3Blue1Brown - 2019-06-30
Fourier series, from the heat equation to sines to cycles. Home page: https://www.3blue1brown.com Brought to you by you: http://3b1b.co/de4thanks 12 minutes of pure Fourier series animations: https://youtu.be/-qgreAUpPwM Some viewers made apps that create circle animations for your own drawing. Check them out! https://www.reddit.com/r/3Blue1Brown/comments/cvpdn7/make_your_own_fourier_circle_drawings/ https://isaacvr.github.io/coding/fourier_transform/ Thanks to Stuart@Biocinematics for the one-line sketch of Fourier via twitter. As it happens, he also has an educational YouTube channel: https://www.youtube.com/channel/UCKOiJd9YCbv7LeL2LFOGiLQ 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 https://youtu.be/qS4H6PEcCCA The Coding Train https://youtu.be/Mm2eYfj0SgA Jezmoon http://www.jezzamon.com/fourier/index.html 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" ------------------ These animations are largely made using manim, a scrappy open source python library: https://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 it 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: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u 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. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
Fun fact: Fourier series were used to produce this video and the audio too. The pictures and audio waveforms were analyzed by a computer and turned into numbers that describe the vectors. Playing back the video is a matter of replaying those vectors. Compressing the audio and video to make it take up less space and less bandwidth is a matter of stopping the analysis when more vectors don't add more visible or audible details anymore.
So Fourier is the reason we can stream media as effectively as we can? That's amazing.
I knew it! Haha
With the audio, isn’t PCM sampling the standard compression technique?
I can see Fourier used for video because of the low FPS rate 30 or even 40) but I can’t figure out how it could work with audio... yes it’s easy for an static waveform but actual waveforms in real life are not static and need to be sampled/analysed at a higher rate... maybe I’m saying something stupid ^_^
@JaBig * Sampling and compression aren't the same thing. PCM basically means measuring a value at regular intervals. For example an audio CD has music in PCM format that's sampled 44100 per second, using a value between -32768 and 32767 to represent the signals of the recording.
Fourier transformation is used to divide the signal into a number of frequency bands, and determines how much each frequency band contributes to a short fragment of the signal.
If you know how much each frequency contributed to the signal, you can synthesize the original signal from the values. Recording the values of the frequency-domain version of the signal takes up the same amount of space as the time-domain version of the same signal, but lossy compression algorithms reduce the precision of the recorded frequency-domain numbers to reduce the data rate that's needed to store and transfer the signal. This is the principle on which every single audio/video compression method is based.
And in answer to your question: would it work better on a more pure signal that's more predictable: yes it does! If you would want to transfer a single tone as a compressed audio stream, it can be represented perfectly (or almost perfectly) because the simpler the signal is, the less data is needed to describe it accurately. So even a very-low bit rate data stream can accurate represent a single-frequency tone. But of course music or audio in general is more complex so you need more bandwidth. But with smart algorithms, the human ear can be tricked into thinking that even a low bandwidth data stream in mp3 sounds just as good as a PCM audio steam. Your mileage may vary of course: some ears are better than others.
For video, the same principle is true: pictures are divided into "macroblocks" of 8x8 pixels. Colors and brightness are separated and then a Fourier transformation is done over the horizontal and vertical pixels of each block, and smart algorithms are used to represent the data in fewer bits by reducing precision.
@Nguyễn Đức Anh Why are you trying to be edgy? Jealous that you don't understand things like this?
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.
@Anarchy you should go major in mathematics and see how difficult it is and then be unable to talk to anyone about it outside of your sphere at college because no one understands it. then you feel isolated after working so hard and then people like you act like theyre arrogant... its ok tho
@Jack Pistone Pure irony.
You've chosen to take on the study load of two STEM bachelor degrees, buddy. Just one of them would've already put you under inhumane conditions... Let alone two. Of course that's going to take a toll in your social life. It may even prevent you from developing in other areas that help you become a fully independent adult that are important also.
