I've been really interested in drawing stuff using Fourier series lately, especially after taking a Signal Processing course. I think you have the best intuitive explanation of Fourier analysis that I have seen so far. This is really cool.
I like how he explains the Fourier formula: { .. V_i ..} is the orthonormal basis, a_i = < f, V_i >, f = Σ a_i V_i. The abstraction (to the vector space) makes it very clear to understand.
I wish you had a patreon I'd for sure donate 50 bucks a month probably could get several buddies to as well just keep content like this going. Easily one of the best yt channels it will explode if you keep it up.
This is so fucking cool!! As soon as I understood how it worked I went to program it myself, I never really understood fourier transforms before this, but this made it crystal clear and intuitive!
This is a really cool application. I love how rich the applications of fourier gets. Would be really cool to do this with generalized fourier series. Like with another basis other than complex exponential.
SVD - singular value decomposition. I have watched many, many videos on Fourier. Your explanation using the dot product is masterful. So clear and concise. Could you do more math videos? One on a generalized Fourier (SVD) would be incredible. I can see the key is all about projections / correlations, but you just have a knack for getting to the essence of the algorithm. Please keep making videos!
Wow, thanks for helping me understand Fourier analysis! Everything else online seemed too technical. One note: since the integral is multiplied by 1/(2pi), the term can be interpreted as an infinite average of each complex point on the path multiplied by that of the circle. This would probably lead to another intuitive explanation.
13:23 I just spent a semester doing linear algebra, and I never even came across this formula. Everything from span to basis vectors seems to make a lot more sense. *hurray to side effects from great videos*
Cody Griffin Same here, learned to even prove that you can describe the 2D plane with 2 vectors but not the how part, it could be that we talked about orthogonality and the implications only in calculus. I learned sg. new even though I understood the integrals.
What I'd really love to see if you have the time, is an animation of the increasing approximation of the image. Start out with just the 0-index term, and adding in subsequent pairs of terms to provide more detail.
Enjoying this work immensely as it rekindles my DSP interest from the time we ported these functions to the PowerPC (1997-2000) : FFTs Sliding Fourier Transform , Complex FFT /Inverse, Complex FFT /Inverse with Table, Real FFT/Inverse, Real FFT with a Table, Real FFT Inverse Transform with Table and Fast Hartley Transform . While at IBM I worked with my friend Alex and ported all the DSP functions to the PowerPC to accomplish general purpose DSP processing on a RISC Microprocessor.. sinectonalysis.com/dspvec3.htm
Oh my GOD !!!! i had this idea like yday while i was reading fourie for a lesson in uni , and now i just found your video !!!! My god thank you so much for this !!!! I will add you to my thesis ... when ill finish it ! Thanks alot this is critical info for what am i researching !!!!!!!!!
What if you add each circle one at a time, largest to smallest, and see the effect it has on the final curve as you add more circles with increasing precision? You should see the leaf show up out of increasingly better approximations.
7 ปีที่แล้ว
my thoughts exactly, that would be interesting to watch
WTF kind of high school student wouldn't be totally confused in the first 5 minutes? My high school taught us algebra, geometry, a tiny bit about how to add and multiply complex numbers, a tiny bit of trig, and... that's it. Nothing about integrals, vectors, the relationship between the exponential and sine/cosine functions. Hell, even the whole concept of mapping an interval to a closed loop would have been surprising to me, since at the time I thought all functions were of the form y = . (*Note that this was a long time ago. I since earned a bachelor's degree in math. I just wanted to say that some of this material - particularly regarding functions on the complex plane - is material you might not see until *well* into university studies, and only if you're a math/physics/engineering major).
