Truly fantastic, and I love that you shared the code. Time to start pointing all those requests for Fourier series to this video (and to GoldPlatedGoof's)
The algebraic derivation of the fourier series coefficients brought images of your video to mind. I can see f(t) spinning around and the integral allowing us to find the central tendency. Thank you so much for this rich intuition.
Well trivially it would be any complex number for one of the coefficients and 0 for all of the rest. Then it would just be a circle tracing out a circle. However, that's an interresting question for the non-trivial cases.
Today’s video was motivated by an amazing animation by Santiago Ginnobili from pretty much exactly 10 year ago of a picture of Homer Simpson being drawn using epicycles. Please also consider checking out Santiago’s original video and leave a ‘like’; he REALLY deserves it. Same with some of the other other resources that went into this video and that I linked to in the description. As usual, if you’d like to support me in making these videos please consider contributing subtitles in your native language (Russian is taken care of). Please don’t contribute translated titles/descriptions without subtitles (people hate being lured into watching videos with titles in their native language only to find a video in English only. And, b.t.w., I just finished a hellish first semester at uni here in Australia and now should have a bit more time to go for some ambitious TH-cam projects. Fingers crossed. update on epicycle animations by viewers: Tetraedri_ (Hilbert curve with a twist) th-cam.com/video/RiPMhVU1pFo/w-d-xo.html K van der Veen (Hilbert curve) . th-cam.com/video/MDyr2wnJooc/w-d-xo.html Daniel Hader (Mathologer drawn with epicycles) imgur.com/a/7Bt2Bfi Andrew Kepert (Christmas themed) th-cam.com/play/PL9JP5WCX_XJaoF1Fk_byKmDzqr47w9o0g.html MrJarnould (Dingo Ate My Math Book) th-cam.com/video/mBfMWqCh7ZM/w-d-xo.html Rob (Rolling Stones lips and tongue) codepen.io/BobsPen/pen/BPaEwd Darius Tan (Simultaneous drawing of two famous pictures) media.giphy.com/media/4No4L5TONFrGwpxEHZ/giphy.gif . caused a huge responseon the math subreddit www.reddit.com/r/math/comments/8x4exe/more_epicycles/ xfmvx . (batman) th-cam.com/video/vPBcTyjMcYk/w-d-xo.html JCOp (kommodo dragon in python) th-cam.com/video/-TcyFy5s6o4/w-d-xo.html Dani Φi (selfie) twitter.com/DaniPhii/status/1017644093000900608 Csala Bálint (Earth) twitter.com/BalintCsala/status/967496488174718983 bluemon . (bluemon writing) imgur.com/BOYxBIX what else is on (desmos spirograph) www.desmos.com/calculator/qi9dmgzwrp foivos hn (writing Mathologer) ibb.co/b3f1z8 . ibb.co/jkQFsT frogstud (regular polygons, stars, etc.) imgur.com/a/Q1lsXAn Bona Fide (Fish) th-cam.com/video/_wKpKePKoDg/w-d-xo.html AlmondBread (orbiter) almondbread.github.io/orbiter/tracer.html Nico Schlömer ("The epicycler" automatic epicycle drawing program) github.com/nschloe/epicycler Csala Bálint ("The circle machine" automatic epicycle drawing program) circlemachine.github.io Keenan Horrigan (automated drawing program) www.reddit.com/r/processing/comments/7qmhcz/a_leaf_approximated_with_epicycles_source_code/ some other related bits and pieces found by you: twitter.com/i/status/991475476148318208 found by 3zehnutters (really, really amazing simultaneous drawing of "The creation of Adam" by Michelangelo !! lots of other similar animations on the twitter site of the creator.) th-cam.com/video/DfznnKUwywQ/w-d-xo.html . found by Adelar Scheidt
Mathologer Do you think you could compile a code where you imput a line picture and a number of circles and it draws it. IF ANYONE DOES THIS, OR FINDS IT, POST A REPLY WITH A LINK TO THE PROGRAM
How good is this at compression of a line or loop, related to the precision needed for the sizes and turning speeds of the circles. How does this compare to instead using line segments on a hyperbolic plane? Those also amazingly cover all curves! The game Hyper-rogue features an amazing demo of this.
I'd just like to thank you for making videos like this. Math videos like this are one of the reasons I stuck with math when it felt repetitive in school. They are the reason I always knew that I knew nothing, but that I wanted to know more. They inspired me to look into branches I would never have encountered on my own. These types if video are my favorite, really entertaining and fun, rigorous enough that it can hold it's own without too much outside knowledge and intriguing enough to make me want to go out and learn more or in this case try to build the program myself. (Nothing "here's an interesting fact" type of video that also get really popular, it's a matter of taste). Thank you for making these videos and keeping the quality of the math in it so wonderful.
I have been fascinated with the Fourier series for nearly 40 years but this was the first time that I saw it through the vision of epicycles! You brought great joy and wonderment to my life today. Thank-you.
you've outdone yourself sir. jesus. i still can't swallow the fact of how excellent in terms of insight this video really is. you see, i've seen a fourier series before, but up until now i've never seen through it. thank you hundredfold sir! also, whoever makes you this awesome teeshirts is a freaking genius. every next one is at least as witty as the previous one.
Another superb popular math video from mathologer: visually appealing, whimsical and intellectually stimulating. What a splendid introduction to Fourier series. Your work is much appreciated!
A veritable epicyclic tour de force. Thank you. A few years back, I was given a demo of a Victorian era mechanical machine, all brass and hardwood, for creating Fourier waveforms at the University of Manchester and now I have far more intuition for how it works.
Hey Dec, fancy meeting you here! Here's the video of the Henrici-Coradi analyser in action. This is a mechanical FFT (sorry, SFT :) ) machine which I found in bits in a bin at a University where I used to teach. Mathologer, I wonder if you've heard of one - I hadn't, until I worked out where all the bits went together! th-cam.com/video/JvJPjugtenI/w-d-xo.html
At the end of the 19th Century the U.S. Coast Guard generated tide tables for all major ports in the U.S. using an apparatus like that. Two petty officers operated it full time, and they would begin calculations about a year in advance. I believe they used about 11 terms.
