What is a Fourier Series? (Explained by drawing circles) - Smarter Every Day 205
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- Опубліковано 28 тра 2024
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Doga's a super smart dude who writes a Turkish blog "Bi Lim Ne Güzel Lan" that roughly translates roughly to "Science is Awesome Dude". We had a lot of fun working on this together. He would really appreciate it if you checked out his blog. The fun thing is that most of his articles transcend language.
Doga’s Blog (written in Turkish):
bilimneguzellan.net/
Doga’s original Fourier Series blog article that blew my mind:
bilimneguzellan.net/fuyye-serisi/
Click here to tweet him "thanks" for / bilimneguzellan
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Warm Regards,
Destin
Holy cow, Destin how are you today?
Thank you for asking. I'm actually quite tired if I'm honest. I worked all night to get this video done. We also uploaded a new podcast episode last night so it was kind of a confluence of stuff stacking up, (not unlike the sine waves in this video). It's a cool podcast episode though. It's about a rather interesting time I tried to pee in a bottle and was literally stopped by physics. I think you'll like it. www.nodumbquestions.fm/listen/2018/12/9/049-peevnrt
Thank you again for asking how I am. I'm a real person and a lot of time people treat me like a content generation algorithm. Super cool of you to talk to me.... the human. Thank you.
@@smartereveryday I can assure you it wasn't in vain, this was super interesting ! Thank you for making these awesome videos. I hope you get some well deserved sleep tonight!
@@smartereveryday loved the podcast Destin, had me dying of laughter while you shared the pee story.
@@smartereveryday get back to work, bot! who told you that you could pause making videos?! lol
@@Discostew2 I haven't listened to the podcast yet, but I think I know where this is going LOL.
Big fan of Mr. Fourier. He made my life very difficult.
LOL
:))))))))))
Yes, but without him, you wouldn't be watching this.
looooool
@@halonothing1 lol
Well, I guess I know what I'm going to try to program on this Friday's live stream!
Well, I guess i know what I'm going to be watching on Friday!
please do !!
I was just thinking of you! This is a perfect match for your show :D
Destin: "How did you make it?"
Me: "processing? p5js?"
Doga: no
Me: :(
I guess I know who I'm subbing to next!
1:53 "sine ways are probably the simplest kind of waves right? the second most some kind of wave is..."
me: cosine waves
him: a square wave.
me: oh
Hey, you stole my thought.
_sad cosine noises_
Rerin YL dude, stop ruining the fun with your facts
@Rerin YL You don't say?
@@gamingbloopers6055 it's true tho. It's a flawed joke cuz it's not based on facts. That's not ingenious nor fun.
Dr. Doga hasn't looked happy since his pronunciation of GIF was corrected
LMFAO
Ya see Norm, GIF stands for Giraffe Interchange Format, so it has to be pronounced that way...
Ahahaha so true
Even ChatGPT agrees that the hard G pronunciation is more common. Take that, Destin.
In Physics we have a saying: If you have one problem needs to be solved with fourier series, then you have 2 problems.
Imagine how hard it would be then with a sevenier series!
Mr. Jackson I like you
fourier => frequency analysis - just pick the standard modules from the shelves if you are having to do it...
@@wsjacksonjr your last name brings me nightmares (Classical Electrodynamics book by Jackson)
@@corgikun2579 but it's a really really good book if you're having concepts cleared from something much more basic like the Griffith's electrodynamics
I had Dr. Doga for my Physics I and II classes my freshman year. Crazy to see him on this channel.
Same here buddy
Which college is it
Egemen Bağış he said tech
@@egemenbags5465 Georgia Tech
How can we keep in touch with Doga?
I like the part where you asked him to explain it. You can see him stop,
(momentary exasperation) and rethink it to put it into words. We actually saw his brain change gears. Loved it.
Doga (Doğa, more correctly) means "nature" in Turkish. So "nature" tries to understand "nature" by using math :) proud of him!
As a grad student who uses Fourier transforms daily, it is incredibly valuable to watch well made videos like this that take a step back and see the beauty behind the math. Often in the classroom we focus to much on the answer and not enough on the beauty behind the math to get the answer.
If you haven't seen it yet, 3blue1brown is an amazing channel that's full of visualizations that show off the beauty of math.
You need to watch 3Blue1Brown's visualization of the Fourier transform.
UA-cam should recommend these types of videos to everyone.
it did!
