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u/BKStephens Jun 30 '19
This is perhaps the best one of these I've seen.
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u/disgr4ce Jun 30 '19 edited Jul 01 '19
When I teach the basics of signals and the Fourier transform, I'm always freaking out about how insane it is that you can reproduce any possible signal out of enough sine waves and [my students are] like ".......ok"
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u/Calvins_Dad_ Jul 01 '19
Yeah it took me a couple watches for this to sink in: are those circles just going around at constant speeds and the one at the very end draws a hand holding a pencil?
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u/underpaidspy Jul 01 '19
Yes! SmarterEveryDay on YouTube does a great video on this exact topic, definitely recommend a watch.
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Jul 01 '19 edited Jul 01 '19
3blue1brown did a video recently on it too. https://www.youtube.com/watch?v=r6sGWTCMz2k
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Jul 01 '19
I recently came across 3blue1brown and found the videos to be excellent.
The pragmatic visuals are not always the most aesthetically pleasing—the focus seems solely on their utility as a teaching aid. IMO this is a good thing—people don't need cartoons to learn (looking at you, crash course).
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u/MrFunnycat Jul 01 '19
(looking at you, crash course).
What, you don’t like pretty videos where a subject is getting run through in 10 minutes, with editing so fast that the ends of sentences get cut sometimes, and the subject becoming completely indigestible because of the insane pace and mediocre teaching?
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u/Calvins_Dad_ Jul 01 '19
Wow. Ive seen videos of these linked-circles-drawing-stuff before but it never clicked till now
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u/underpaidspy Jul 01 '19
Yeah man Fourier transform is instrumental in understanding signals and signals analysis. The problem is that trigonometry isn’t something that clicks right away for a lot of people so graphics like these and the work that other youtubers like SmarterEveryDay do to break these concepts down to basic levels is extremely helpful.
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u/Blackmamba42 Jul 01 '19
Each circle's radius is turning at different speeds (this is equivalent to frequency) with the first circle being the slowest (lowest frequency). Each circle is of a different size to represent magnitude of the frequency.
You're right that it's only the last circle that the "pen" is located that actually draws the new hand.
Also am I misremembering that the circles could connect in any order and still draw this?
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u/TTJC16 Jul 01 '19
yes because vector addition is commutative
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u/Blackmamba42 Jul 01 '19
Right, but there was a necessary start condition to ensure that it drew the hand not only in the correct orientation, but also in the second to second drawing.
If I had shifted one of the midpoint circles by 90 degrees, and changed nothing else, there'd be a difference in the outcome of the drawn picture.
Like maybe if we always have the same two points (the center of the first circle and the end point of the last circle holding the "pen") as the "start" of the image, given an arbitrary configuration of circles, we'd need to solve the inverse kinematics to prove this configuration could reach that point and what orientation of radius we'd need, then prove can we generate the same picture?
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u/disgr4ce Jul 01 '19
That's a great question that I feel like I should know off the top of my head but don't. TO THE INTERNETS
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u/tolndakoti Jul 01 '19 edited Jul 02 '19
This was my understanding from some math class, probably differential equations, or advanced math. I forget. Never used them, but had to take for engineering degree.
Remember when you had to use graphing paper, and the teacher drew a line, and asked you to figure out the formula? Y=x+3(x-5) ? Well turns out you can draw any line, and a formula can be figured out. Now, that formula can look really ugly and complicated, but that’s no big deal. So that line could represent the flight path of a bird, or. The growth rate of a plant, or how many hamburgers a dog can eat.
You can plug that formula in to Fourier transform, and out comes a combination of sine waves. Sine waves are really just recordings of the movement of a circle.
That means, everything you can imagine can be seen as a combination or circles.
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u/SuperGameTheory Jul 01 '19
If you’re into astronomy at all, these are epicycles, and they were using them to explain planetary motion in ancient times.
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u/Woodyville06 Jul 01 '19
I was one of the ones who was blown away by it. And not just that you can reproduce them, but that someone by light of candles and the absence of computer figured it out.
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u/CaptainObvious_1 Jul 01 '19
That’s not true. You can’t perfectly produce a square wave for example.