I made a 'n'th order Bezier curve plotter in demos. Here's the link: www.desmos.com/calculator/i2lqrhiafz edit: also this, 'n' cubic bezier stitcher: www.desmos.com/calculator/vblnhyu5o4
This explains one of the questions I had last video. I wondered whether there were multiple ways of approximating shapes. But that seems to be true, as there are multiple ways of describing the shape with a path f. One can speed up and slow down the path. I guess we should choose a canonical path for each shape, which goes at a constant speed. Wouldn't that define a subspace of the infinite dimensional space used to approximate them? The subspace of constant speed paths? I'll have to look up what we know about that.
I'm only 7 minutes into this, but I have to stop and ask do people come out of high school now with a solid grasp of complex numbers and functions that operate in the complex plane? I have a bachelors of computer science and its been 10 years since I was in high school, but this math is slightly over my head. I can follow it, but I have to pause and think about it a bit. Doesn't feel like stuff I would have been able to follow at all straight out of high school.
Robertlavigne1 The channel Welsh Labs has a series called imaginary numbers are real which is great for both newbies and to refresh rusty math heads. Also I would recommend 3 blue 1 brown series on linear algebra. In combination gives a lot of good input on the subject.
Thanks for the recommendations!! I've been through 3blue1brown's linear videos, although its been a few years now. I will go check out the welsh labs series.
I’m currently in the final year of “high school” in the UK - what we call Year 13, taking the A-Level Further Maths qualification. I did get taught the general content in the video (at the 7min mark), the complex numbers, argand diagrams and functions w/ e^iθ straight in school, no self studying business going on. I suppose the prerequisite stuff is accessible to the few who did get to take any sort of ‘higher’ math classes during high school
When you ask a computer to plot something, usually the algorithm is to plot some points and then see how much the function is changing around those points. More points are plotted around where the function is changing a lot in the hope of getting an accurate picture through interpolation. For something this complicated, I really should have either raised the bar for when the computer would say it's done, or at the very least asked it to start by throwing down lots of points. Unfortunately I forgot to do both and as a result I plotted a slightly different picture in each frame. So my bad.
Now I see. So, we could call it a Display Bug, since it happened past the curve computation and 'on the way to the screen', right? Anyway, great videos! Don't give up the quality you presented in your first one, although your freestyles are interesting and fun as well! :) Your approach to the subject can't leave me indifferent.
If you plot them in the plane one after another on their respective domains, you get the original drawing. I set their domains by dividing [0,2pi] into equal pieces, one for each function.
Great video! My only question would be for the explanation for the dot porduct and the integral. He made it sound trivial , but for me it kind of doesnt make much sense. Why is the dot product of two functions, the same as the integral? The integral is the area under the curve but the dot product is a sum. Let's just consider that p*q is just another function, then how can those little rectangles be the same as an infinite sum of values (dot product of two functions)? Can please someone explain that? Thanks in Advance!
At 8.40 isnt it only valid fro f(t) periodic over T = 2pi ? what if f isnt, is the periodic infinite series still defined? what if/ when does this infininite serioes not converge?
Congratulations for your video ! I am trying to write my own code, in a difference language, following your rules. I am using only three functions, just to see if it works. The first function occur from 0 to 2*(π/3), the second from 2*(π/3) το 4*(π/3) and the third from 4*(π/3) to 2*π. I want to produce 101 Fourier Coefficients. Finally i am receiving an array of sums of products of (1/2π) * [integral of (functions * e^(-n*i*t))] * e^(n*i*t), were n= 0:100. For example : C(1)=(1/2*π) * [ integral (f1 * e^(-1*i*t) over (0 , 2*π/3) ] * e^(1*i*t) + (1/2*π) * [ integral (f2 * e^(-1*i*t) over (2*π/3 , 4*π/3) ] * e^(1*i*t) + (1/2*π) * [ integral (f3 * e^(-1*i*t) over (4*π/3 , 2*π) ] * e^(1*i*t) C(2)=(1/2*π) * [ integral (f1 * e^(-2*i*t) over (0 , 2*π/3) ] * e^(2*i*t) + (1/2*π) * [ integral (f2 * e^(-2*i*t) over (2*π/3 , 4*π/3) ] * e^(2*i*t) + (1/2*π) * [ integral (f3 * e^(-2*i*t) over (4*π/3 , 2*π) ] * e^(2*i*t) and so on until C(101). The array is like this : C(-50) , C(-49).....C(0).....C(49) , C(50) Then add up all like this : C(-50) + C(-49)+..+ C(0) +...+ C(49) + C(50) and plot from 0 to 2*π. But, i am only receive the graph of the third function. I mean, from 4π/3 to 2π, the graph (the path) is the correct, but from 0 to 4π/3, is zero (a line over x axes with y values of 0) Any body can help please ?