When I first heard about Fourier analysis, I thought the teacher was saying "Four-year analysis", that it was something you only learned in a four-year series of courses. Didn't make much sense, glad I was wrong.
Loved it. Like 3Blue 1Brown, your videos make the "magic" and beauty of maths accessible to innumerate but curious viewers such as myself through the use of clever animations. And they don't get much cleverer than the ones in this episode.
What is wonderful about the mind is that it can grasp that something is truly awesome without understanding it. I am awed. Now I am inspired to try and understand.
I decided to watch youtube because I was stuck on my history of math project. (Simulating Plato's universe modle) turns out this video was exactly what I needed! Thanks Mathologer!
As a historian of science who occasionally teaches bits of early modern astronomy in my survey, I can say that you did an amazingly good job of getting the history down. It's way too easy to see Copernicus as more revolutionary than he really was. He switched the places of the earth and sun but left the Ptolemy's math the same--as you so accurately show. What I didn't know is that Ptolemaic math and epicycles had such power and were still in use. Thanks for a great video. This is definitely one of your best videos.
No, Copernicus resolved planetary motion into *uniform* circular motion, as Aristotle bade. He dispensed with the hated non-Aristotelian equant and non-uniform circular motion. And produced a more complicated, but no more accurate model. And the user still had to convert from heliocentric to geocentric coordinates, as well as calculating the heliocentric position of the earth. In modern terms, Occam's razor is embodied in Fisher's f-test. More parameters without substantially increased accuracy = a worse model.
For years I had NO concept of how the Fourier Series "worked" , but use it in total ignorance! Thank you for the graphical explanations; suddenly I understand !
This channel features some of the most beautiful mathematics on the internet, explained intuitively and, miraculously, without throwing rigor out the window. I applaud you.
OMG!! Thanks a lot! I've been looking for months for such a video. I've always wanted to know why Fourier series are defined this way, and I already had an intuition, but your video helped me a lot to actually understand it! 3Blue1Brown managed to explain Taylor series to me, you explained Fourier series to me: this is a major step in my learning of maths!
I studied Fourier analysis just after linear algebra. I remember the sudden realization of the connection between the two disciplines. If you treat functions as vectors, a Fourier series is just the function rewritten in a different base. The periodic components of the Fourier series actually form an orthogonal base for the vector space, which is what makes it easy to compute the coefficients.
Hi I studied fourier analysis for 3 years in college. These visuals finally give me some sort of intuition and imagination for the math that was before a complete mystery voodoo magic. Much appreciated. Liked and subscribed.
Fourier Transforms never fail to astound me, but so far I hadn’t seen them introduced from the aspect of epicircular movements. I have very much enjoyed this video, thanks a lot.
Your explanation started with a story and gradually dwelled into the details. Very nice. I think this helps to incite curiousity. It took a while before I digested why we used algebraic expression with complex coefficients to express a normal function. I wish I saw your video when I was a kid, it could have saved many hours. Good job!
One nice thing in this representation is that it also shows a lot about the physics, why having more terms with significant magnitudes means more energy in the system, and why phase is free (hence why it make sense to use a polar representation instead of cartesian). It shows how ringing is a chopping of higher order coefficients.
This is truly breathtaking ! I feel betrayed by my educators for making Fourier Transforms boring and unintuitive, I can now finally grasp my head around and get a better understanding of it's application.
Ohhh.. so this is how the jump was made form geocentric epicycles to suncentric ellipsis. In school we were told that if we make the sun the center then everything works out beautifully, but no explanation of how this conceptual jump was made, and I never got around to studying it further. Very cool!
You wanted some comment on technicalities and limitations of Fourier analysis. Those technicalities and limitations were what got Cantor started on what eventually became set theory and cardinality of sets. And it was all because he thought it was because the question was how many bad points there were on a curve. Lebesgue eventually got to the generally accepted answer (I don't know that there's a "right" answer) which is that the discontinuities of the curve you're trying to draw need to be a set of measure 0. These were done to figure out how to deal with the "bad functions" that Fourier series unleashed on the world. Before Fourier, a continuous function was one you could draw without lifting your pencil from the paper and a differentiable one was one without corners. (I really don't know why the Cauchy definition existed. He had read Fourier, but I don't think anyone had created Fourier series of discontinuous functions yet. Check dates, but I think Cantor started that. 1846?) Anyway, by 1900 we had the Peano space-filling curve and the Koch snowflake (1903?) and the Cantor set (an uncountable set with measure 0), even though they were considered "monsters".
Already if you have just one jump discontinuity at a point x, the Fourier series will always converge to (f(x-) + f(x+))/2 where f(x-) and f(x+) are the limits from the left resp. from the right. So in any such point it won't converge to f(x) unless f(x) is by chance equal to (f(x-) + f(x+))/2. For example if you take f(x) = { 1 if x > 0, 0 else } then f(0) = 0 but the Fourier series will converge to 1/2 in x=0.
absolutely great video!! I have had a little bit of an insight into the epicycle topic by the physicist Alexander Unzicker. He often uses the geocentric model with epicycles as a comparison with several scientific models, that are not falsifiable, because they can image all measurements by just adjusting the modle one step further. Seeing the mathematical background here was very impressive. Thanks for that!
I've always wanted to know how the OCR software in my PDF program, or the live trace function in illustrator could work, but all of this makes it so clear. You build into the software the ability to have pixels, then have something that can draw a vector line over the pixels. In the case of the OCR, it compares to a font library, and boom you're done. And for Illustrator's live trace, it's pretty much the same thing, only it can match colour, based on the physical information. OK, maybe it's not necessarily doing it exactly this way, but the math makes sense, and why software engineers have such a strong basis in math makes even more sense. And to be honest, when I think of it as a series of formulas, that can get /that/ good with not a ton of middle bits going on, it makes sense that my home computer has had enough power to process those calculations for years now, which is why the boundaries of the software seems to stretch so much every upgrade cycle. I can have my PDF software do this "enhance scanned documents" thing where it discards the background info all together, traces the text, converts it to a font so that it can be read by a screen reader, and exported as a file that's a fraction of the size in very little time. I flat out couldn't understand how the software would "learn" where the curves are, but let's think about it. If the English language only has 26 letters, there's only a finite number of ways that those letters could be made. If the software knows that the tracing is doing this particular thing, then assume it's that letter, and you've got a half decent spell check built in, your OCR doesn't even have to do that much work. And that also means that for any language that's got a standard set of letters, building that software shouldn't be that bad, nor should the making the docs accessible be that big a deal. When you see the cool implementations that someone's software they wrote to do something cool and fun, and IT ALL MAKES SENSE, that's a moment that I frequently live for. I've always been intimidated by higher maths. I struggled through Calculus with a C, and never really looked back. But the way you explained this with your visuals really got through the fear, and got me to see this in different light.