Instead they push their algorithm towards flat earth. lmao.
ben shapiro wants to know your location
@@ov3rkill
it's fact , earth is flat
@@ov3rkill that`s becouse the earth is realy flat in the place where youtube office is
Leaning about the harmonic series rn. learning that not only can you make just about any sound just by adding simple sign waves together but you can also draw by adding sign waves is absolutely blowing my mind.
Your channel is truly one of UA-cam's gems. Keep up the terrific work!
"It looks like a whip".... (Starts studying whips)
@@DemirSezer not no more. But yeah.
Hi dustin, have you herd about of powered paragliding? Its pretty cool, can you do a video about how the wing works? Thank you for your'e time.
Whips have all sorts of uses...
4:23 "I made this amazing art tool and you're correcting my pronunciation?"
Graphics Image Format. why would it be Jif? we don't say jraphics.
@@we4selradio591 This
Unless i am wrong he was joking when he said "do you know its pronounced jif"
And if he wasn't joking then he is flat out wrong and it would be a correction at all
@@we4selradio591 jpeg, sonar, laser, scuba, pin, NASA and many more... google their meaning. If you want to use that "rule", then you are going to lose every one of those cooler sounding acronyms.
I always pronounced it with a hard g. The difference being that no one said diddly because they knew what I meant and it was irrelevant to what we were working on. None of my classmates flunked out of engineering school due to different pronunciations.
Welcome to the comment section, where we have:
90% about the GIF pronunciation
10% regarding the Fourier series
I know right
In which category should we classify your comment lol
Destin opened the can of worms by commenting about it. If he had let it pass, people wouldn't comment.
I love it when you explain the science behind things. Those of us who watch the science, engineering communicator channels, do so to get a basic understanding of how. certain formulas or topics work. The way you explain things is so helpful.
His reaction to your correction of how to pronounce "gif" was perfect lol
Jif
It's not gif, it's gif.
@@SomeDumDum01 gif gif gif, to the day i die!
the internet was build on gifs... not jifs :D
@@SomeDumDum01 Irrefutable proof that its a hard G, you have to spell it with a j to get the idea across. XD
@@MouseGoat giraffe, ginger, German, gif.
That is such and awesome visualisation of the Fourier series! It also makes me wonder what your logo sounds like. You could play each of these circle bundles as a musical note that is made of all of the sine waves, so your logo is a chord of 4 notes. I wonder if it would sound nice?
You're right! After all that I forgot it was a collection of sine waves and would totally have an associated sound. I hope Destin finds it, even if it is just 4 tones.
Ima gonna hazard a guess that it sounds awful.
it would sound like noise
@@TrentSheather any type of wave can be considered "sound" not just a pure sine wave.
Ok you are right and wrong. The thing you will hear is the TEXTURE of the sound, thats really interesting. But you can perfectly change the pitch to whatever you like by changing the speed of reproduction, since frecuency is how fast it plays. So it can sound really cool
Thanks for you and Doğa. Love from Turkey 🇹🇷🇹🇷
This demonstration of circles and the wave is what made it click for me. I've seen Fourier transforms and explanations for them, seen how waves add and also seen it used to describe that circle thing that can draw any image. What i never saw until now is how these are related and how a speaker really works. like "it moves with the deep frequencies and then during that motion it also moves faster with the high tones" is what i've heard, and that's good and all. I never knew how a computer would actually compute such a wave, but it makes sense now! You stack the circles and track the Y value. There's probably some elegant way to do it easier in code, but it all makes sense now!
2:35 Those "wipers" are called "epicycles" in Ptolemaic astronomy.
Exactly... I also remembered that medieval astronomers were forced to use epicycles in order to explain the periodicity of the planet's orbits as seen from the Earth, and assuming the Earth in the center of the universe...
How are they used in astronomy
@@trevorjaster4072 They aren't. They were used, 500+ years ago, in order to explain the motion of the planets, as they assumed, at that time, that the Earth was at the center of the Universe.
@trevor: Epicycles were used to explain retrograde (or backwards) motion of the other planets in the Earth-centered Ptolemaic astronomy.
And now the phrase "adding epicycles" generally means to make some theory work by adding absurd complexity.
Doga's face when you said jif had me in tears
It is truly a great visualization to see how stacking/mixing sine waves actually produces different wave shapes.. Too cool..!! Love your Channel..!!