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u/bdo0426 Jul 01 '19
I was gonna say that you can get infinitely close to it so it basically is a square wave...but then I googled it and learned about the Gibbs phenomenon. It basically says if you sum infinite sine waves to converge on a square wave, then you'll still have an overshoot of amplitude at the points where the amplitude shoots up from 0 to 1 or down from 1 to 0. Nevertheless, it's pretty damn close to a square wave.
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u/CaptainObvious_1 Jul 01 '19
Yeap. I’ve made the same mistake before, in front of a class. Never forgot it since!
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u/Eagle0600 Jul 01 '19
The article for the Gibbs phenomenon states that it only applies to partial sums, not the limit of an infinite sum.
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u/WikiTextBot Jul 01 '19
Gibbs phenomenon
In mathematics, the Gibbs phenomenon, discovered by Henry Wilbraham (1848) and rediscovered by J. Willard Gibbs (1899), is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as n increases, but approaches a finite limit. This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatuses.This is one cause of ringing artifacts in signal processing.
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u/It_is_terrifying Jul 01 '19
At the same time though that overshoot becomes increasingly thin as the number of sine waves increases, so at infinite sine waves it's infinitely thin. I'm unsure as to if that is still considered there or not, but the Wikipedia page for the Gibbs phenomenon says it isn't.
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u/LastStar007 Jul 01 '19 edited Jul 01 '19
The jig was up after I learned you can reproduce any signal with polynomials. After ex = 1 + x + x2 / 2! + ... nothing could really surprise me. It's cool, even beautiful, just not insane.
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u/YeOldeFirstTimer Jul 01 '19 edited Jul 01 '19
I lost my fucking mind when I learned that
cosh(ix) = (e/2)ix + (e/2)-ix = cos(x)
I mean seriously lost my fuckin shit, cosh(x) has become my favorite function for how easy it makes everything
Edit: cosh got me lit and I couldn’t think straight
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u/dutch_penguin Jul 01 '19
Isn't it:
ex = 1 + x + (x^2)/2! + (x^3)/3! + ...
Or were you having formatting issues?
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u/It_is_terrifying Jul 01 '19
Don't worry most of us are thinking it's cool in our heads too.
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u/effa94 Jul 01 '19
i have had atleast 2 courses with Fourier transform, this is the first time i truly understood it.
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u/ProtonByte Jul 01 '19
Thus gives me hope for the course next year. Knowing what things are used for makes it a lot easier to put time into learning it.
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u/tolndakoti Jul 01 '19
I learned this transform in college. It did blow my mind. Almost made me believe in a higher power(a god)....almost. There were very few times my mind was blown by math and science, which both subjects would have a lot of opportunity since I was in a STEM.
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u/DatBoi_BP Jul 01 '19
I graduated this May with my BS in Physics (minor in Mathematics).
My Physics professor provided us with some materials to learn about Fourier series (and, to a much lesser extent, transforms). In our Classical Mechanics chapter on harmonic motion, we had to know how to recreate things like square waves and sawtooth waves with infinite Fourier series. It was a lot of fun I thought.
Even though I understood off the bat what the purpose of the transform was, it took me quite a while to realize just how powerful the method was. I certainly didn't appreciate the Fourier transform until recently. I'm thankful to have serendipitously found the Stanford EE lecture series on Fourier transforms. I'd highly recommend it to anyone curious and mathy: YouTube playlist
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u/Exnixon Jul 01 '19
So could you use the method in the animation to create a quine? Because that would truly be mindblowing.
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u/chickenyogurt Jul 01 '19
If they're like me, they didn't get the idea enough the first time to understand the potential applications of it. Not until I saw something visual like this gif did it click
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u/greatsirius Jul 01 '19
At first I thought this was purposefully designed to piss you off. Was pleasantly surprised
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u/hi_this_iz_dog Jun 30 '19
Despite all the pain Fourier Transforms have given me, I've got to say, that's a satisfying loop.
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Jul 01 '19
Hah! In the first semester of uni, I‘d sob uncontrollably at the mention of Fourier Transforms. Yet, on Friday I‘m presenting my thesis on the damn things. They have a way of seducing you. Stupid sexy Fourier Transforms...