Thank you very much again for this amazing video. I didnt knew the connection of complex Fourier approximation with plain graphs. Very very interesting. Well, the only that I knew was Fourier approximation for real valued functions.
Excellent exposition of a difficult to grasp concept (Fourier). I have put a link to this video on the maths page of my own educational website (aka my handle). Keep up the good work and ill mention you to all of my students (UK)
Great video! I now have a vague idea about what a Fourier transform is, how to do it, and what to do with it; whereas before it was just some black magic that transforms a time-domain signal into the frequency domain.
Mr Mathologer Sir. All your videos are absolutelly Phenomenal i love them. And i actually want to point it out (even though it's about another one of your videos) that due to your explanation about Eta and Zeta function i finally understood why only the Harmonic Series is a sum that you can neither Supersum (as you said) nor assign a value and not hearing all the hmm "explanations" in other videos that the harmonics Series diverges cause i don't know some random guy in simpsons said so.
Mathemagical excellence. The first viewing, I was impressed by the techniques used to abstract planetary systems by the Ancient Observers, and the further abstractions that made some degree of practical applications useful numerically. Second time I began to see the correspondence between e-Pi-i resonance imaging of multi-phase superimposed frequency interference positioning Image condensation and QM-Time Principle In-form-ation. Third iteration and the correspondence with Math-Phys-Chem and Geometry in Spacetime sequences of Time Timing in Eternity-now connection of modulated QM-Time Principle went straight to the fourth realization of Superspin Temporal Superposition-point Singularity positioning is Quantum Operator Fields Modulation Mechanism of probabilities in potential possibilities Holographic Universe. This is about nine years worth of associations with all the other videos on TH-cam, (looking past the language and labels at the actual epicyclic resonance of superimposed Timing-spacing), so the story behind the stories of Actuality here-now-forever can be found on your smartphone, by observation. Amazing Mathologer Mathology is the basis of an amateur researcher's methodology. Thank you.
People still too often think today of epicycles as reflective of a "primitive" understanding of the nature of periodic motion, influenced as we are by Kepler's discovery of the ellipse as the basic form of planetary motion, a form in turn elegantly explained by Newton's theory of gravitation. But in fact, the intuition that periodic shapes should generally be reducable to summations of perfect cyclicity was absolutely spot-on, as Fourier showed mathematically (and which nearly one's every interaction with a computer exploits on a massive scale). The way the epicycle analysis was _correct_ is as interesting and impressive as the discovery of how it was too powerful to make an explanatory theory in astronomy.
This is such a good explanation. Until now the subject has totally eluded me but I really feel like I have some understanding now. Thanks for posting. I've commented this before but I really do think your videos should be on TV, it would be such a welcome break from the brain-rot that takes up so much of the broadcast schedule. Hopefully a commissioning editor will watch your video and read this comment.
This is beyond brilliant, dear Mathologer! I studied Fourier Series and I knew how to approximate any periodic function by a Fourier Series. But this is the first time I have seen the most elegant connection between the Ptolemaic epicycles and the Fourier Series! And as for drawing Homer's face with a bunch of complex exponentials: this feat should belong to the permanent collection of the world museum of culture! Wow, wow, wow!
Fascinating. I'm glad you did this one that is so much like one or two I saw about square orbits by "All things physics", only with more maths details on Fourier series. I bet they saw your video first! I can't do animations but just formulae on Excel - and have one set up to make spirograph patterns and can just change some figures at the top of some columns to change the shape, number of points, etc. It's based on the toy version with its limited cog number ratios - but on Excel you can input any numbers. The problem with trying to make the square orbit was if you make the 'hole distance' on the small wheel further out towards the edge of the wheel the orbit corners are too pointy and sides too curved and if you put the 'hole' closer to the centre the square sides are straighter but the corners more rounded, and at some point there is a compromise between square corners and straight sides. Anyway that other video inspired me to get the Spirograph out and draw loads more patterns, and my Mum's cousin appreciated a card covered in them for his birthday! I also printed out some computer generated patterns and left them at church for children to colour in when they are bored or have nothing better to do...! At least that is kind of educational for them if it gets them thinking about such mathematical patterns. It would be nice to see pictures of Fourier transforms of the sounds that various musical instruments make. Wouldn't they just be spiky plots of the distruibution of frequencies / harmonics. Some sounds look fascinating on audio files (I use Wavepad aduio editing) and sometimes I look at the Fourier transform time plot which shows how the proportions of each frequency change over time. (Maybe because I want to find out the frequency of an annoying background whistle or hum!)
One of the things that motivated copernicans (not sure if Copernicus himself) that gets forgotten is the fact that the Sun seems to be in every planet's orbital plane. When observations were accurate enough, one can notice that for each planet, the positions of that planet at any two moments in time were coplanar with the Sun, but not with any of the other planets (especially Earth). Each planet moving in its own plane that always includes the Sun is pretty suggestive of the Sun being some kind of centre.
I really like the functional analysis approach to this, as it shows that Fourier series are just an instance of a much more general fact. The idea is that the functions e^int form an orthonormal basis of the continuously differentiable functions from [0,2pi] to C. So the coefficients in the Fourier series are just the inner product of your function with the basis elements. Making all of this rigorous is not easy, but I just really like this way of thinking.