3blue1brown and Mathologer both have wonderful videos on this subject
I have never left one of your videos without a smile on my face and this is no exception. Thank you so much for doing what you do.
so true
One of the best parts about this is when Destin says that, just as a complex Fourier Series is the sum of all its simple shapes, a very complex engineering project can be the sum of relatively simple concepts and parts. Nice analogy!
This is what kills me at university at the moment...
I took PDEs about a year ago, ughhh that class was a pain, good luck!
Watch 3b1b video on it
Just echoing Eden's comment: 3blue1brown has an excellent video on the furier transform here on youtube.
@@willfrank961 Thank's guys!😅
Me too, Im doing it for the second time 😅
"...let's challenge him !"
Destin shows a logo of SmarterEveryDay. I'm like, "Come on, man, you can't approximate that with Fourier series. That's a multi-valued function!"
Doga constructs a graph with 4 parametric functions, each approximated with Fourier series. I'm like "Oh... o_0
I stand corrected." :D
Here too. I am really annoyed with myself for not knowing about doing Fourier representation in 2d like that.
Apparently you can make a courier series that approximates an image without doing the X and Y sines separately. Instead you use e^2iπt. As t progresses, e^2iπt goes around a circular path. You can then add several of these circular paths together.
3blue1brown has an excellent video regarding this topic.
@@nanamacapagal8342
Excuse me, but I think you missed the point. Series with e^2pi*it terms is just another representation of series with sin(2pi*t) and cos(2pi*t) terms. Both of these can only be used to represent single valued function of t.
Say, you have a circle of radius 1 centered on an origin of cartesian coordinate system. It cannot be represented by a single valued function. It can be represented by implicit function x^2 +y^2=1, but if you try to express y in terms of x, equation splits into two: y1 = sqrt(1-x^2), y2 = -sqrt(1-x^2), representing "high" arc and "low" arc of the circle, respectively. Circle is a double valued function inherently. On the other hand, you can represent it as parametric function: y=sin fi, x=cos fi.
The trick here was representing that logo as a set of parametric equations, and then using Fourier series to approximate them, not an original graph, which is multi valued.
@@Hexanitrobenzene oh so that's what you meant by multivalued
Sorry my brain was a bit fuzzy when I wrote that comment
First SmarterEveryDay video I've seen and even the promo was cool lol. This is amazing. Subscribed
Finally, thanks to you, Destin, and Doga, I can visualize additive functions with a Fourier series! Thank you!
Just want to point out that he's not a student! He's Dr. Doha!
We are all students of science. Okay, I had to :)
Dr. Doha. Thank you.
Doga, with a G according to the video
its Doğa
That animation of making the SED logo with various instances of N should be your intro.
Cool idea
6:40
I love it, my teacher from algorithms first told me about this and i am absolutly amazed. Keep going!!!
This is true brilliance - to be able to take the most complex functions in the universe & simplify them to where a child can understand! Our family loves your work, Destin! Thank you for being such a great teacher!
I'm no mathematician or coder (MD by trade) but managed to get a square wave going - much like the one above with all the circles whipping around - using Javascript & p5 library. It was actually easy . The series is basically sin(wt) + sin(3wt)/3 + sin(5wt)/5 ................ t is your time step , w = freq . The more terms the squarer. Getting the graphics looking good & moving was the tricky bit.
Ahh, interesting. My guess is if you used varrying lengths of the n coefficient in sin(nwt) / n, you could derive any organic shape in nature.
Also add the starting phase of each harmonic sin(wt+phi1) + sin(3wt+phi3)/3 + sin(5wt+phi5)/5 ecc
Destin, I want you to know how much of an inspiration you are to me. Every time I see a video of yours pop up I instantly feel so happy. These past few months have been a huge struggle for me. Just 15 minutes ago I was feeling so down and unmotivated but once I saw your video on my feed... I don't know how to explain it, but I just felt this instant relief for some reason. I feel happy and motivated now and it's all thanks to you. I hope you have a wonderful day. Great video!
Mind blowing, drawing that logo with a Fourier! Thank you! And a special thanks to Doga. That totally made it click for me as well. I must've watched >90% of your videos, and only now I realize you have a podcast.. Im really happy you do, though, and I subscribed immediately.
Fascinating. I exercise to your videos and time flies. It’s so fun.
I pronounce it "ga-jif" to make sure I cover all my bases.
How to annoy every geek with only one word.