I've been there, done that. I burned myself out in the last year of my undergrad course. At that point you can't really drop out, and it started to sink in that I was a slave all along, in a system designed by a small group of people (It took me years to recover from that).
A technologically "advanced" society doesn't appear out of nowhere. It's rich billionaires like the Carnegies investing in education that get to decide what our curriculums look like. There was a time once, and still fairly recent, where our ancestors did not go to school from a young age. They weren't as heavily indoctrinated with information that is only useful if you wish to become a scientist or engineer. As such, I only became an engineer because we're told from a young age to view these jobs with the utmost respect. They plant axioms in our brains like "Science is truth" and "God cannot be proven".
I'll ask the following question. What certificate is worth sacrificing basic fucking human needs? You probably don't even have the time to have a partner (someone that can make you feel less isolated), wind down and relax, or study subjects you actually DO find interesting out of personal interest and not because it's part of a curriculum you never had any say in.
Of course, I don't know your personal situation. But you're already having a tough time in your third year, and the work load is just going to increase. If you feel like you're going to lose your sanity, I recommend dropping one of the bachelors and sticking with the one you see a future in. Spend your remaining time in becoming a wholesome person, which will bring you stability in the long run. You can always take on a graduate degree later on in life, if you feel ready for it by then.
Joy New I’ve studied Mathematics at college. Thanks for your comment.
@Anarchy I deeply appreciate your perspective. I really do. I am actually doing fairly well. I have an office position in the navy for nuclear science when I graduate and my life is great. Living with a bunch of guys actually from highschool that I met in college. Lol. It’s still extremely difficult but I know it will be worth it. Especially when lesser difficult degrees begin to be rendered useless. Adderall is my god.
@carpe diem nerd
"Taylor made polynomial" and then not acknowledging the pun, I love it
He also dropped "foray into Fourier" on us.
I wouldn't have even caught the "tailor made" pun if I hadn't seen this comment, because I was falling asleep while watching this and thought he literally meant to say "Taylor" and didn't even notice he had said "tailor made". I just heard "Taylor" and was like "zzz yeah Taylor series right got it zzz"
@Hextator LOL you're super cute
Egg of females Inside
@NASA limbu ehh what?
@Max oh. ohhhh OHHHHHH MY GID
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.
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.
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!
Totally
@Shrey Tiwari same pfp.almost😂
Pretty amazing for a guy with “only” a Bachelor’s in Mathematics from Stanford.
@Aryaman Mishra He only has a Bachelor’s in Math and comprehends the stuff better than most PhDs !
@TBG Visualizing a Fourier Transform is an incredibly useful way to understand it !
I’m imagining students watching this and furiously taking notes while I’m here at 2am thinking “oooh circles make shapes”
Whoever first invents a time machine, please go back in time and show Fourier this video. Absolutely amazing!
@Agustin Aviles Maybe he made this series because of this comment :3
Imagine how he visualised this inside his head back in 1800s.
@FiasaPower I was wondering the same... What a genius could imagine this?
Amazing minds.
That obviously never happened, so time travel will never be invented. QED
Saurabh - they already did.
friend: how well can you draw?
me: how many arrows you got?
friend has left the chat
@Leopold Simmons - one, it's got a little graphite tip
WAIT WAT KIND OF ROAST WAS THAT?!
I did not expect this many likes lmao
Lmfao
Holy shit someone needs to make one of these as a spirograph type thing
me in the beginning of video: "Time to get smarter!"
me halfway through: "Oh look the circles! it draws da picture! :O"
So you succeeded!
Math is even more beautiful when someone teach like this.
@Robert Evans unfortunately lot of people look at screen instead of closing eyes and listening
Watching the motions of those 300 arrows reminded me of swirling stars in space, and I wondered if those moving swirling stars and galaxies have their own Fourier Transform composition.😊
@Moujesh Agrawal ,
Math is fun when no grades are at stake
True
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!