since we have to have an input image anyway, and also have to do a lot of processing before we can even start with the fourier, why can't we just cheat and skip the fourier and draw fake circles and just draw the pixels in the rectangular plane based on their coordinates in the input file. and make the drawing process animated. visually it would look the same.
Hello sir , can you make a video on ,Banach-Tarski paradox . I am a maths student and I AM doing a maths project , so need your help. I am not understanding the transformation of the sphere in 3d to another sphere of same size and shape. Thank you sir. Doing bsc. In maths 1 year.
I feel the order of the drawing circles doesn't matter but I have never verified it. Can anybody confirm or point to any proofs/references if this is true or false?
At 31:06 you mention that you could use a potato to graph. I mainly use potatoes for hash functions.
If only we had infinite elbows, we could draw anything.
I love all your videos, but this video has been my favourite! Please keep this format alive.
"I did not intend to do this derivation, but i got a little excited" I know what you mean dude. i know what you mean.
I've been really interested in drawing stuff using Fourier series lately, especially after taking a Signal Processing course. I think you have the best intuitive explanation of Fourier analysis that I have seen so far. This is really cool.
I like how he explains the Fourier formula: { .. V_i ..} is the orthonormal basis, a_i = < f, V_i >, f = Σ a_i V_i. The abstraction (to the vector space) makes it very clear to understand.
I LOVE your videos. You do an amazing job of explaining mathematical concepts.
I've come up with a great drinking game for GoldPlatedGoof videos:
Drink every time he says "So the question then becomes..."
without words about how cool and informative this vídeo was. Thank you!
I wish you had a patreon I'd for sure donate 50 bucks a month probably could get several buddies to as well just keep content like this going. Easily one of the best yt channels it will explode if you keep it up.
40:35 Hello!
FYI. Mathematica has a function for z -> {x,y} now, it's ReIm[].
This is so fucking cool!! As soon as I understood how it worked I went to program it myself, I never really understood fourier transforms before this, but this made it crystal clear and intuitive!
This was an absolutely amazing video: Congratulations to your great skills in maths and in 'Mathematica'!
Holy fucken shit, That's awesome !
I knew this kind of thing was possible, but to see it like that blew my mind.
I want more !!!! MORE !!!!!
This is a really cool application. I love how rich the applications of fourier gets. Would be really cool to do this with generalized fourier series. Like with another basis other than complex exponential.
Truely amazing video breaking down the maths behind the epicycles. Thanks!
SVD - singular value decomposition. I have watched many, many videos on Fourier. Your explanation using the dot product is masterful. So clear and concise. Could you do more math videos? One on a generalized Fourier (SVD) would be incredible. I can see the key is all about projections / correlations, but you just have a knack for getting to the essence of the algorithm. Please keep making videos!
This is a seriously underrated beautiful video that taught me so much
Wow, thanks for helping me understand Fourier analysis! Everything else online seemed too technical. One note: since the integral is multiplied by 1/(2pi), the term can be interpreted as an infinite average of each complex point on the path multiplied by that of the circle. This would probably lead to another intuitive explanation.
Take a look to this video
th-cam.com/video/Mf8z3nm7prQ/w-d-xo.html
It is also epicycles
The visualization of 2000 circles reminds me of an octopus tentacle.