I held one of some seminar lectures today (we each got a paper) where this idea is generalized to the graph setting. Because the modulations (the e^int) are also the eigenfunctions of the onedimensional Laplace-Operator we can define a Fourier-Transform for Graphsignals (functions on graphs) by taking the inner product of the signal with the eigenvectors of the so called Laplacian-Matrix. citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.367.6064&rep=rep1&type=pdf
*Proof for integral(exp(i*z*t)dt on [0,2π])=0 for every non-zero integer z:* The integral of a complex curve, i.e. a function from some interval I to the complex numbers (in this case f(t)=exp(i*z*t) for t in [0,2π]) is defined as integral(f(t)dt on I)=integral(Re(f(t))dt on I)+integral(Im(f(t))dt on I) where Re(u+iv)=u and Im(u+iv)=v are the real and the imaginary part of a complex number u+iv. Since Re(exp(i*z*t))=cos(z*t) and Im(exp(i*z*t))=sin(z*t) for all t in [0,2π], we get I:=integral(exp(i*z*t)dt on [0,2π]) =integral(cos(z*t)dt on [0,2π])+integral(sin(z*t)dt on [0,2π]). Substituting u=z*t and dt=1/z*du on the right-hand side leaves us with I=1/z*integral(cos(u)du on [z*0,z*2π])+1/z*integral(sin(u)du on [z*0,z*2π]) =1/z*(sin(z*2π)-sin(0))+1/z*(-cos(z*2π)+cos(0)) =1/z*0+1/z*0 =0.
Awesome video again. Love your dedication and work that you put in to make the animations for us to see and enjoy. I'm not too fresh on my calculus, but the integral going from 0 to 2pi being equal to zero (I think) comes from the fact that as we go from the upper area (above the x-axis) and down again (below x-axis) results in the two halves of the area cancelling as the bottom will have flipped signs.
Yeah, we also have to first establish that the integral of the complex function is the sum of the integral of the real part + i times the integral of the imaginary part and then use that the integral fo sine and cosine is 0
I really enjoyed this video. It inspired me to take a look at Fourier series again which I hadn't done since my college days almost 50 years ago now. One comment on the use of imaginary numbers in the calculations: That's great but I think it is reasonable to point out that this is a computational trick and the calculations can be done without the use of imaginary numbers. ETA: I look at the ratio of down votes to up votes on many of the videos I watch. The only videos with large number of views that drop below 1% are very good IMO. So well done to Mathologer from that perspective. But what didn't the people that down voted the video not like? It's a math video. Presumably they understood that going in. So why did they think this particular math video deserved a down vote or do they just cruise the internet looking for videos to down vote no matter what the content.
The response of a FIR digital filter can be viewed as a system of epicycles too, the coefficients are directly the constants, constrained to be real. The vector from origin to the moving point is the complex response of the filter as the input frequency goes from 0 to fs (corresponding to 0 to 2π)
Since you can change the speed at which the pen travels through lines I guess you could recreate any greyscale image by having the darkness controlled by how long the pen spends in the pixel (guess it would be a spray can). Have 3 Fourier machines and you could create a colour image. Someone make this, I'm too lazy! [Edit] How about a 3D Fourier machine which traces a mesh? You could have three of them spraying a texture to a mesh! How about 4d? recreate this video with the timestamp of each frame as it's position of the 4th dimension's axis and show the machine modelled as a 3d projection.
Truly fantastic, and I love that you shared the code. Time to start pointing all those requests for Fourier series to this video (and to GoldPlatedGoof's)
Great to see a genius complimenting another genius c:
Glad you like it :)
The algebraic derivation of the fourier series coefficients brought images of your video to mind. I can see f(t) spinning around and the integral allowing us to find the central tendency. Thank you so much for this rich intuition.
Do you think that there could be an epicycle that draws itself?
Well trivially it would be any complex number for one of the coefficients and 0 for all of the rest. Then it would just be a circle tracing out a circle. However, that's an interresting question for the non-trivial cases.
That's probably the clearest explanation of the discrete Fourier transform I've ever seen.
Today’s video was motivated by an amazing animation by Santiago Ginnobili from pretty much exactly 10 year ago of a picture of Homer Simpson being drawn using epicycles. Please also consider checking out Santiago’s original video and leave a ‘like’; he REALLY deserves it. Same with some of the other other resources that went into this video and that I linked to in the description.
As usual, if you’d like to support me in making these videos please consider contributing subtitles in your native language (Russian is taken care of). Please don’t contribute translated titles/descriptions without subtitles (people hate being lured into watching videos with titles in their native language only to find a video in English only.
And, b.t.w., I just finished a hellish first semester at uni here in Australia and now should have a bit more time to go for some ambitious TH-cam projects. Fingers crossed.
update on epicycle animations by viewers:
Tetraedri_ (Hilbert curve with a twist) th-cam.com/video/RiPMhVU1pFo/w-d-xo.html
K van der Veen (Hilbert curve) . th-cam.com/video/MDyr2wnJooc/w-d-xo.html
Daniel Hader (Mathologer drawn with epicycles) imgur.com/a/7Bt2Bfi
Andrew Kepert (Christmas themed) th-cam.com/play/PL9JP5WCX_XJaoF1Fk_byKmDzqr47w9o0g.html
MrJarnould (Dingo Ate My Math Book) th-cam.com/video/mBfMWqCh7ZM/w-d-xo.html
Rob (Rolling Stones lips and tongue) codepen.io/BobsPen/pen/BPaEwd
Darius Tan (Simultaneous drawing of two famous pictures) media.giphy.com/media/4No4L5TONFrGwpxEHZ/giphy.gif . caused a huge responseon the math subreddit www.reddit.com/r/math/comments/8x4exe/more_epicycles/
xfmvx . (batman) th-cam.com/video/vPBcTyjMcYk/w-d-xo.html
JCOp (kommodo dragon in python) th-cam.com/video/-TcyFy5s6o4/w-d-xo.html
Dani Φi (selfie) twitter.com/DaniPhii/status/1017644093000900608
Csala Bálint (Earth) twitter.com/BalintCsala/status/967496488174718983
bluemon . (bluemon writing) imgur.com/BOYxBIX
what else is on (desmos spirograph) www.desmos.com/calculator/qi9dmgzwrp
foivos hn (writing Mathologer) ibb.co/b3f1z8 . ibb.co/jkQFsT
frogstud (regular polygons, stars, etc.) imgur.com/a/Q1lsXAn
Bona Fide (Fish) th-cam.com/video/_wKpKePKoDg/w-d-xo.html
AlmondBread (orbiter) almondbread.github.io/orbiter/tracer.html
Nico Schlömer ("The epicycler" automatic epicycle drawing program) github.com/nschloe/epicycler
Csala Bálint ("The circle machine" automatic epicycle drawing program) circlemachine.github.io
Keenan Horrigan (automated drawing program) www.reddit.com/r/processing/comments/7qmhcz/a_leaf_approximated_with_epicycles_source_code/
some other related bits and pieces found by you:
twitter.com/i/status/991475476148318208 found by 3zehnutters (really, really amazing simultaneous drawing of "The creation of Adam" by Michelangelo !! lots of other similar animations on the twitter site of the creator.)
