hum " 60 Hz " Thus why in order to activate TRUE NMRFA of molecules you have to use Pure source of DC energy Alternating Current " Hum "contaminates a pure Sine wave frequency.. US War department R&D 1946 Infrasonic Waveform Weapons technology.

[@sudoLife]

The way you connect the concepts is so amazing! I can't even begin to imagine how much effort you put in your own education.

[@NandhiniChandramoorthy]

What an elegant explanation! I wish someone had taught me the concept of Fourier transforms with such clarity when I was an engineering undergrad . Leaving aside the excellent visualization for a moment, just this insightful way of looking at fourier transform- winding frequencies and all, makes it understandable finally. You, sir, are truly a genius !

[@abc7297]

When you understand the main idea of Fourier Transformation finally in a youtube video instead of in a whole semester in university.

[@jimjim3979]

It's even better sometimes on KZbin

[@orti1283]

I can relate to that feeling too, but to be fair, it probably finally sunk in because you had built notions and worked with them before despite not understanding them to this level of intuition

[@marcoottina654]

this video should be proposed in EVERY FUCKING CLASS

[@johnevon8220]

@@orti1283 good point

[@crl122486]

I actually watched this before studying this in uni. For new people, this will shave off about 80 hours haha

[@vaibhavtiwari8700]

"chaos and chaos and chaos chaos chaos, and whup things line up pretty nicely" *repeat* -- My life

[@GubeTube19]

I read this comment exactly at the same time he said it in the video lol

[@emilandersenaudio]

love the WHUP

[@TaborPrzemyslaw]

If you think about it, events in life are kind of like different frequencies that sum up

[@980616]

Good one!!

[@snehadash7117]

I love how lucid and elegant and visual 3b1b videos are...How each step is derived from the previous one and how we build up to the final result from almost absolute zero through a very intuitive and systematic sequence of steps❤ We are blessed to have this channel Loved every bit of this video's brilliance!

[@winterbird4069]

I’m into my third year of one of the most prestigious programs at one of the most prestigious universities in my country, and even there they fail to simplify the core concept as well as you do. The visuals and examples you give are 100x more comprehensible than what my profs have tried to explain. Thanks for helping educate the world in a better way :)

[@beenaplumber8379]

I'm a retired neuroscientist and now a part-time middle school teacher. I teach digital music production, and I rely heavily on Audacity software in my class. The noise reduction algorithm relies on Fourier transforms (which I have never understood adequately), and this video has helped tremendously! Thank you so much! What an intelligent way to reduce unwanted noise! Far better than a simple noise gate. Nice to finally understand the way it works. Sorta...

[@ahreurink]

Don't you use fourier transform in interpreting EEG in neuro science?

[@beenaplumber8379]

@@ahreurink In my research, I never used EEG readings. I know they do use a complex noise reduction algorithm, and I'll bet it is the fourier transform. My work involved more of the molecular biology and behavior, and my labs mostly used animal models. Getting an EEG reading on a poor little mouse is not easy.

[@MadScientist267]

Until you end up with the NR gurgles lol

[@LunizIsGlacey]

@@ahreurink Wow, neuroscientist to digital music middle school teacher is not a career path I would’ve guessed! Your interests and skills must be very diverse.

[@kidzbop38isstraightfire92]

They teach digital media production in middle school? That seems so niche lol

[@willypataponk]

In 20 min you've explained what my Analysis IV teacher failed to do during a year. You're fantastic!

[@Turbomachinsky]

Absolutely, positively, the most stellar and intuitive introduction to Fourier transforms I have ever seen. You have far better explained, in just 20 minutes, what my uni lecturer ever could in a semester back in the day.

[@RedLamentations]

I wish I could express how thankful I am to you in a way that is more than just words. You have continued to provide free, quality content that has helped me to understand so many concepts in my engineering classes, and in most cases you’ve done a better job in an hour than the professors have done in a months worth of classes. From the bottom of my heart, thank you

[@simonblaha]

1:50 Seeing a mathematical representation of a Dminor7 chord is all my jazz-mathematician soul ever needed.

[@androidiscool7437]

♪ D minor 7 with 3 Blue 1 Brown ♪ - D F A C Intensifies

[@calyodelphi124]

I just came back to this video again after a year or so just to re-watch it because thanks algorithm, and I came down into the comments to make this exact same comment only to find this one. :D

[@Alex-ud6zr]

I take it you enjoy the Black Saint and the Sinner Lady then?

