Pure Fourier series animation montage
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- Опубліковано 1 лип 2019
- Because why not?
Learn the math behind this: • But what is a Fourier ...
If you're curious about the number of vectors used for each animation:
- Eighth note: 100
- Sigma: 200
- Britain: 500
- Fourier drawing: 300
- Nail and Gear: 200
- Treble clef: 100
- Hilbert curve: 300 (relatively small given the detail, which is why it looks puzzley)
- Seattle: 400
These animations are 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 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:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with UA-cam, 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
After the Fourier series video, I was left with a lot of "extra footage", and much of it is so mesmerizing that it really felt like a shame not to put it out there in some form.
nice
thank you
Cool!
Ty man
Awesome!
I still can't believe that each of those vectors are rotating at a constant speed. Blows mind man...
It helps if you just try to focus on one
Yep
They are not rotating at a constant speed. They have different frequencies.
@@mina871000 i think he meant each is rotating at their own constant speed
@@andrewc1036 You're right.
Are you a wizard?
an SED comment with no likes and replies?
@@SreenikethanI its so rare
Wow
He is definitely a gift from the universe....
He is.
So, what kind of art do you like the most?
Me: It's complicated
Would that be a Vermicious Knid?
the most*
@@JorgetePanete duck! /whooosh
@@PhilBoswell It wasn't even directed to you, Mr. Ihavereddit.
One could say it's complex
I think even , Fourier himself would have been much surprised to see this !!!
yes todays image quality in video would shock him
Respect for this guys who made this discoveries and tooday because of them we have a nice life. I always say that math is the building block for magic.
@@henrmota finally someone using the word respect how it's supposed to be
I respect people who do
I've been looking for a video this satisfying Fourier-s.
This is it, you've won
This comment is probably the best thing I've ever seen, ignoring the video it was posted under.
Thank you Fourier good joke sir
My life is complete. Finally, I can rest in peace now.
Flunked your French AND Math classes, didja? It's pronounced "four yay", not "for yur"...
The motion is so complex. It feels like its living.
its a complex plane after all...
@@auxencefromont1989 get out
The coherent action of the vectors almost explains the coherence of the trillions of cells in a complex life form.
@@mjtsquared it's called "emergent behavior". when many simple things combine to make a complex thing
Living things are just overly complex non living things doing chemical reactions
If you make a screensaver with this I’ll buy.
Please, this. Please please please please. Please -> +infinity
...
worth it :)
Imagine showing this in a waiting room... Would be awesome!
@@frollard, it won't if it's already rasterized.
How about my screensaver thingy I created in Processing? github.com/hugopeeters/processing/tree/master/attraction
You're doing such wonders for the field of mathematics... Look at the revival of the interest in math... I really think it was, in no small part, because of you and people like you. Appreciate your work.
Watermark it before reddit takes it without crediting you
Anyone interested enough to click on a Fourier series animation can recognize that signature 3blue1brown style 👌
I think everybody saw it here first anyway lol
@@katjam Fair point but it's visually striking enough to go mainstream in a "look at the pretty shapes" context rather than "let's appreciate the beauty of mathematics" one. Most likely without a source credit.
@hyper finally someone said it
@hyper no it doesn't. Most subreddits strive hard to credit OG content. But with the volume of content being uploaded it's sometimes just not possible. UA-cam has similar issues too.
If there were an auditory analogue to this it would be the best sleep music ever
There is an auditory analogue, any audio clip can be decomposed into sine waves just like images can. It doesn’t sound too pleasant however!
Objects in Motion Fair point - I should’ve specified something more melodic - stacking ostinati/ polyrhythmic figures of different lengths, stuff like that
@@ObjectsInMotion
I imagine it sounds awful as-is but what about an algorithm to select and enhance harmonics of an ambient tune - a Fourier synth!
Here is what I was referencing: ua-cam.com/video/3IAMpH4xF9Q/v-deo.html
@@ObjectsInMotion mp3 and similar (audio) signal compression techniques work along similar lines ( cosine-transformation - a special case of the Fourier-transformation ( which is related to the Fourier-series, where the period - in principle - is stretched to infinity ) - just in small sections of 8 samples at a time.
So does mpg, mp4, DivX/ XviD etc. for video signals ( 2D ), which work on blocks of 8 x 8 pixels of the full image ( typically).