th-cam.com/video/DfznnKUwywQ/w-d-xo.html . found by Adelar Scheidt
Mathologer
Do you think you could compile a code where you imput a line picture and a number of circles and it draws it.
IF ANYONE DOES THIS, OR FINDS IT, POST A REPLY WITH A LINK TO THE PROGRAM
How good is this at compression of a line or loop, related to the precision needed for the sizes and turning speeds of the circles. How does this compare to instead using line segments on a hyperbolic plane? Those also amazingly cover all curves! The game Hyper-rogue features an amazing demo of this.
do you like my username?
I made one
imgur.com/6KCpftO
imgur.com/BOYxBIX
Mathologer, take a look at this video from Disney Research: watch?v=DfznnKUwywQ
I'd just like to thank you for making videos like this.
Math videos like this are one of the reasons I stuck with math when it felt repetitive in school.
They are the reason I always knew that I knew nothing, but that I wanted to know more.
They inspired me to look into branches I would never have encountered on my own.
These types if video are my favorite, really entertaining and fun, rigorous enough that it can hold it's own without too much outside knowledge and intriguing enough to make me want to go out and learn more or in this case try to build the program myself. (Nothing "here's an interesting fact" type of video that also get really popular, it's a matter of taste).
Thank you for making these videos and keeping the quality of the math in it so wonderful.
That’s exactly what i feel like, when watching videos like this. Thanks
I have been fascinated with the Fourier series for nearly 40 years but this was the first time that I saw it through the vision of epicycles! You brought great joy and wonderment to my life today. Thank-you.
That's great :)
you've outdone yourself sir. jesus. i still can't swallow the fact of how excellent in terms of insight this video really is.
you see, i've seen a fourier series before, but up until now i've never seen through it.
thank you hundredfold sir!
also, whoever makes you this awesome teeshirts is a freaking genius. every next one is at least as witty as the previous one.
Your approach of showing the circles animations first and then explaining the circles using complex exponentials was very intuitive. Thank you!
Another superb popular math video from mathologer: visually appealing, whimsical and intellectually stimulating. What a splendid introduction to Fourier series. Your work is much appreciated!
Glad you liked the video and thank you for saying so :)
Wow, somehow I have never completely understood why the inverse Fourier Transformation is calculated as it is .. thanks to this video now I know.
A veritable epicyclic tour de force. Thank you.
A few years back, I was given a demo of a Victorian era mechanical machine, all brass and hardwood, for creating Fourier waveforms at the University of Manchester and now I have far more intuition for how it works.
Hey Dec, fancy meeting you here! Here's the video of the Henrici-Coradi analyser in action. This is a mechanical FFT (sorry, SFT :) ) machine which I found in bits in a bin at a University where I used to teach. Mathologer, I wonder if you've heard of one - I hadn't, until I worked out where all the bits went together!
th-cam.com/video/JvJPjugtenI/w-d-xo.html
At the end of the 19th Century the U.S. Coast Guard generated tide tables for all major ports in the U.S. using an apparatus like that. Two petty officers operated it full time, and they would begin calculations about a year in advance. I believe they used about 11 terms.
Dude the way you explained how Euler’s formula works was the best. I couldn’t have passed Signals and Systems without you.
That's great :)
Thank you Mr. Ologer for this video!
Starrgate papapig
Algebra autopilot,
That killed me.
Great job love the video.
This video should be mandatory in schools. Excellent as ever.
Another wonderful video! Although I'd seen much of this elsewhere, I never made the connection between Fourier series and epicycles. Very inspiring!
When I first heard about Fourier analysis, I thought the teacher was saying "Four-year analysis", that it was something you only learned in a four-year series of courses. Didn't make much sense, glad I was wrong.
that's probably the best fourier explanation on the site!
Loved it. Like 3Blue 1Brown, your videos make the "magic" and beauty of maths accessible to innumerate but curious viewers such as myself through the use of clever animations. And they don't get much cleverer than the ones in this episode.
What is wonderful about the mind is that it can grasp that something is truly awesome without understanding it. I am awed. Now I am inspired to try and understand.
I decided to watch youtube because I was stuck on my history of math project. (Simulating Plato's universe modle) turns out this video was exactly what I needed! Thanks Mathologer!
As you can see from this like/dislike ratio, we really REALLY appreciated all the gritty details, proofs, and calculations. Thank you!
Not at all gritty, but a satisfactory level of hand waving.
As a historian of science who occasionally teaches bits of early modern astronomy in my survey, I can say that you did an amazingly good job of getting the history down. It's way too easy to see Copernicus as more revolutionary than he really was. He switched the places of the earth and sun but left the Ptolemy's math the same--as you so accurately show. What I didn't know is that Ptolemaic math and epicycles had such power and were still in use. Thanks for a great video. This is definitely one of your best videos.
Glad you approve :)
No, Copernicus resolved planetary motion into *uniform* circular motion, as Aristotle bade. He dispensed with the hated non-Aristotelian equant and non-uniform circular motion. And produced a more complicated, but no more accurate model. And the user still had to convert from heliocentric to geocentric coordinates, as well as calculating the heliocentric position of the earth. In modern terms, Occam's razor is embodied in Fisher's f-test. More parameters without substantially increased accuracy = a worse model.