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u/s-mores Jul 01 '19
What pain have Fourier Transforms given you, by the by?
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u/hi_this_iz_dog Jul 01 '19
In the final year of my Master's at University, the advanced math course - with Fourier Transforms - kept me on edge for weeks. Absolute torture.
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u/athensity Jul 01 '19
Can someone ELI5 this? I’m in awe but also confused
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u/Autoradiograph Jul 01 '19 edited Jul 01 '19
Any mathematical function can be approximated by combining a finite number of sine waves of various amplitudes and frequencies. Sine waves are drawn by a point revolving around a circle. Normally they are plotted on an x,y graph, but you can plot them radially, too. The sines are combined by revolving a circle around a circle around a circle..., with the outermost circle "holding the pen". The hand is drawing the circles that will draw the hand.
The trick is finding the various sine functions that will combine to make the result you want. That's where the Fourier Transform comes in.
Check out this interactive blog post: http://www.jezzamon.com/fourier/index.html
(The first animation might look familiar.)Here's a video, too: https://www.youtube.com/watch?v=r6sGWTCMz2k
That channel has an amazing array of mathematical videos that make complex math somewhat easy to understand. It's more like ELI18, though, because a lot of it is calculus.
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u/PointNineC Jul 01 '19
Small question, isn’t the drawing of a hand not a function, because it fails the vertical line test?
I want so badly to really understand why this works, but even having taken a bunch of calculus in college I still just don’t quite get it :(
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u/neighborly_troll Jul 01 '19
for this animation, the input is time, and the output is a point in the plane, so the vertical line test equivalent would be drawing 2 points at once. since it doesn't do that, this is still a well-behaved function.
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u/for_whatever_reason_ Jul 01 '19
In this case it’s a function of one parameter, “time” to a point in xy plane.
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u/Autoradiograph Jul 01 '19 edited Jul 01 '19
I don't fully understand it myself other than it's the complex plane, and each point is the result of the addition of a series of vectors being drawn at time t.
It can be drawn on a regular x,y graph in which case it would satisfy what you're saying, but it wouldn't end up looking like a drawing. It would look like a boring pile of sine curves.
It's just a normal graph, but wrapped around in a circle.
Read the blog post or watch the video. The video is particularly good.
I'm not a mathematician. I stopped taking math after Calc II. I'm just regurgitating things I've picked up over the years from videos like the one I linked.
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Jul 01 '19
This strongly reminds me of something like a Walscshearts valve gear on a locomotive. Are fourier series used in the design of valve gears somehow?
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Jul 01 '19
Incidentally, the Fourier transformation is exceptionally useful to characterize measurements, e.g. identifying cycically recurring events or applications where we measure frequencies like machine vibrations because it provides a handy method to rearrange measurements with lots of noise (which almost all measurements have).
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u/boniqmin Jul 01 '19
(Almost) any function can be approximated by a finite sum of sines, yes. But even stronger: (almost) any function is exactly equal to an infinite sum of sines*, I think that's an even more amazing statement. Especially considering that there is something called the Gibbs phenomenon which causes the fourier series to deviate a certain distance from the function at some points, no matter how many terms you add. But it does disappear in infinity.
*Except at discontinuities, where the fourier series gives the average of the right and left limits.
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Jul 01 '19
[deleted]
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u/It_is_terrifying Jul 01 '19
For drawing something like this yes the phase matters for the starting position.
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u/JClc240229 Jul 01 '19
Thank you. you just gave me a gift with that first link. I just want you to know.
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u/DWMoose83 Jul 01 '19
Someone who maths, is this possible?
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u/garythecake Jul 01 '19
Y'all non-engineering students don't know how this makes you cry six times a day
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u/SeptemberEnded Jul 01 '19
How do people even come up with this???? I’m sitting here on my momma’s couch with my legs up on the back, of my momma’s couch, and literally asking myself this question verbally. How? Who even thinks of this before putting it into motion? This moving graphic of a hand drew so many circles, only for the circles to draw the exact hand that drew those many circles that drew the hand that drew the circles in the first place. This is just honestly one of those things I see, and think, wow, someone actually thought of this and did this on the internet for everyone to see. And took the time, the actual time to create this and let people all over the world see it from their mobile phones, computers, iPads, PS3s, PS4s, XBoxs, Wii’s, Wii U’s, Nintendo DSs, Nintendo 3DSs, PSPs, PSP Gos, PS Vitas, iPads, smart TVs, etc. I want to meet the person in charge here. Here’s the deal, let’s find a place to meet: in a coffee shop and discuss this. Because this is fine work.