TABLE OF CONTENTS:
0:00 Eighth note
1:23 Capital Sigma
2:27 Great Britain
4:03 Fourier's portrait
6:04 Nail and Gear
7:43 Treble clef
8:56 Approximation of Hilbert curve
10:24 Seattle outline from Frasier
I think that last one is the Seattle outline from the sitcom, Frasier
@@nickheredia1341 Thanks for telling me! ^^
Anchor? Read the description! Nail and gear!
Also CGP Grey logo...
it's the island of great britain not the UK
now this is the weird algorithm bait i came here for good job
You see something in this good...
Those sweet Gibbs effects. My students are gonna love this. Thanks for sharing your work.
I love how the circles get aligned in tight corners
Go to cuba!
Wilbraham-gibbs plz
Watching your videos makes me feel we were just banging rocks together 35 years ago when studying Fourier Series in Electrical Engineering! It was all chalk boards and acetates on overhead projectors back then. What you're showing in these videos is stunning. Thank you.
I stepped away for a minute and left this video playing. When I came back, I noticed one of the vectors wasn't rotating and seemed to be moving independently of the others. Then I realized I was looking at my mouse pointer.
Lol
One of the most satisfiying videos on the platforms. The music, the animation, everything about it is so soothing. Keep up the good work ❤
First off, just imagine if algorithms didn't exist and you had to set each one of these vectors manually. Secondly, it's amazing to think that these "drawings" can technically be listened to, since it's just a wave function after all.
*wait...*
*ok i need to know how to do that*
@@akasakasvault7597 ikr lol
What a thought!!
Well, you multiply wave functions by an imaginary constant and technically every drawing uses the same functions just different constants, so can you really say than you can listen to them?
But I guess it'd be less interesting because we can only choose *either* the real part *or* the complex part of it...
Those moments when the arrows all line to make a long, straight line are especially oddly satisfying, within an already oddly satisfying video
Those were 12 minutes I'm really glad I spent watching this. Absolutely awesome!
It's awesome to really see sinc interpolation around the corners. I was having a difficult time wrapping my mind around what that meant for complex valued functions but this clears it up
The music is sublime and the animation is exquisite.
I have forked manim and am going source diving.
Thanks for your incredible content Grant.
I'm so thankful for the attention to detail in each of your videos. Your work makes the world a more beautiful place because it changes the way we think about how everything works, and how shifting ones thinking can lead to a totally different understanding of the mundane. Inspiring.
You should make this a looping 24/7 live stream and just occasionally toss SVGs into the program to add to the collection.
Man even these vectors are better at drawing than I am!
Seriously though this stuff is unbelievably interesting! I’m sure Fourier would be proud to see his likeness drawn using his maths
Excuse me? Why a bunch of robotic arrows can draw better than me? Hmm... Math witchcraft is getting fancy...
>robotic arrows
not even that, just normal arrows rotating at constant speed
Well robots aka computers are made to do jobs with high precision. It is one of their properties. I guess u knew that.
What is Math..? Does Math give Life to form..! Is it a vision to our Understanding..? Or is it just a bunch of Numbers, is multiplying a expedient way of adding is dividing an expedient way of subtracting, are equations expedient way of doing both...
What are Robots, a computer program of Zeros and Ones...
#mathcraft
Crazy to think that any arbitrary image can basically be represented by an incredibly complicated chord with infinite notes at different magnitudes.
Better yet, it can be approximated with arbitrary precision from finite notes.
Actually it’s not a chord - it’s a single note with timbre specified by proportions of harmonics.
And the image needs to consist out of a single line
@@maciejkubera1536 can you elaborate on this further?
@@ckcost8714 A single note played on a musical instrument is itself not a pure sine wave, but it consists of many sine waves with specified frequency relationships (f, 2f, 3f etc.). The same here - the image is drawn by adding pure circles (which can be seen as "pure" frequencies in two dimensions). You could call it a chord, but normally a chord is a collection of notes played on an instrument so and every note has it's own several pure frequencies. Greetings!
My notification for this video cut off at the "12," and I thought it was going to say "hours." I immediately clicked on it.
It's so wild that every vector has constant angular velocity throughout the entire drawing.
I felt amazed by the beauty of what I have seen. It's mesmerizing how this is possible. Such a caotic system ruled by very "simple" math. Watching how it runs was feeling the pure sense of beauty. Twelve minutes well lived.