[@calyodelphi124]

@@Alex-ud6zr I'm not terribly well-versed in specific jazz artists/bands, unfortunately. I've just had a few years' worth of band experience in high school and subsequent music theory osmosis vis a vis Adam Neely more recently. ^^;

[@Alex-ud6zr]

​@@calyodelphi124 Coolio. Well, if I have but 1 album to recommend, it would be The Black Saint and the Sinner Lady as it 1. fits the description, and 2. across the 2000 albums of which I have listened to and logged, it stands as the most perfect representation of music that I have personally ever had the pleasure of listening to. :>

[@wwkx]

I'm studying engineering and I've been passing exams like calculus and acoustics where the concept of Fourier Transform (and Laplace too) should be well understood. But nobody ever explained it clearly or tried to, they just gave us the formula for the series and that's it, we basically had to accept it blindly. Researching on my own just gave me back analytical processes to get to the final integral, but without explaining the meaning of each step and just putting even more confusion in my head. I also like to make music so I've been playing around with equalizers and the like for a long time and I ended making up my own idea of how the frequency analysis actually works. But now in just 20 minutes you managed to fill one of the biggest gaps in my knowledge and I'm so happy you made this video, like from 14:29 on I really started seeing rainbows all around lol. Thanks a lot for this vid, I can't wait for the next ones

[@emmanueloverrated]

Same experience as yours. I had to accept the transform without understanding what was the thinking process of M. Fourrier. Pretty sure he was teaching it that way, when alive. Upper grades in mathematics would be much more interresting with explanations like these. Professors are lost in the abstractions and have lost (or never had?) the art of transmitting the knowledge.

[@Ayplus]

wwwKx same here buddy

[@Ayplus]

Truth

[@elherediaenc]

Truth has been spoken here! This video is helping me to understand Computational Photography concepts (Processing,etc...)!

[@tricanico]

jshowa o maybe it's important to read a little bit of Shakespeare here and understand we are humans After going through that, i believe it's important to understand stuff for the future of humanity, don't you find it odd that math is taught so badly? And that people that studied math a couple hundred years ago not more were chased and burnt to death? Something's going on here. But even more practically: don't you think that when you understand stuff you just are faster and better doing everything?

[@lucasfreitag9794]

Genius analogy ! I often rewatch your videos to freshen up my intuition. It is so nice to have your animations and ideas to better understand difficult concepts taught in University.

[@juanrossi3]

I'm speechless with this work of art. The explanation of the underlying principles is amazingly clever, and the visual production is outstanding. I can't imagine the amount of hours needed to make all of this. It's an honor to have watched your video!

[@gonzalobera96]

Your videos sir, are a gift to humanity

[@berbudy]

so true, I'm so glad this guy exist

[@JorgetePanete]

berbudy exists*

[@SameerAnsari-yj1tt]

So Trueee...!!!

[@mokopa]

I often wish youtube had a better way of indicating HOW MUCH I like a video. This one would have scored high.

[@franciscofarias6385]

yeah, by youtube standards i liked this video and the one with the rat taking a shower equally

[@briancooke1906]

right! If I click like really hard, it just likes.

[@Israelpwn]

About 8 years ago, YT had a 5 star ranking system. I preferred it a whole lot more.. However, that meant you couldn't save the video in a "liked" list unless you favourited it.

[@AaronRotenberg]

Mikky They got rid of the five star rating system because the vast majority of ratings were either one star or five stars. People are hesitant to give a "bad" (read: less than five stars) rating to anything that has no significant flaws, even it is fluff rather than solid content.

[@SeraphimKnight]

Remember when youtube had star rating for videos? I always like that system better.

[@ruaraidhmcdonald-walker9524]

Beautifully done. Went looking for a simple way to explain this to younger engineers who can use it but don't "feel it" and you sir have just nailed it. Perfect.

[@danishali8335]

Brother, You deserve an applause, because of the Explanation and the Modeling of the video. I really appreciate your work, and all of us are thankfull to you, God bless you

[@Pear.Jerry.]