Brilliant, I’m going to use this 😂😂😂
Jyff is also a good one it approximates the spelling as a word G I F JYFF
Zhaiff for life! Zhaiff for life! Zhaiff for life!
"Gzheyf" - just to make sure it's wrong for everyone.
Mathologer made a video explaining this with even more in depth math, if anyone is interested. He analyzes a function that can draw Homer Simpson.
ua-cam.com/video/qS4H6PEcCCA/v-deo.html
It's really great! Highly recommended.
Yeah AND BTW 3BLUE1BROWN ALSO HAS A GREAT VIDEO ON THIS TOPIC AS WELL, GO CHECK THAT OUT! XD
Saw ML earlier than this.. anyway i would like to have this as a toy.. a physical thing, gears i ca reconfigurate
Thanks a lot for making this video and the animation. It does help me a lot in understanding how a series of sine waves can be resulted in a square wave. Really thanks a lot.
The look you got for the correct pronunciation of gif (thank you!) that was blistering. You rock, Destin.
He says "gif". I like him.
Fancy seeing you here!
Joe Scott and Destin corrected him to gif and I don’t know how to feel. Btw it’s pronounced “gif”
Very appropriate, because he is gif-ted.
@@NautilusGuitars THERE'S DOZENS OF US!!! :p
Its Gif, with a hard G. It Fundamentals book agrees with me
As a musician and a synth addict who understands how adding mere sine waves atop another produces different textural sounds, this video makes me excited.
Love what I learned here. A glimpse into a whole world of math and waves and their potential. Not my academic discipline; which is why I really appreciate such a visual and intuitive walk-through. Thank you!
This is going out to my audio engineering buddies. It's super interesting to see a whole different visualization of waves we like to mess around with in synthesis.
I've used fourier series in numerical methods but this video made my mind blow away...brilliant.
Great video, did a bit of digging on the subject and I discovered reference to a famous paper by J.W. Cooley and J.W. Tukey from 1965. Their work utilised Fourier analysis and led to a radical increase in computing speed by exploiting the binary notation inherent to computers and the symmetry of sine waves. This leap in computing power is what enabled the effective storage and recall of analogue recorded sound via digital bits of information.
They re-discovered a way to quickly compute discrete Fourier transform solutions. (O[n log n]). Gauss had discovered the method in the early 1800's, even before Fourier published his work. Then people forgot.
Bullshit
This was incredible! I can actually, finally, visualize the usefulness of, and the mathematics behind Fourier math!
You just BLEW my mind dude!! The simplest building blocks, like circles, can create ANYTHING!
I love that when you give the sources you use at details :)
7:48 "Makes a great gift." Don't you mean "jift"?
I have a very nice jrafics card in my computer.
"Choosy programmers choose .gif!"
dude... i scrolled down to make this comment and you'd beaten me to it. have a thumbs up.
When you use a pun about peanut butter to dictate how to pronounce a word
Being second-language myself, I only knew it's pronounced "jif" from this video!!! like wtf
Omg! This is one of the best things I've watched on UA-cam!! Thank you.
it's cool revisiting some of destin's old content as I progress further into my engineering degree. i first watched this vid in high school, and the idea of fourier series seemed like peak math, phd level stuff, and now that I'm in my junior year, the mathematics behind fourier series is relatively simple, but visualizing it is still as magical as it was when I first learned about it 4 years ago
This is beautiful. Such a cool visual representation. It definitely would have helped me to see this when I was learning fourier series in calculus
Wow. I wish we had visualizations like this when I was in school. These videos must inspire young engineers and science students.
That’s absolutely rad dude!! I’m still wrapping my brain around it but dand it’s cool!!! Keep them coming!!!
This is without doubt the best way to teach the Fourier series. I saw this and it clicked immediately after hours of confusion studying books
This is such an intuitive way to understand Fourier series. Wish we were taught stuff in this way.
Some of us were. Not sure what has happened in the last 30 years, but maybe it is coming back through these visual tools. Imagine though that Fourier and his contemporaries had to 'see' this to make it work.
It's just the visualized addition of some major complex fourier components in the complex plane animated with time. So it's just a random analysis result!
If you understand complex numbers (incl Euler's formula), cross correlation and linear combination/algebra, then you can understand fourier series fully.
Most of which wasn't explained in this video at all!!
This concept is the key to sound design and synthesis. It's mindblowing knowing that all it takes is sine waves to emulate a real sound or make a sound you've never heard before.