Reminds me of DNA moving through a cell. Remember those animations of how Ribosomes work? Building and folding proteins, depositing molecules in cellular machinery? It looks like this. The coiled chains whipping around, coordinated chaos coinciding. Makes me think that the motion of objects such as these can be described by similar principles.
That's genius! What if the DNA actually represents fourier series coefficients? More research needed... DNA manipulation is possible :)
This thread’s on a whole new another level
@@raunakdas4646 did you get it sir?
Seeing the Gibb’s phenomena play out in 2D is oddly satisfying. Well done!
Imagine how it'd be in 3d...
Can you please give time stamp
@@sagargrampurohit373 Look for sharp turns, you will see the alignement of the circles, and the squiggles around which actually only grow bigger with the number of circles.
@@sagargrampurohit373 0:28 for example
@@sagargrampurohit373 1:05 is a better one imho
#1: Take note.
#2: We need sum time.
#3: Lands End to John O'Groats.
#4: How meta.
#5: Hello Internet!
#6: You're in treble.
#7: I saw that jig, Mr. Hilbert.
#8: You ought to be proud, Kelsey.
You should make this into a series of videos, Grant. It's pretty fun trying to guess what these drawings are just from seeing their Fourier constructions up close.
number eight is actually "frasier".
@@zixuan1630 Played by Kelsey Grammer.
I’m too high for this and it makes me think of DNA and protein folding.
You are definitely my long lost brother (or sister).
Yeah! absolutly. Even cooler than fractals!
More like enzymes IMO.
So true
Wasn't high while watching but now I'm gonna GET high bc I absolutely think it will be great lol
The best thing about you is not just that you explain math, but you also show the audience the "math in action" which is probably the best thing anyone can do. And it's not just math what I'm talking about, this way of teaching in which the student gets too "see" and "feel" the subject in motion is probably is the best motivation for him/her to continue his/her study in the field. It just gives purpose to the field, it provides significance to the otherwise boring equations. You are giving people a purpose, and that's a big thing.
Thank you.
I’m absolutely mesmerized by these animations. These vectors are weaving together like magic, and I constantly find myself staring at the vector cluster itself rather than the resulting image. Amazing concept and amazing animations!
@3Blue1Brown
It would be cool if you could make a tutorial for how we could set it up so that we can use any picture (SVG at least) to generate such an animation. Code and all. Nice for experimentation and to get a better intuitive understanding, great as introduction to the concept of Fourier series before someone learns the formal mathematics. I had the unfortunate displeasure to learn Fourier while being integrated in other subjects from non-mathematicians. I learned the transformations, some of the rules and that you can make any periodic curve with enough sinusoids but I never learned the intuition behind it. I think that now I am STARTING to understand the principle behind it: complex numbers in essence being rotational vectors, combining enough of those rotating vectors (and maybe a few constant vectors) with different rotational frequency and length to draw any figure and after one period it starts again at the same place for the entire figure. I can imagine that if you play with simpler figures at first and build it up that you get a better understanding of the nuances.
Did you ever find a tutorial for the code after all?
I put it in 0.25 speed just to multiply the duration of this! I can watch for hours! Thank you for another awesome math content!
The power and beauty of math at work right here.
That Hilbert curve stole my heart 😍
Notification squad for the win. Love your work, this is extremely satisfying to watch!
This is awesome. Love your channel man! loved the linear algebra course.
I love the dynamics of these animations, especially the last one. The way the chain of vectors goes from gentle spirals to lightning-fast whips to frenetic chaos and back, it's absolutely mesmerizing.
I feel oddly amazed by the very small imprecisenesses throughout the images it creates. I think it tells me so much about math and nature itself. Thanks a lot. Your videos are pure jewel.
First! Love your videos! BTW, your videos serve as some of my highest level math knowledge. Thank you so much for your contributions for math education.
Math meditation.
I love you.
Next level Etch-A-Sketch
I ABSOLUTELY NEEDED THIS IN MY LIFE THANKS FOR MAKING THIS A THING
Grant: Just so you know, you could make a 20-minute video like this every week and you'll have 100k subscribers within a month (if not less). The only caveat: if you monetize, put the ad at the very beginning or the very end.. nothing destroys satisfaction more than having it interrupted!
Imagine the path you take in your life gets recorded
And then it’s expressed like this
Now I feel like my entire life is controlled by clockwork lol
very *specific* clockwork
What if the universe is just a 3d function, with nonagintillions or bazillions of operations and variables changing their values at a constant.