I study Applied Mathematics and I think of myself as of someone knowing quite a bit of mathematics and also (from what others say) having some sort of skills in transferring this knowledge in a pretty understandable way to other people. But compared to you... Well - YOU ARE SIMPLY A DIDACTIC GENIUS! The whole 'scenario' of each episode, the way you move from the things which are very easy towards the more difficult ones, the examples that you pick, the interpretations that you find, the visualizations... Each video is just a mathematical masterpiece! Thank God there was a moment in your life when you thought that your mission is to show the people the beauty of maths on KZbin. I cannot think of anyone doing it as good, as you do. :)

[@erikziak1249]

This is one of the highest quality content on KZbin.

[@orangeheartguy]

I absolute love how you represent these complex mathematical ideas with such beautiful illustrations. I also appreciate the touch of explaining a concept you know people might be confused by taking time to explain all the components, especially when you were talking about the winding of the signal around a circle. I also appreciate how you break down the formulars by building the understanding of the viewer first then, like building blocks reconcile all that has been taught. I literally search for your channel + the topic name when I want to get a beautiful understanding of mathematical concepts. Thank you for all you do. I'm very sure a lot of college students like me deeply appreciate you. The fire of my love for math burns brighter after your videos. After watching this, I'm definitely going to check if you have one on z- and Laplace transforms.

[@maitenoda1730]

I'm stunned by the clarity this video brought me. Thanks so much for puting materials like these in a free platform. Your didatics is amazing. Congratulations!!

[@AleksandarBosakov]

15 years ago I was a chemistry student and I had to take exams on linear algebra and calculus. And I specialized in spectroscopy. That was a long time ago, now I'm in another sphere - computers, and I've forgotten a lot of details, though not the basic concepts. Let's just say, your videos made me connect a lot of dots in my mind. Incredible work, man. See, it did happen for me to teach - there was a schoolgirl next door, whom I sometimes helped with the lessons. Once she told me that she gets higher grades if I explain the lesson to her and that I was born for a teacher. Yet it's guys like you, that I take my lessons from.

[@trottsky87]

I am an electrical engineer and I studied Fourier Transforms in university and remember not understanding how it works. In my job I actually use Fourier Transforms to view the frequency domain, but again tools calculate it for me. When I viewed this video, it was the FIRST time in my life I finally understood how Fourier Transforms work! I absolutely love this video! Awesome job!

[@xavierbolivar-hazboun1347]

I've been coming back to this video every couple of years as I get through my university career and makes more sense everytime. Cheers!

[@azursmile]

This video really helped me out of a bind where I was expected to understand FT with hardly any intro. Thank you so much!

[@vishnukumar4531]

This video gives me chills . How could a person come up with such elegant explanations 🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥🔥

[@dl0.0lb]

Dang - I stopped taking math in 10th grade after my AP Math teacher took me out of the program because I didn't like doing all the extra homework. After going to school for music but becoming a programmer, I continued to learn more advanced math as needed. I've had random times where I've used the Fourier transform, but never really understood it or why it worked. This made it crystal clear and it feels good to fully understand what is going on both conceptually and in the notation. Thanks a ton for your time putting this together.

[@trenvert123]

The way you explain it makes it much more interesting. I went from wanting to know what it is, and not getting a good definition from Wikipedia, to wanting to create one myself. Thanks!

[@teachies902]

this is awesome!! i've been working on a project involving decomposing waveforms into their harmonics, and these videos clear up so much.

[@MisterMcHaos]

Sometimes - just *sometimes* - one needs access to a "double like" button. Well done, 3B1B.

[@Renanbmx123]

I agree my friend !!!!

[@JorgWagner]

In this special “sometimes case” I would be tempted to hit that double button multiple times.

[@zbzb-ic1sr]

I wish grant had a video on tensors, been browsing the internet and I can't get an intuitive feel on tensors.

[@HattoriHanzo62]

Yep, one per turn :-)

[@oafhauohguoihgakds5151]

actually on windows you can do it by pressing alt+f4

[@dombowombo3076]

I would love to see a video about the Laplace - Transformation. I´m very gratefull you are out there making videos about topics I´m often not able to fully understand myself. :)

[@AdityaSharma001]

I just cannot express how amazing your explanations are, because that praise will still be less. So I just want to thank you a ton for helping us and making a better knowledgeable future.