24:48 is the best animation illustrating how the Fourier Transform plays out.
For years I had NO concept of how the Fourier Series "worked" , but use it in total ignorance!
Thank you for the graphical explanations; suddenly I understand !
This channel features some of the most beautiful mathematics on the internet, explained intuitively and, miraculously, without throwing rigor out the window. I applaud you.
:)
OMG!!
Thanks a lot! I've been looking for months for such a video. I've always wanted to know why Fourier series are defined this way, and I already had an intuition, but your video helped me a lot to actually understand it!
3Blue1Brown managed to explain Taylor series to me, you explained Fourier series to me: this is a major step in my learning of maths!
11:25 I see a little silhouetto of a man
Scaramouch, scaramouch ..... :)
will you do the fandango
Thunderbolt and lightning very very frightening me ... :)
At that point of the video you've been looking at the man for 11 minutes :)
GALILEO!
@@JustAlex96 GALILEO!
GALILEO!
GALILEO!
GALILEO, FIGARO!
@@francescosorce5189 Magnifico
JUST WOW! GREAT! AWESOME! WONDERFUL!!!
Wow this video was definitely a lot of work! Thank you for putting in your time to help and educate us :) mathematics is beautiful
I studied Fourier analysis just after linear algebra. I remember the sudden realization of the connection between the two disciplines. If you treat functions as vectors, a Fourier series is just the function rewritten in a different base. The periodic components of the Fourier series actually form an orthogonal base for the vector space, which is what makes it easy to compute the coefficients.
Daniel Eriksson are there more bases for functions? If so, which ones?
This is QUICKLY becoming my favorite youtube channel !
I wish I could like this video an infinite number of times.
Hi I studied fourier analysis for 3 years in college. These visuals finally give me some sort of intuition and imagination for the math that was before a complete mystery voodoo magic. Much appreciated. Liked and subscribed.
Fourier Transforms never fail to astound me, but so far I hadn’t seen them introduced from the aspect of epicircular movements. I have very much enjoyed this video, thanks a lot.
Amazing stuff! I feel I can never get enough of videos talking about fourier transforms :D
Your explanation started with a story and gradually dwelled into the details. Very nice. I think this helps to incite curiousity. It took a while before I digested why we used algebraic expression with complex coefficients to express a normal function. I wish I saw your video when I was a kid, it could have saved many hours. Good job!
This is so astonishingly beautiful! Really!
Just wow! I will never look at Fourier Transforms the same way ever again!
I think this might be the best video you've made so far. One step closer to deciphering life the universe and everything...
Definitely worth sticking around to the end, I loved the final reveal.
Mathaloger always no1 mathamatics channel in the world
Entertaining, amusing and highly illuminating. The visual representation at the end has, well, sort of, switched-on the cerebral lightbulb!
Came here for Simpson's and math I don't understand. Ended up finding the video that actually let me understand the Fourier transform. Brilliant work.
Mission accomplished :)
One nice thing in this representation is that it also shows a lot about the physics, why having more terms with significant magnitudes means more energy in the system, and why phase is free (hence why it make sense to use a polar representation instead of cartesian). It shows how ringing is a chopping of higher order coefficients.
Loved the explanation of how to calculate c_n! Thanks for this video :)
This is truly breathtaking ! I feel betrayed by my educators for making Fourier Transforms boring and unintuitive, I can now finally grasp my head around and get a better understanding of it's application.
This is really beautiful ....this must be what Fourier must had in his mind while deriving it.
This is a wonderful explanation I have never seen before, I wish I had seen it years ago.
Thank you!
Ohhh.. so this is how the jump was made form geocentric epicycles to suncentric ellipsis. In school we were told that if we make the sun the center then everything works out beautifully, but no explanation of how this conceptual jump was made, and I never got around to studying it further. Very cool!
You wanted some comment on technicalities and limitations of Fourier analysis.
Those technicalities and limitations were what got Cantor started on what eventually became set theory and cardinality of sets. And it was all because he thought it was because the question was how many bad points there were on a curve. Lebesgue eventually got to the generally accepted answer (I don't know that there's a "right" answer) which is that the discontinuities of the curve you're trying to draw need to be a set of measure 0.
These were done to figure out how to deal with the "bad functions" that Fourier series unleashed on the world. Before Fourier, a continuous function was one you could draw without lifting your pencil from the paper and a differentiable one was one without corners. (I really don't know why the Cauchy definition existed. He had read Fourier, but I don't think anyone had created Fourier series of discontinuous functions yet. Check dates, but I think Cantor started that. 1846?) Anyway, by 1900 we had the Peano space-filling curve and the Koch snowflake (1903?) and the Cantor set (an uncountable set with measure 0), even though they were considered "monsters".
Already if you have just one jump discontinuity at a point x, the Fourier series will always converge to (f(x-) + f(x+))/2 where f(x-) and f(x+) are the limits from the left resp. from the right. So in any such point it won't converge to f(x) unless f(x) is by chance equal to (f(x-) + f(x+))/2. For example if you take f(x) = { 1 if x > 0, 0 else } then f(0) = 0 but the Fourier series will converge to 1/2 in x=0.
This mathematician is so awesome
The best video you’ve ever made! This is so good! Thank you
absolutely great video!! I have had a little bit of an insight into the epicycle topic by the physicist Alexander Unzicker. He often uses the geocentric model with epicycles as a comparison with several scientific models, that are not falsifiable, because they can image all measurements by just adjusting the modle one step further. Seeing the mathematical background here was very impressive. Thanks for that!
Fourier analysis has always been pure magic to me.
Awesome video! I appreciate how obvious you’ve made the beauty of these ideas. Fantastic work!
I've always wanted to know how the OCR software in my PDF program, or the live trace function in illustrator could work, but all of this makes it so clear. You build into the software the ability to have pixels, then have something that can draw a vector line over the pixels. In the case of the OCR, it compares to a font library, and boom you're done. And for Illustrator's live trace, it's pretty much the same thing, only it can match colour, based on the physical information.