I'd say he Grant-ed you the insight you needed.
Who would win?
a) Many artists who have received PHDs in art.
b) Some rotating vectors
@Queta Arbuste wow thats a looooooooooooooooooooong poet
Some arrows and round bois
@Sunny the Great No PhD in Art, MFA is the terminal degree. And don't disrespect the arts at all either, as an Art Math double major both deserve immense respect not only for their societal contributions, but also for both being quite rigorous.
@Hồng Hải Nguyễn Khắc lmfao
This question led me to think me now from where exactly does the artist starts(initial value) 🤦🤐unlike the series of Fourier .,🙃...GOD 's Plan 😄😄😇🤘🏾
If a pile of Fourier series can draw one picture, it can draw a sequence of pictures. When is the anime coming out?
@ElectroCow what if the anime doesn't have a manga?
@shayan moosavi It has light novels though.
Let's take this to the logical conclusion: a 3D movie defined as an octonion fourier series. three dimensions for space, three dimensions for color, and two dimensions for audio waveform, exclude the real components. THIS NEEDS TO BE DONE!
@Xander Rice You gotta start it! Let us know your progress
no.
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
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.
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.
A lot of Stanford/MIT/UCB etc Computer science graduates have all 3.
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
It's because of people like him that i learned about math more
Primer omg Primer I’m a big fan!
10:08, a beautifully wordless proof of e^-ix + e^ix = 2 cos(x). Thank you for this!
i actually didnt think about that, ur a smart guy
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).
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.
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.
No Taylor, Lagrange, Descartes and Pascal? Damn...
Shamefully missing Thales among the immortals as well. Video dropped from 10/10 to a 9.5/10 just for that.
"by using the power of the spin, one can unlock many more possibilities than previously thought possible" -gyro zeppeli
The Spin can draw, but only if it manipulates enough vectors. Thus the stand “Ball Breaker 3 Leaves” or “Tusk Act 3: Leaves” would be able to draw with great precision.
Perhaps if I was being original, I guess the stand would be “Photograph” since it can accurately copy drawings like a photograph copies real life by reflections.
That moment when Rotating vectors draw better than you 😑
DarthZackTheFirst r/woooosh
Hey, but do you have 300 arms to rotate 360 degrees, each at a constant speed? I don't think so
I mean, they usually do, lol...
That moment when people specify rotating vectors better than someone who probably doesn't even know what rotating vectors are (-_-)
Define "better". Then make a rotating vectors chain to write down the definition, so we forgive you...
This video is being recommended to me by my university maths department!
All the best from Berlin! Awesome work
"It sets a good foundation for the ideas that will come up later in the series, like the laplace transform..." So... when are you gonna continue this series?
14:50 my brain just melted.
realizing what happens when I see my computer screen. The CPU will calculate thousands of these vectors in less than 1/60 second to generate an image every .016 second. And it's using imaginary numbers! I'm going to love relating vectoring to video games when I'm tutoring :)
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...
3:42 “To get a new Taylor made function...”
Does that supposed to be a pun, or you genuinely want to say tailor-made :v
You'll never know :)
Her in bed: He is probably thinking of other wommans.
He: How many arrows do i need to paint the Eiffel Tower?😄
oo
thats a good meme :D
just replace Eiffel Tower with idk anime girls or so ;)
0:32 Where's all the love for the mighty Nail and Gear?
OrigamiPie Flaggy Flag
The mighty Nail and Gear has flown in space, and now in 3Blue1Brown...
"The power of rotation is limitless, trust in that!"
AKA
Tusk Act 4, for art
Im only click the video searching a jojo reference.
I dropped out in 10th grade 25 years ago and your videos have inspired me to go back to school.
I definitely recommend https://www.khanacademy.org/ as well, it's an amazing little miracle in education. Great source materials, many many excercises which usually explain why are your answers wrong, progress monitor, predefined study paths... you even collect achievements and other tiny fun details.