[@sdsa007]

This is one of the best visualizations ever! and there was another Shakespearian one you made that is also marvellously insightful about vector spaces, awesome works of educational art!

[@rdhunkins]

Thank you for this! I have a BS in engineering, an MS in Computer engineering and none of the math, or signal processing classes I've taken explained this in several semesters as clearly as you did a few minutes. It all makes sense now. I finally get it. Bravo! Standing Ovation!

[@softwarefac]

Every lecturer in my life has failed to learn me what Fourier transforms is but you did it in just 20minutes. Amazing!

[@bibekluitel6113]

Loved the way how you explains and how you go deeper in the relevant field.

[@Sunny-qe5el]

I had a hard time understanding "Signals and Systems" with Fourier Transform for frequency domain representation of signal. But, everything is so clear now. I am so grateful to have found this video.

[@purplehippo27]

I've had 5 years of undergraduate/graduate maths education and this is the best introduction to Fourier Transforms I've ever seen. Amazing. I must've done a billion Fourier Transforms and I've never seen it explained in this way. Plus the fact that you were able to derive the integral definition in a way that I feel anyone willing to learn it will understand easily is incredible!!! Keep it up man 👌

[@qwerty11111122]

I love how you went through building up the expression that would later be the "fourier transform" function. It's been my thing to make sense of equations. They're not just variables that you can plug and chug. Each term in a formula stands for something that should exist in the real world, and knowing what it means just fills my heart. Keep on fighting the good fight, Grant

[@101_huzefajohar7]

I can’t express in words how grateful I am to you for helping me clearly understand and visualise the logic and derivation of such a comlplex mathematical topic. Youre really doing an incredible job! There is no comparison of the quality of third class education given to us in Indian colleges and what amazing content you are giving for free!

[@donlevinson4645]

3b1b, really, you are some type of an angel. You fix the irreversible damage that highschool and academics cause you when they "teach" you math in most possible boring and non- intuitive way that there is. I cannot imagine the amount of effort you put into these types of videos and I really wonder where YOU studied math in such a way that makes it so clear and second-nature to you. Thank you a lot from the depths.

[@MrxstGrssmnstMttckstPhlNelThot]

Gotta convert numbers to a base with more fours in it. So it's fourier. For example: Take 12057 make it base 9. Which gets 17476. Then 17476 to base 16 is 4444. That number is the fouriest. Good job! ;)

[@deepanshuchoudhary4598]

I am literally speechless, I don't know how could I thank you for this stuff. I wish our teachers would have taught us in that fun way, where maths feels natural.

[@n.p.9997]

Did you really understand all that ?

[@deepanshuchoudhary4598]

@@n.p.9997 Yeah, bcoz of mathematical background.

[@buh357]

Honestly, I watched for around 3 minutes and was already blown away by your explanation. WOW.

[@eicydee3212]

These videos are amazing, thank you! Heard about all these topics in lectures while studying, but never really understood them, managed to pass the exams anyway, got my degree and never needed this math again. And these videos make me really understand how these things work and how everything just lines up if one looks at it with the proper mindset. Really a blessing for today's students and everyone who is just interested in it.

[@fosheimdet]

It's hard to overstate how valuable your videos are. They breathe life to difficult concepts which are often just presented without any motivation/explanation and expected to be taken for granted.

[@mostafaabdein459]

Dude, Your work is so great that it seems almost impossible to me, you manage to simplify the most complex concept to a degree that was -for me- unimaginable, great job

[@kaejolie]

I have a PhD in Engineering and I never had a teacher explaining Fourier transform so intuitively. We have been defrauded by the academic institutions. Also, I have not seen a textbook explain this concept so elegantly like you did. I want my money back!

[@stkyriakoulisdr]

From Neural Networks to Fourier Transform to Linear Algebra and Calculus... A Like is not enough really for all the work you are doing

[@markfisherectuals]

sponsor them on patreon, it'd really help them, and us, in making more of these beautiful videos, more often.

[@7gamezone]

SOOO TRUE!!!!!!!!!!!!!!!!!!!

[@sahandsoltanieh5186]

I confirm that :)

[@EDoyl]

Every time you upload a video I think: "eh, I've studied that before and these are supposed to work as an introduction for an unfamiliar audience, I'll listen to it in the background or something". And then every time it's a mind blowing approach. When you say something like 14:43 and I realise where you're going with it I'm always surprised and amazed at how elegant and cool you've made the topic.