OK, maybe it's not necessarily doing it exactly this way, but the math makes sense, and why software engineers have such a strong basis in math makes even more sense. And to be honest, when I think of it as a series of formulas, that can get /that/ good with not a ton of middle bits going on, it makes sense that my home computer has had enough power to process those calculations for years now, which is why the boundaries of the software seems to stretch so much every upgrade cycle. I can have my PDF software do this "enhance scanned documents" thing where it discards the background info all together, traces the text, converts it to a font so that it can be read by a screen reader, and exported as a file that's a fraction of the size in very little time.
I flat out couldn't understand how the software would "learn" where the curves are, but let's think about it. If the English language only has 26 letters, there's only a finite number of ways that those letters could be made. If the software knows that the tracing is doing this particular thing, then assume it's that letter, and you've got a half decent spell check built in, your OCR doesn't even have to do that much work. And that also means that for any language that's got a standard set of letters, building that software shouldn't be that bad, nor should the making the docs accessible be that big a deal.
When you see the cool implementations that someone's software they wrote to do something cool and fun, and IT ALL MAKES SENSE, that's a moment that I frequently live for. I've always been intimidated by higher maths. I struggled through Calculus with a C, and never really looked back. But the way you explained this with your visuals really got through the fear, and got me to see this in different light.
You have to do another video on Fourier Analysis! Truly awesome
that shirt is great lol
Yes, definitely a great find. Got it from here: www.teepublic.com/t-shirt/2066458-mathflix
Too bad the Fourier series showed on this shirt is not quite elegant…
You are a genius at unraveling the mystery of mathematics
Thank you for putting LIFE into FOURIER SERIES!
Thank you very much again for this amazing video. I didnt knew the connection of complex Fourier approximation with plain graphs. Very very interesting. Well, the only that I knew was Fourier approximation for real valued functions.
That type of conciceness and well thought and organized instruction is what I strive to be as a teacher. Sublime. :)
Excellent exposition of a difficult to grasp concept (Fourier). I have put a link to this video on the maths page of my own educational website (aka my handle). Keep up the good work and ill mention you to all of my students (UK)
Thank you for the video! All of you friends are super awesome!
I'd like to add to the thanks below... the making and sharing of this video is top-notch stuff!
Great video! I now have a vague idea about what a Fourier transform is, how to do it, and what to do with it; whereas before it was just some black magic that transforms a time-domain signal into the frequency domain.
Just beautiful. I've seen this video many times along the years enjoying it always.
Your video truly bring me in to the wonderful world of Fourier Transform, Thank you!
Mr Mathologer Sir. All your videos are absolutelly Phenomenal i love them. And i actually want to point it out (even though it's about another one of your videos) that due to your explanation about Eta and Zeta function i finally understood why only the Harmonic Series is a sum that you can neither Supersum (as you said) nor assign a value and not hearing all the hmm "explanations" in other videos that the harmonics Series diverges cause i don't know some random guy in simpsons said so.
Amazing! Great video. I love to learn new connections like these, thanks!
Mathemagical excellence.
The first viewing, I was impressed by the techniques used to abstract planetary systems by the Ancient Observers, and the further abstractions that made some degree of practical applications useful numerically.
Second time I began to see the correspondence between e-Pi-i resonance imaging of multi-phase superimposed frequency interference positioning Image condensation and QM-Time Principle In-form-ation.
Third iteration and the correspondence with Math-Phys-Chem and Geometry in Spacetime sequences of Time Timing in Eternity-now connection of modulated QM-Time Principle went straight to the fourth realization of Superspin Temporal Superposition-point Singularity positioning is Quantum Operator Fields Modulation Mechanism of probabilities in potential possibilities Holographic Universe.
This is about nine years worth of associations with all the other videos on TH-cam, (looking past the language and labels at the actual epicyclic resonance of superimposed Timing-spacing), so the story behind the stories of Actuality here-now-forever can be found on your smartphone, by observation.
Amazing Mathologer Mathology is the basis of an amateur researcher's methodology. Thank you.
This video is beyond the words that I can come up with at this very moment.
People still too often think today of epicycles as reflective of a "primitive" understanding of the nature of periodic motion, influenced as we are by Kepler's discovery of the ellipse as the basic form of planetary motion, a form in turn elegantly explained by Newton's theory of gravitation. But in fact, the intuition that periodic shapes should generally be reducable to summations of perfect cyclicity was absolutely spot-on, as Fourier showed mathematically (and which nearly one's every interaction with a computer exploits on a massive scale). The way the epicycle analysis was _correct_ is as interesting and impressive as the discovery of how it was too powerful to make an explanatory theory in astronomy.
This is such a good explanation. Until now the subject has totally eluded me but I really feel like I have some understanding now. Thanks for posting.
I've commented this before but I really do think your videos should be on TV, it would be such a welcome break from the brain-rot that takes up so much of the broadcast schedule. Hopefully a commissioning editor will watch your video and read this comment.
:)
And that's why the square wave contains tons of harmonics...
Dammit, I love it!
This is beyond brilliant, dear Mathologer! I studied Fourier Series and I knew how to approximate any periodic function by a Fourier Series. But this is the first time I have seen the most elegant connection between the Ptolemaic epicycles and the Fourier Series! And as for drawing Homer's face with a bunch of complex exponentials: this feat should belong to the permanent collection of the world museum of culture! Wow, wow, wow!
The history was much appreciated, thank you!
Fascinating. I'm glad you did this one that is so much like one or two I saw about square orbits by "All things physics", only with more maths details on Fourier series. I bet they saw your video first! I can't do animations but just formulae on Excel - and have one set up to make spirograph patterns and can just change some figures at the top of some columns to change the shape, number of points, etc. It's based on the toy version with its limited cog number ratios - but on Excel you can input any numbers. The problem with trying to make the square orbit was if you make the 'hole distance' on the small wheel further out towards the edge of the wheel the orbit corners are too pointy and sides too curved and if you put the 'hole' closer to the centre the square sides are straighter but the corners more rounded, and at some point there is a compromise between square corners and straight sides.