@@naufanaurezan Well the universe can probably be completely described mathmatically. You could probably make an equation which describes the entire universe, including the equation itself... I think.
Just stop it, none of these comments are even remotely profound.
@@erikpoephoofd I've done some research on this topic, and what you're talking about with an "equation which describes the entire universe" has a name. The name is called the Theory Of Everything. Although it doesn't technically doesn't have to be an equation, it is too much alike what you described for me not to mention it. Now apparently, it is unlikely for a theory of everything to be discovered, as often when something seems to connect two of the fundamental forces of the universe, something new is made which separates them. I read an article on this some time ago and can't remember all the details, but I remember it saying that it is not likely that there will be a Theory Of Everything, but there are a number of different theories out there which get "close enough"
My favorite moment is at 10:13, when all the vectors are lined up vertically when at the center of the drawing. As others have mentioned, this is just beautiful and illustrates Fourier concepts in a way that I've been waiting my entire life to see, without realizing it. Bravo.
That looked so beautiful! Nice music too! It was cool seeing those arrows & circles moving with the line drawing. I can imagine Bjork making a music video like this.
There's some kind of connection to complex systems here, it feels like.
before it starts i already know this 12 minutes will be my next hour or so
What amazes me the most is that Fourier series are nearly 200 years. Your video illustrate that from apparently simple theorems you can get an almost mystic experience.
I've personnaly spent years trying to mentally picture this precise phenomena, how fourier series can draw closed shapes, and here it is. I just did not anticipate how beautiful it would be.
Science is hard work, art is hard word, and somehow you manage to do both. Thank you for your videos.
My god, this is beautiful. I'm not a mathematician but this brings me to tears. I am amazed by how arrows, each with different rotation speeds be able to make smooth, and even straight lines.
Like god’s work!
This channel should be renamed 3blue1brown ASMR.
This video has become somewhat of a safe space for me. When I'm really overwhelmed or anxious or just plain tired, I come here and I enjoy the beautiful music and the motion on screen and it makes me feel calm.
I also thought this video could totally be an Internet checkpoint
Vectors are so pleasing! Thanks for all you upload and do and special thanks for this *bonus* content! I'm a musician and your explanation of FFT has brought me back to college in the best of ways.
Visual ASMR
VSMR?
SO COOL
I am very satisfied by this.
an elegant caricature that portrays what life is.. thank you for such a mesmerizing work
The fact that there are people out there smart enough to make these really puts me in my place
Great animations, fourier/4 would recommend
I observed that all the sinewaves together act like a realistic whip. Is Fourier used to study somehow the motion of a whip?
Ooooooooooo nice Idea. i think i could, and I think its possible it is
i think it's not exactly the same. a whip sends a motion wave from one string end to the other, with the open end breaking the wave at the tip. in this case the whip moment is when the vectors of different rotational frequency's phase is matching, moving as one. also the individual parts of a whip change direction due to the moving wave, those vectors can't, they turn continuous.
This is incredible. Love this channel so much! :)
Elegant, soothing and meditating, everything in its right place
Could some one make an app so that you draw a shape and it would generate the Fourier series rotations?
@@huverdoose thank you my friend
Seeing Britain, I'm encouraged to ask "how long is the coast of Britain" and does this have anything to do with fractal dimension? Can you estimate the fractal dimension from the coefficients in the Fourier series somehow?
3b1b did a video on this topic. Fractal dimensions
The fractal dimension depends on how much you continue to see detail as you zoom in. With these drawings I think the lines would look straight if you zoomed in enough because they are created by rotating vector in space. To have infinite detail you would need infinite vectors.
That is pure awesomeness! Thank you for putting that up ❤
This is an absolutely mesmerizing video. Great job as always!
Hello! Love your channel!
@@unflexian Hey thanks! Sorry I haven't been uploading recently! Working on my thesis... There is a lot more to come when I am done with my PhD.
*SPOILERS*
Shapes I'm proud of guessing quickly: Great Britain, Hilbert curve.
Shapes I'm disappointed that I didn't guess: Nail-n-Gear, Seattle skyline.
How did you do?
Could not help but liken the movements to flock dynamics.
It's almost hypnotizing.
Wow. Every vector is rotating independently, yet together they all draw something so beautiful. Mesmerizing. This is why I love math.
Great video, loved the Nail and Gear!
when you were five years old did you think you'd be making things this beautiful?