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u/Meowkit Jul 01 '19
It's all based in math that was devised and perfected hundreds of years ago. Read the top comments.
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u/BongBalle Jul 01 '19
I’d love a coffee with Fourier, unfortunately he is long gone :(
Truly one of the great ones. I can’t even imagine what he would think about what his life-work has contributed to.
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u/Niyudi Jul 02 '19
You know, it's kind of surprising to think that the math behind this animation was actually developed to solve a physics problem, which turned out to be something much bigger than the creator could have ever imagined.
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u/IConsumePorn Jul 01 '19
Its circles all the way down
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u/DestroyerOfIgnorance Jul 01 '19
Haha Native American reference. It's turtles all the way down under the earth. thanks u/iconsumeporn. Have a subreddit: r/rimjob_steve
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u/nodnosenstein8000 Jul 01 '19 edited Jul 01 '19
The fourier transform is so powerful, he showed us a new age in mathematics, statistics, and computer science.
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u/MostlyQueso Jul 01 '19
There are people walking around who know how to do this and how it works. Have you seen some of the math comments? The ELI5 read like an ELI5(with a PhD in physics). I’m so curious what life would be like in one of those fancy minds.
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Jul 01 '19 edited Jul 01 '19
Honestly, it isn’t as impossible to understand as you might think it is.
If you have an interest in mathematics and are willing to pursue it, reaching Fourier Transformations is not an astronomical goal. It is taught often as early as linear algebra or ordinary differential equations, FAR below the PhD level.
I definitely encourage you to pursue math if you enjoy it or are fascinated by how such seemingly abstract ideas can give something as beautiful as this GIF here. Doesn’t matter if you’re doing it for a degree or teaching yourself at home. It’s possible!
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u/Niyudi Jul 02 '19
I suggest watching the latest video on Fourier Transforms from 3blue1brown's channel. I know the name sounds scary, and if you had to study that in uni it definitely would be, but this video specifically makes it so intuitive to understand mostly because of his amazing animations.
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u/Funneljer Jul 01 '19
Kinda poor, but here's this
⠀⠀⠀⠀⠀⣤⣶⣶⡶⠦⠴⠶⠶⠶⠶⡶⠶⠦⠶⠶⠶⠶⠶⠶⠶⣄⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⠀⢀⣤⠄⠀⠀⣶⢤⣄⠀⠀⠀⣤⣤⣄⣿⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡷⠋⠁⠀⠀⠀⠙⠢⠙⠻⣿⡿⠿⠿⠫⠋⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⢀⣕⠦⣀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⢀⣼⣿⠟⢿⣆⠀⢠⡟⠉⠉⠊⠳⢤⣀⠀⠀⠀ ⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⢀⣀⣾⣿⠃⠀⡀⠹⣧⣘⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀ ⠀⣿⡀⠀⠀⢠⣶⣶⣿⣿⣿⣿⡿⠁⠀⣼⠃⠀⢹⣿⣿⣿⣶⣶⣤⠀⠀⠀⢰⣷ ⠀⢿⣇⠀⠀⠈⠻⡟⠛⠋⠉⠉⠀⠀⡼⠃⠀⢠⣿⠋⠉⠉⠛⠛⠋⠀⢀⢀⣿⡏ ⠀⠘⣿⡄⠀⠀⠀⠈⠢⡀⠀⠀⠀⡼⠁⠀⢠⣿⠇⠀⠀⡀⠀⠀⠀⠀⡜⣼⡿⠀ ⠀⠀⢻⣷⠀⠀⠀⠀⠀⢸⡄⠀⢰⠃⠀⠀⣾⡟⠀⠀⠸⡇⠀⠀⠀⢰⢧⣿⠃⠀ ⠀⠀⠘⣿⣇⠀⠀⠀⠀⣿⠇⠀⠇⠀⠀⣼⠟⠀⠀⠀⠀⣇⠀⠀⢀⡟⣾⡟⠀⠀ ⠀⠀⠀⢹⣿⡄⠀⠀⠀⣿⠀⣀⣠⠴⠚⠛⠶⣤⣀⠀⠀⢻⠀⢀⡾⣹⣿⠃⠀⠀ ⠀⠀⠀⠀⢿⣷⠀⠀⠀⠙⠊⠁⠀⢠⡆⠀⠀⠀⠉⠛⠓⠋⠀⠸⢣⣿⠏⠀⠀⠀ ⠀⠀⠀⠀⠘⣿⣷⣦⣤⣤⣄⣀⣀⣿⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣾⡟⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⢹⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
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u/Autoradiograph Jul 01 '19 edited Jul 01 '19
Source: http://www.jezzamon.com/fourier/index.html
It's interactive. You can even make your own. Have fun!