Very much enjoyed part 2 & 3. I think it was good you’ve done it in such detail, thx.
This is TH-cam Gold!
Highly impressed. Thank you very much!
Fantastic vídeo!!! Astonishing explanation
Thank you so much!!! This explanation revolutionize my way of understanding Fourier analysis.Great!!!
Wow, thank you for this video!!
13:23 I just spent a semester doing linear algebra, and I never even came across this formula. Everything from span to basis vectors seems to make a lot more sense.
*hurray to side effects from great videos*
Cody Griffin Same here, learned to even prove that you can describe the 2D plane with 2 vectors but not the how part, it could be that we talked about orthogonality and the implications only in calculus. I learned sg. new even though I understood the integrals.
I thoroughly enjoyed this. I'd love to see more videos diving this deep into a topic.
What I'd really love to see if you have the time, is an animation of the increasing approximation of the image. Start out with just the 0-index term, and adding in subsequent pairs of terms to provide more detail.
Thank you for this journey
i love this format of video!
That thumbnail though...
Math Man McGreal the nerdiest
Enjoying this work immensely as it rekindles my DSP interest from the time we ported these functions to the PowerPC (1997-2000) : FFTs Sliding Fourier Transform , Complex FFT /Inverse, Complex FFT /Inverse with Table, Real FFT/Inverse, Real FFT with a Table, Real FFT Inverse Transform with Table and Fast Hartley Transform . While at IBM I worked with my friend Alex and ported all the DSP functions to the PowerPC to accomplish general purpose DSP processing on a RISC Microprocessor.. sinectonalysis.com/dspvec3.htm
Nice video!!! Brings me closer to really understand fourier.
So damn good! Thank you very much.
Oh my GOD !!!! i had this idea like yday while i was reading fourie for a lesson in uni , and now i just found your video !!!! My god thank you so much for this !!!! I will add you to my thesis ... when ill finish it ! Thanks alot this is critical info for what am i researching !!!!!!!!!
If you don't node it, code it...!
"I believe now would be a good moment to do a sanity check" okay sure. *rolls 2d10*
Take a look to this video
th-cam.com/video/Mf8z3nm7prQ/w-d-xo.html
It is also epicycles
He's watching H3H3 at 34:57 :-)
You got me :)
papa bless
Very cool man. At first I was put off by the time mark but by the end I actually wanted to know more. :D
Very awesome video.
Any chance you can release your leaf svg so we can play around with it? I'd like to try my hand at creating the code without part 3.
We miss your videos, king.
Just astonishing...
He never defines what V is.
Awesome video! :D
Example of SVGs path drawing using Fourier Transform : othmanelhoufi.github.io/fourier
What if you add each circle one at a time, largest to smallest, and see the effect it has on the final curve as you add more circles with increasing precision? You should see the leaf show up out of increasingly better approximations.
my thoughts exactly, that would be interesting to watch
WTF kind of high school student wouldn't be totally confused in the first 5 minutes? My high school taught us algebra, geometry, a tiny bit about how to add and multiply complex numbers, a tiny bit of trig, and... that's it. Nothing about integrals, vectors, the relationship between the exponential and sine/cosine functions. Hell, even the whole concept of mapping an interval to a closed loop would have been surprising to me, since at the time I thought all functions were of the form y = . (*Note that this was a long time ago. I since earned a bachelor's degree in math. I just wanted to say that some of this material - particularly regarding functions on the complex plane - is material you might not see until *well* into university studies, and only if you're a math/physics/engineering major).
This was amazing!!! I want to try this myself now thanx
Holy shit that was a really good video and explanation! I wonder if this could be done on the HP Prime :)
can anybody explain where the conjugate comes from at 15:20
awesome channel
6:15-6:38 thanks for that insight. :)
Can you leave a link to your code so we can generate Fourier curves with our own images?