[@DeathBringer769]

That's why mental laziness and complacency is always a potential threat to any of us, even more so when you think "oh, I've already learned that" in your head, or you go "oh, I can't ever be mentally lazy; I have a college education and studied hard." I've been guilty of the same in the past sometimes but I keep these ideas in mind and actively try to be better. Remember: you ALWAYS have something more to learn. NEVER stop trying to learn more every day (or at least improve/further clarify your current understanding on subjects at the very least.) Having worked or studied hard in the past is no reason to get complacent in the *present* and think you're "done" already (that's just "self-justified" laziness, in my view.) That's the attitude I try to take at least ;)

[@EDoyl]

Thanks for the input, Deathbrewer

[@ameymeher4559]

I almost cried watching this video. Such great explanation! Everyone deserves this

[@nyahhbinghi]

it is a thing of beauty

[@parkerstroh6586]

bloody hell me too ahahha

[@kin_1997]

Beautifully explained, second time watching the video and its starting to make sence, will be back again to further try and grasp the concept just as intuitively you make it seem to be. Your effect is forever appreciated 3B1B.

[@RandomMusingsOfLowMelanin]

THIS IS SO BEAUTIFUL

[@stonail665]

Give this guy a Fields Medal for discovering the best way of teaching Mathematics.

[@BritishBeachcomber]

I've been designing electronics using the FFT for more than 40 years but this is one of the best explanations I've ever seen.

[@vibrato17]

Sometimes I lose hope about the future with all of the news going on, and then I remember there are people like you, Mr. Sanderson. You're taught us so much and reignited curiosity in millions of people and we'll try to make the world a better place with it.

[@ImaginaryHuman072889]

actually, the fact the graph shows the spike at 0 when the sine wave is shifted up (because you don't want to deal with negative values) makes perfect sense, because this effectively is telling you that there is some 0-frequency (or "DC" in electronics) component to the signal. anyway, great video. as an EE who took all sorts of classes on this stuff in college, I still learned some stuff because these explanations and visualizations are amazing!

[@50srefugee]

My life would likely have been very different if these videos had been available to me when I first attempted calculus fifty years ago. I learned a set of rules for working with equations. But when I hit integral calculus, I had only the dimmest understanding of what those rules MEANT, and found I could go no further. With these videos, I might well have been able to climb over that wall, and see much further from its summit.

[@paulbarton4395]

Even the calc books are better nowadays, bigger with better pictures and more geometrical explanations, nicer formatting (but more $$$ also)...I've seen math texts from 50 years ago, they don't look fun

[@aoliveira_]

@@paulbarton4395 I don't like "fun" books. They look like childish.

[@aoliveira_]

Sometimes is good that the books are difficult. Sometimes you are understanding just because you feel that you are following but in fact you are not understanding at all. Trying to figure out difficult things makes you realize and correct that.

[@paulbarton4395]

@@aoliveira_ I was referring to the books formatting, under the assumption that the content is correct and rigorous. Books from 1950 have tiny, cramped text and bad images. LaTex formatting has improved everything, and as for visuals, our ability to generate 3-dimensional plots can help build intuition.

[@polygontower]

@@aoliveira_ Fun in this context means more understandable and intuitive

[@lourias]

If only this video had been published back in 2011... I kind of understood the sequence, but you adding the SOUND frequency portion now makes PERFECT SENSE!

[@aryamsthapak7095]

"Kind of...This is...a bit of a lie, but it's in the Direction of the truth" -A great mathematician

[@hassanakhtar7874]

Great teacher*

[@hichamsabah31]

A sign of wisdom.

[@TheMadHermit]

I took a class on this in College and I still had no idea what Fourier Transforms were when I finished. The first 2 minutes and 22 seconds of this video made it abundantly clear. Well done! Clear, visual, best-in-class video explanations like this, that can be watched over and over again from the comfort of your home is a game changer and will disrupt the traditional and expensive College education system as we know it. I think Khan Academy is on the right track. If they start giving out degrees it's Game Over.

[@tricanico]

The vision of the future

[@MaximeCAMELOT]

That's so true mate !