School is slow. Consider learning online. Just find a direction.
im only in fifth grade and i suck at school because my i use my right brain istead of my left brain because on my left brain i think in balck and white and only do regular math and english. but on my right brain i do algebra, arithmetic, music, art, dancing and think in colors
I'm your 1000th like
aww
thats really nice
Just for those trying to do the exercises, dont get stuck on trying to write cn as an/2 + ibn/2 because cn is actually an/2 - ibn/2 ,and from there you can get the correct conclusion.
let's watch this for the 6th time, this is like when you go back and watch rock lee vs gaara
How is it that you and I have exactly the same interests.
Having a PhD, I watch your videos and nothing is new to me. But man I feel like I am learning a lot. GREAT JOB and Thank you.
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!
Your visualization skills alone deserve a Nobel prize :O
Nobel prize for optics?
@hello from C-137: as in Fields Medal? Indeed.
Agreee👍👍
Agreed!
and you yourself deserve another
I'm nodding my head and pretending I know exactly what he's talking about
I have no idea what any of this means. But it's so beautifully explained and satisfying to watch, that I can't.... stop.... watching
After watching all of videos I can finally say this man desrves a nobel prize for making things easier
19:39 what happens when I by some unlucky coincidence have two constants which are the same?
The integral would give me f(t)*e^a*i dt=c1+c2
But I wouldnt know that and would guess the solution for c1 is c1 + c2
"immortals of math"
playboi you didn't even include Hilbert and Grothendieck
rip yung hilby with the bucket hat
So in the beginning, I was somewhat curious about why the "straight lines" in the Fourier drawings were never quite straight. Now that I have seen the video, that fact makes the Fourier series even more beautiful.
I just want to say i really appreciate how this is a video in a series, however the creator goes out of his way, and loses views, just to make it easy for us to understand, honestly i would've fully expected 'So go watch the first few videoe in the series so you know whats going on' , you
know, the youtube way, from another creator. greatly appreciated
Sir, you are amazing. There are no more words that I can tell to thank you.🤗🤗🤗
that list of immortal mathematicians is imcomplete
it doesn't have you in it
@48956l dang, a toxic kid
Wrong, it doesn't have ME in it and that's a problem.
@48956l educating uneducated people like you is probably better than publishing a paper.
it doesn't have any Arab or Muslim mathematician even Alkhawarezmy the one who invented Algebra and introduced Algorithms
I can’t imagine how can someone come up with so beautiful animations!!! Really love it!!!
(I don’t even talk about the content and the teaching methodology!!! Lovely!!! Brilliant!)
This, Johnny, is a phenomenon known as the “spin”
Hey Grant, could you do something about Laplace Transform and its applications in solving differential equations and in topics such as Control Engineering?
My brain is dripping out of my ears.
This is UHmazing.
you're the best, the most useful channel on YT, congrats, you got a big fan now
"ah shit, here we go again"
-me about to spend the next indefinite amount of time pondering maths
LOL That's me exactly! :D
wow. congrats on your world changing comment
Begins grinding gears.
there goes sleeping
22:11 Fourier kind of looks sad, that's all I'm going to say.
😂😂
Learn Engineering - 2019-07-04
What an amazing explanation, this video deserves 10 million views!
Science done right - 2021-01-09
@Best of the World Music Soykut 1 know that
Peter N. J. - 2021-01-24
@xl maybe because a million Is less than a billion
animegeek96 - 2021-01-24
It has almost 5 million views now
cproteus - 2021-01-27
@VineFynn I think this mathematical series is metaphorical to each of us, independently rotating in our own little sphere, but added together making some mysterious gestalt tracing through time and space..... so, why wouldn’t it benefit everyone to grasp a concept so fundamental to parsing the effects of the actions of the sums of seemingly disparate parts....? Even purely philosophically?
daltanionwaves - 2021-01-28
More like 10 hundred millions