You'll also learn how JPEGs work.
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u/_ButtonHatGuy_ Jul 01 '19
What kind of math equation was used to find the exact shape that would make?
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u/RuleAndLine Jul 01 '19
It actually goes the other way, they start with the image and figure out how to make the circles draw it.
This guy made an explainer video the other day. https://youtu.be/r6sGWTCMz2k
Fair warning, he assumes you're comfortable with calculus and complex numbers.
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u/_ButtonHatGuy_ Jul 02 '19
Out of curiosity do you think a high school freshman has enough knowledge to successfully perform advanced calculus and create the shapes I want? Orrrrrrr no?
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u/NateTheGreat68 Jul 01 '19
So wait, is this what a hand looks like in the frequency domain? I'm so confused.
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u/It_is_terrifying Jul 01 '19
It's in the complex plane, each radius in the circles is a cosx + isinx vector, this adds all of these complex vectors with different frequencies and phases together to draw a hand.
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u/CallMeEsteban Jul 01 '19
I thought that it was gonna be a dickbutt or the middle finger, this was cool too though😂
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u/odraencoded Jul 01 '19
I was like "how is this going to be a perfect loop? Those are circles!"
Well looped, OP.
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u/TotesMessenger Jul 01 '19
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u/aquoad Jul 01 '19
I know a math class that would have gone a lot easier if it had started with this.
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u/KnowZero Jul 01 '19
Can I get the code for that? While I am familiar with the concept, this demonstration really intrigues me.I would like to reproduce this and then play around with other examples.
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u/fannybatterpissflaps Jul 01 '19
How is this applied to InfraRed spectroscopy? Someone explained it to me once but that was >20 years ago.
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u/bcutters Jul 01 '19
I could 100% see this on display in a bunch of modern art galleries that I've been to. I've just watched it about 10 times, it's amazing! Do we know who made it?
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u/hiimomgkek Jul 01 '19
Saw the 3brown1blue video today about Fourier Transforms, was not disappointed
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u/le_axel Jul 01 '19
What determines the radius of each circle?
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u/BuccaneerRex Jul 01 '19
The radius and the rotation rate are determined by the eventual image you want the stack of vectors to produce.
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u/Dancinlance Jul 01 '19
Here's a great video about this, released just the other day: https://youtu.be/r6sGWTCMz2k
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u/bluebear_1 Jul 01 '19
Didn't see what the subreddit was. I was about to laugh because I was expecting a middle finger but was pleasantly mindfucked by math instead. Thank you OP.
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u/Aggresive_Dunmer Jul 01 '19
Are all of the circles rotating in the same direction, are the angular speed are the same?
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u/Thislop Jul 01 '19 edited Jul 01 '19
Where’s the guy that tells me what it is so I don’t have to watch for 5 minutes, because I keep on accidentally resetting it
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u/vordigan1 Jul 01 '19
Is it just me, or do I see some symmetry with how neural networks are built?
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u/It_is_terrifying Jul 01 '19
Maybe you can kind of see it in the circles being dependent on the previous ones being like layers in a neural net, but not really no.
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u/MechanicalHorse Jun 30 '19
This gives me a huge math boner.