I am trying to find out what the actual equation of the leaf would be after getting it to draw. How would I go about doing this?
I made a 'n'th order Bezier curve plotter in demos. Here's the link:
www.desmos.com/calculator/i2lqrhiafz
edit:
also this, 'n' cubic bezier stitcher:
www.desmos.com/calculator/vblnhyu5o4
4:37 that sin(t) tho lol
Great video
Damn this is cool
Yay!!! Going to bed and then this happens!
Too good!
Do the points round the shape need to equidistant in any way, like one point per centimeter shape, or similar?
How can anyone dislike this?
Samir Geiger - like this .
This explains one of the questions I had last video. I wondered whether there were multiple ways of approximating shapes. But that seems to be true, as there are multiple ways of describing the shape with a path f. One can speed up and slow down the path.
I guess we should choose a canonical path for each shape, which goes at a constant speed. Wouldn't that define a subspace of the infinite dimensional space used to approximate them? The subspace of constant speed paths? I'll have to look up what we know about that.
here from mathologer
Amazing.
I'm only 7 minutes into this, but I have to stop and ask do people come out of high school now with a solid grasp of complex numbers and functions that operate in the complex plane? I have a bachelors of computer science and its been 10 years since I was in high school, but this math is slightly over my head. I can follow it, but I have to pause and think about it a bit. Doesn't feel like stuff I would have been able to follow at all straight out of high school.
Robertlavigne1 The channel Welsh Labs has a series called imaginary numbers are real which is great for both newbies and to refresh rusty math heads. Also I would recommend 3 blue 1 brown series on linear algebra. In combination gives a lot of good input on the subject.
Thanks for the recommendations!! I've been through 3blue1brown's linear videos, although its been a few years now. I will go check out the welsh labs series.
I’m currently in the final year of “high school” in the UK - what we call Year 13, taking the A-Level Further Maths qualification.
I did get taught the general content in the video (at the 7min mark), the complex numbers, argand diagrams and functions w/ e^iθ straight in school, no self studying business going on.
I suppose the prerequisite stuff is accessible to the few who did get to take any sort of ‘higher’ math classes during high school
not from where i'm from, we only learn this in colege.
Zenytram Searom How old are you in college? That term has different meanings depending on the country.
Amazing! Enjoyed every minute. But why was the center piece of the leaf ('stem') wiggly during the animation?
When you ask a computer to plot something, usually the algorithm is to plot some points and then see how much the function is changing around those points. More points are plotted around where the function is changing a lot in the hope of getting an accurate picture through interpolation. For something this complicated, I really should have either raised the bar for when the computer would say it's done, or at the very least asked it to start by throwing down lots of points. Unfortunately I forgot to do both and as a result I plotted a slightly different picture in each frame. So my bad.
Now I see. So, we could call it a Display Bug, since it happened past the curve computation and 'on the way to the screen', right? Anyway, great videos! Don't give up the quality you presented in your first one, although your freestyles are interesting and fun as well! :) Your approach to the subject can't leave me indifferent.
Take a look to this video
th-cam.com/video/Mf8z3nm7prQ/w-d-xo.html
It is also epicycles
if you can make it look 3d by moving each circle apart abit it'll be FOURIER'S TENTACLE!
awesome
where can i buy the book?
Nice.
Interesting...
I was just thought about that you can make a video about fourier after watching ur last vid
i don't quite understand what it means when you show the table of the piecefunctions. can you explain what it is exactly?
If you plot them in the plane one after another on their respective domains, you get the original drawing. I set their domains by dividing [0,2pi] into equal pieces, one for each function.
Great video! My only question would be for the explanation for the dot porduct and the integral. He made it sound trivial , but for me it kind of doesnt make much sense. Why is the dot product of two functions, the same as the integral? The integral is the area under the curve but the dot product is a sum. Let's just consider that p*q is just another function, then how can those little rectangles be the same as an infinite sum of values (dot product of two functions)? Can please someone explain that?