[@stephenvoncrven4319]

i think you are somewhat hiased. these types of explanation is mostly apreciated by people.who already got ass burnt by college courses and were forced to solve problems/projects/whatever your uni is up to, then forced to somehow "make the notions work for them". at THAT point those 3 minitues made your brain snap and say "that's it!" and probably lots of electronics, signal, automation notions came together (look at how similar are those patterns to equilobrium criteria in auyomation control). are we sure it will work the same the other way around? without that ass burning college part?

[@danbahadurgurung8593]

me too bro. understanding concept to the core is what i also seek, but tradional college is just a degree machine, where u gobble all formulas and vomit in paper. no idea what u studied in ur entire degree. i also love khan academy. :). do u know another channel or videos which can help great in college level math. like partial derivative,

[@TheTacticalDood]

@stephen von crven Exactly!

[@Mnmunky]

This is an unbelievable video, explaining what I leaned a million years ago rote, into actual understandable terms. Thank you! When my son gets to this point in his math journey, I’m going to sit down with him and we will watch this together.

[@Quasar.Chaser]

I was going to read and learn about Fourier transforms to answer one of my assignment problems. But then I thought "Hmm maybe 3blue1brown has a video on fourier transforms that can help me get the intuition for what I'm about to try to learn" and voila, here I am and I sure am glad about making this choice. Incredible explanation as always! Grant, you make me want to keep learning more and more math!

[@Nachtgrabb]

I have a master degree in Mechatronics. I had lectures in Signal and picture processing. I learned how the transformation behaves, its properties and what happens if you change the function, shift, strech it etc. But nobody gave us an analogy to understand the tranformation formular itself. Thx man. Now I have a new perspective to this topic.

[@thedayb4tomorrow]

I've been subscribed to you for a while now and have thoughts since the beginning that you have a very rare talent for explaining mathematical concepts but this video might well be the most awesome explanation of a mathematical concept I've ever seen. I've recently left academia after a PhD in Particle Physics so now I don't get nearly as many opportunities as I used to to get excited about Maths and Physics but every time I see that you've published a new video I get excited :-) And of course as many others here, I really hope that people will start using material like this in lecture theaters. It gives so much more of a sense of the beauty of mathematics than writing on a blackboard ever could...

[@anushka.narsima]

omg from my math class i just remembered fourier transforms as 'some complicated looking integral formula that seems important' but I finally understand this wonderful topic now. Your videos make me value mathematical concepts so much!

[@beck5429]

This has been a weak spot in my understanding of math since university and you have explained this concept in such a beautiful way, I feel much more confident with Fourier Transforms now! Watching this to prep for a job interview that involves low frequency acoustic data analysis from a microphone array. Crazy stuff, your video really helped.

[@jayanths8526]

Life seems so simple and beautiful when I watch your videos. Thanks..

[@Rebbyrag]

As an electrical engineering graduate I SERIOUSLY wish this had come out while I was still in college, SUPER insightful and it all comes together and makes SO much more sense now. I always thought of it as a shorthand "switch" back and forth between frequency and time domain, but the polar graph and Euler's equation makes SO much more sense.

[@dp0813]

Josh Garber well, technically it IS a shorthand switch between freq & time domains, bc interpretation & manipulation are easier in one or the other domain, depending on the circumstances. But you're right; the explanation in this video is superb & goes beyond just freq & time domain.

[@tharindusathsara3414]

Thank you for this incredible animated explanation of the Fourier Transform! It's so clear and intuitive, making such a complex concept easier to grasp. Your videos truly make learning math a joy! ❤❤🔥

[@ramchillarege1658]

You are amazing. Being a EE and having worked through this in my younger days, one would think one has some reasonable notion of this. But, there is more. I was just watching lectures by Brad Osgood (Stanford) earlier today and enjoying the discussions on convergence. Then I accidentally watched your video. Your presentation is excellent and the illustration of "wrapping the function around a circle" with the graphics brilliant. Thank you. I will be back to hear more of your lectures.

[@BecozPro]

Honestly as a maths student in my 3rd of 4th year, watching your videos reignites my passion for it. Thank you, you're doing a genuine service to us all :)

[@KaylixYT]

The work done in these videos is really some of the most encouraging, and pragmatic use of the Internet.