Anyway that other video inspired me to get the Spirograph out and draw loads more patterns, and my Mum's cousin appreciated a card covered in them for his birthday! I also printed out some computer generated patterns and left them at church for children to colour in when they are bored or have nothing better to do...! At least that is kind of educational for them if it gets them thinking about such mathematical patterns.
It would be nice to see pictures of Fourier transforms of the sounds that various musical instruments make. Wouldn't they just be spiky plots of the distruibution of frequencies / harmonics. Some sounds look fascinating on audio files (I use Wavepad aduio editing) and sometimes I look at the Fourier transform time plot which shows how the proportions of each frequency change over time. (Maybe because I want to find out the frequency of an annoying background whistle or hum!)
One of the things that motivated copernicans (not sure if Copernicus himself) that gets forgotten is the fact that the Sun seems to be in every planet's orbital plane. When observations were accurate enough, one can notice that for each planet, the positions of that planet at any two moments in time were coplanar with the Sun, but not with any of the other planets (especially Earth). Each planet moving in its own plane that always includes the Sun is pretty suggestive of the Sun being some kind of centre.
My god, your videos just keeps improving, nice job, keep doing it.
Very intuitive video. Please make another one for the Laplace transform and for the discrete cases like Z transform.
I really like the functional analysis approach to this, as it shows that Fourier series are just an instance of a much more general fact. The idea is that the functions e^int form an orthonormal basis of the continuously differentiable functions from [0,2pi] to C. So the coefficients in the Fourier series are just the inner product of your function with the basis elements.
Making all of this rigorous is not easy, but I just really like this way of thinking.
I held one of some seminar lectures today (we each got a paper) where this idea is generalized to the graph setting. Because the modulations (the e^int) are also the eigenfunctions of the onedimensional Laplace-Operator we can define a Fourier-Transform for Graphsignals (functions on graphs) by taking the inner product of the signal with the eigenvectors of the so called Laplacian-Matrix. citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.367.6064&rep=rep1&type=pdf
Chemical Brother sounds like some of these EEs who are into signal processing ought to be mathematics majors too! :^)
Thanks Mathologer for one more awesome video!
Great video! You have some talent, making math as much fun as you do!
A surprise to be sure!
*But a welcomed one*
/r/prequelmemes ?
Hello there
General Kenobi
reddit is leaking
* taking over
FTFY
I really enjoy the little chuckles after his jokes
"Circle of ideas", pure poetry.
*Proof for integral(exp(i*z*t)dt on [0,2π])=0 for every non-zero integer z:*
The integral of a complex curve, i.e. a function from some interval I to the complex numbers (in this case f(t)=exp(i*z*t) for t in [0,2π]) is defined as
integral(f(t)dt on I)=integral(Re(f(t))dt on I)+integral(Im(f(t))dt on I)
where Re(u+iv)=u and Im(u+iv)=v are the real and the imaginary part of a complex number u+iv. Since Re(exp(i*z*t))=cos(z*t) and Im(exp(i*z*t))=sin(z*t) for all t in [0,2π], we get
I:=integral(exp(i*z*t)dt on [0,2π])
=integral(cos(z*t)dt on [0,2π])+integral(sin(z*t)dt on [0,2π]).
Substituting u=z*t and dt=1/z*du on the right-hand side leaves us with
I=1/z*integral(cos(u)du on [z*0,z*2π])+1/z*integral(sin(u)du on [z*0,z*2π])
=1/z*(sin(z*2π)-sin(0))+1/z*(-cos(z*2π)+cos(0))
=1/z*0+1/z*0
=0.
I loved 3blue1brown's videos about the Fourier transform, but actually I like yours more
It seriously might be the best one at all
Awesome video again. Love your dedication and work that you put in to make the animations for us to see and enjoy. I'm not too fresh on my calculus, but the integral going from 0 to 2pi being equal to zero (I think) comes from the fact that as we go from the upper area (above the x-axis) and down again (below x-axis) results in the two halves of the area cancelling as the bottom will have flipped signs.
(the integral bit) Well, it's a bit trickier since the e^it takes on complex values and not just real ones :)
Yeah, we also have to first establish that the integral of the complex function is the sum of the integral of the real part + i times the integral of the imaginary part and then use that the integral fo sine and cosine is 0
great video, and ur giggle makes the cherry on the top
This is the best and most important math video I have ever seen! Wow!
Glad you like it :)
one of the best videos ive ever watched, holy crap well done
This video is the best video I have ever seen you, thank you very much
I really enjoyed this video. It inspired me to take a look at Fourier series again which I hadn't done since my college days almost 50 years ago now. One comment on the use of imaginary numbers in the calculations: That's great but I think it is reasonable to point out that this is a computational trick and the calculations can be done without the use of imaginary numbers.
ETA: I look at the ratio of down votes to up votes on many of the videos I watch. The only videos with large number of views that drop below 1% are very good IMO. So well done to Mathologer from that perspective. But what didn't the people that down voted the video not like? It's a math video. Presumably they understood that going in. So why did they think this particular math video deserved a down vote or do they just cruise the internet looking for videos to down vote no matter what the content.
The response of a FIR digital filter can be viewed as a system of epicycles too, the coefficients are directly the constants,
constrained to be real. The vector from origin to the moving point is the complex response of the filter as the input frequency
goes from 0 to fs (corresponding to 0 to 2π)
This video had so many 7-gun salute moments, I ran out of ammo... hats off.. zen master!!!!
Very nice crystal clear. Thank you sir.
Reminds me of a chaotic pendulum! Awesome!
That was freaking awesome. Thanks very much.
Since you can change the speed at which the pen travels through lines I guess you could recreate any greyscale image by having the darkness controlled by how long the pen spends in the pixel (guess it would be a spray can). Have 3 Fourier machines and you could create a colour image.
Someone make this, I'm too lazy!
[Edit] How about a 3D Fourier machine which traces a mesh? You could have three of them spraying a texture to a mesh!
How about 4d? recreate this video with the timestamp of each frame as it's position of the 4th dimension's axis and show the machine modelled as a 3d projection.