Thanks in Advance!
My 7 year old son: Epic!
Even better: Epicircle! Or Epicycle! ;)
So where is the Homer?
41:45 actually green seems clearer
when are complex numbers not 2 dimentional Real numbers?
thanks sir
Does it work for paintings
What was your degree?
SupahNova if you mean university degree then AFAIK he has a PhD in mathematics.
@@abdelhameedhassane6929 obviously
Разложить f в Фурье, потом перевести в e. Всё.
I wonder what's so special about whole numbers.
Hi. I think this video is awesome.
And so are you. Bye.
Woah
At 8.40 isnt it only valid fro f(t) periodic over T = 2pi ? what if f isnt, is the periodic infinite series still defined? what if/ when does this infininite serioes not converge?
Same question
I prefer to use 0-Tau for my function. ^_^
Congratulations for your video !
I am trying to write my own code, in a difference language, following your rules.
I am using only three functions, just to see if it works. The first function occur from 0 to 2*(π/3), the second from 2*(π/3) το 4*(π/3) and the third from 4*(π/3) to 2*π.
I want to produce 101 Fourier Coefficients.
Finally i am receiving an array of sums of products of (1/2π) * [integral of (functions * e^(-n*i*t))] * e^(n*i*t), were n= 0:100.
For example :
C(1)=(1/2*π) * [ integral (f1 * e^(-1*i*t) over (0 , 2*π/3) ] * e^(1*i*t) + (1/2*π) * [ integral (f2 * e^(-1*i*t) over (2*π/3 , 4*π/3) ] * e^(1*i*t) + (1/2*π) * [ integral (f3 * e^(-1*i*t) over (4*π/3 , 2*π) ] * e^(1*i*t)
C(2)=(1/2*π) * [ integral (f1 * e^(-2*i*t) over (0 , 2*π/3) ] * e^(2*i*t) + (1/2*π) * [ integral (f2 * e^(-2*i*t) over (2*π/3 , 4*π/3) ] * e^(2*i*t) + (1/2*π) * [ integral (f3 * e^(-2*i*t) over (4*π/3 , 2*π) ] * e^(2*i*t)
and so on until C(101).
The array is like this : C(-50) , C(-49).....C(0).....C(49) , C(50)
Then add up all like this : C(-50) + C(-49)+..+ C(0) +...+ C(49) + C(50) and plot from 0 to 2*π.
But, i am only receive the graph of the third function. I mean, from 4π/3 to 2π, the graph (the path) is the correct, but from 0 to 4π/3, is zero (a line over x axes with y values of 0)
Any body can help please ?
since we have to have an input image anyway, and also have to do a lot of processing before we can even start with the fourier, why can't we just cheat and skip the fourier and draw fake circles and just draw the pixels in the rectangular plane based on their coordinates in the input file. and make the drawing process animated. visually it would look the same.
43 MINUTES??????????????? GOOF
Hello sir , can you make a video on ,Banach-Tarski paradox . I am a maths student and I AM doing a maths project , so need your help. I am not understanding the transformation of the sphere in 3d to another sphere of same size and shape. Thank you sir. Doing bsc. In maths 1 year.
Any chance of including the Mozilla documentation in the description?
developer.mozilla.org/en-US/docs/Web/SVG. Thanks for the reminder, I added it to the description as well.
18:56 you wrote v_0 where it should be v_1
@19:18 1st year college student? That book is tough! I think I had an easier time going through the first chapter of Oppenheim >.>
he's so genius in maths that he began to think that is normal for all people
what language is this?
I feel the order of the drawing circles doesn't matter but I have never verified it. Can anybody confirm or point to any proofs/references if this is true or false?
That's just sum, in regular math, which is here, it works this way.
Dude you sound quite a lot like Quentin Tarantino.
From mathloger i am coming
I do not understand de weydio.