I find myself asking: If, when drawing a given shape, we are allowed to vary the speed along the trajectory, how to choose the speed to get as "simple" a Fourier series as possible ("simple" probably means minimise the effects of high-frequency modes)? I guess that could turn into a calculus of variations problem if made precise...
I think that wouldn't remain a Fourier series anymore. I am not going into definitions, but varying the frequency or the amplitude of a given vector adds a lot of parameters (that too space-varying functions) that add layers of unnecessary complexity.
I just found my favourite video to fall asleep to. Subscribed.
Honestly this is so relaxing
I assume the vectors are drawn in the order of the terms of the Fourier series -- but why are the magnitudes of the vectors approximately but not strictly in decreasing order?
usethefooorce I believe it has to do with how the computer computes the paths. It starts out with crude sketches first and then refines them with more vectors.
Because that's how the math works. Sometimes some of the lower order terms have less of a contribution to the total than the higher order terms do. For instance if the curve being approximated was e^2πit + 1/8 e^10πit obviously you wouldn't include any of the unused terms.
But I don't know what to do with those tossed salads and scrambled eggs. They're calling again...
Good night seattle
This is beautiful on so many level....The power of mathematics drawing familiar forms showing us that, the laws of mathematics and physics are the foundation of our universe is built on, but also our mind that are initially a product of it. love it
There is a kind of flawless beauty which mathematics holds..to see that beauty through the mind's eye and capture it the self eye, Sanderson you do a great job. Sometimes I feel my life is indebted for you
Reminds me of DNA extraction experiment at shcool..
Why is it that the high-frequency modes have decreasing amplitudes?
The convergence of the Fourier series is a hard question: en.wikipedia.org/wiki/Convergence_of_Fourier_series
This
Is
Mindblowing
I studied this, but never understood it until now.
Thank you
I love all your videos, but this is my favourite.
Also is there some interesting link between the behaviour of the oscillators (vectors) and the fractal dimension of the final work?
All these curves appear to me to have a fractal dimension of exactly 1 (so they're not fractals, just ordinary curves). Some of them have intricate bends and twists, but when you focus in on them, they all straighten out into smooth curves and occasional corners. (That said, you certainly could draw a fractal curve in this way, although I at least can't think of anything to say about the lengths of the vectors in that case.)
Maybe looking at rotation rate and displacement angle of each term as a function of the term number as the collection tends to infinity, or something like? In the case of self-similar shapes (idealized fractals) there should be a repeating period evident somewhere?
@YT user 597863 : A truly fractal curve always has infinite length. If you think of a fractal curve as being approximated by successive non-fractal curves, as in the animation at commons.wikimedia.org/wiki/File:Von_Koch_curve.gif for example, then the approximating curves get longer and longer, with the limiting fractal curve having infinite length. But you can't say that the fractal dimension is increasing as you do this; each of the approximating curves has a fractal dimension of exactly 1, until suddenly the limiting curve has a fractal dimension greater than 1 (about 1.26 for the example at my link).
I know for a fact that there is a link between the lengths of the vectors and the differentiability of the infinite series, so this does not sound unplausible in the slightest.
@@SmileyMPV : Do you remember anything about the relationship? In principle, _every_ property of the curve follows from the sequence of lengths; the question is how to tease it out of that data.
Can we extend this idea to 3D ?
(or even higher dimensions)
Pretty interesting idea! But it needs more "complex" numbers...
Well I guess number with more different parts than complex numbers
Well there are quaternions. I believe those have three different "imaginary units". But i do not know if we can do something like fourier transforms with quaternions.
Stunning animation as always.
The music is even more satisfying!!
I loved how you used ONLY sinewaves to make the background music 😍
I wonder how much it would change the picture to just remove one vector
if you remove the last vector it loses some detail, if you remove the first one, it's a translation right back to 0, 0. for anything in between, it's kinda intuitive that the longer the vector you remove, the weirder shapes you get
This would be taking a notch reject filter. Like Gorgi said, it depends on how big that original vector is. For most vectors other than the slowest frequencies, it would probably only generate a ringing effect.
This is the type of content i was searching for :D
Why is this so relaxing!
Could you imagine making a spirograph set this advanced?
sir please make videos on laplace and z transform
You've taught me that Fourier is important for procedural generation of data, I just don't know how yet. This makes me think about it. Thank you.
This video is a masterpiece and a true gem.