Something that really made me appreciate the power of the Fourier Series was my 9 hour Fourier optics practical I did this year. Essentially, you collomate a laser beam and put it through a hexagonal matrix-hole’d piece of plastic. You’d pass this through a Fourier lens and it’d leave you with an effect that essentially allowed you to seperate higher order frequencies from lower order frequencies going from the centre outward. Through using another filter, you can then block those said frequencies, and make “false” images. You can change the shape of a sticker-star’s shadow cast onto a ccd camera. Blew.my.mind!
Destin, your videos are so fabulous.
This is so beyond me. I understand the drawing part in concept only. But when I took physics I, we used slide rules and in physics II we used the brand new HP 35 calculator.
This kind of things should be in the youtube rewind 2019
Exactly
Yesssssss!
Math YT is a subculture
After years of learning Fourier series in college now I know why it is important. Thanks Destin.
Amazing helpful visual demonstration. Thank you!
Dude, I never understood Fourier series in college but this video did the job. Hats off dustin, you are the best. (and obviously the genius in this video)
What a beauty!
This video just explained a thing I'd been trying to understand, so thanks! Now I' a bit smarter :D
“This transcends language” 😭😭😭 👌🏼💯 this was that cool, love it!
I'm studying Fourier series in Differential Equation. This video really help me to understand and also have a good visualization to understand that how sine waves produces in different waves shape . Thank you...
Destin you are an upstanding person. Great video!
I would like to see the function written out for the smarter everyday logo
A function can only have one y-value for every x-value so it's not possible to write it as a function (afaik)
@@DiapaYY
That's not true. A parabolic or other even order polynomial function has 1 value of y for 2 or more values of x.
Also multiple values of x in a sinusoidal function can return the same y value.
@@DiapaYY The functions of the x-coordinate(s) of the planar curve, as well as the y-coordinate(s) can definitely be written out. When combined, you have something called a vector-valued function. However, you would probably need a lot of paper to write out a good approximation.
There are different kinds of functions, most people only know about y=f(x) (if they know about functions at all), but there are also parametric functions like x=f1(t) and y=f2(t), so the coordinates aren't dependent on eachother, but on a third value t that isn't a coordinate (you could look at it as "time" for example). Then you can define both functions and draw any curve you like, even with mutlipley values for the same x value. That's also what was done here. the functions for fourier functions usually look like this:
x = f(t) = a0 + a1*sin(ωt) + a2*sin(2ωt) + ... b1*cos(ωt) + b2*cos(2ωt) + ...
Every additional step adds another pair of sine and cosine terms.
@@DiapaYY The "y value" in this case is a complex number. Indeed, the Fourier transform is inherently in the complex domain. If he did it the way he did the real valued examples, the vertical axis is the real part. The reverse of that might be a more common convention.
Ahhhhhhhhhhhhhhhhh... I finally get it after years of graduating college.
ikr?
Join the club!
Welp! I guess you are not forever a loan.
This visualization (which I've seen a couple of years ago) is cool but doesn't help me that much. I think what helped me the most to understand Fourier series is Winamp and its visualizations (in the 90's), combined with learning how to generate sampled sound from basic notes, and playing with an FFT algorithm. I still don't fully understand Fourier series.
It's very unfortunate that nobody in our uni never attempted to discuss the reasons for Fourier series to appear. As a student I've felt lost as what the heck this whole thing is about
I love watching smart people explain stuff . the smarter they are the humbler they seem to me.
It's so amazing and simple at the same time!Thanks,guys!
This is one of the most mind blowing videos I've seen about math
Felix FTV
I never comment on any videos but I just had to for this video.. I remember doing Fourier Series in my dorm, using Matlab and I am absolutely struggling with Fourier Series and am having the absolute worst time trying to plot them, then one of my roommates who is studying physical therapy (the highest math he took was college Algebra) walks in and goes "ohh that's 'just' a line graph". Never been so mad in my life, had to forward this video to him.
teach him some lesson . lol . make him realise his major is comic infront of pure mathematics
He was right
@@arthurmead5341 u wot
Ahh yes, beautiful MatLab
@@danbahadurgurung8593 that attitude is one of the things wrong with academics.
One field of study is not better than another. I am sure there are aspects of PT that would confuse a mathematical major.
This is one of the most amazing videos I've ever seen. Thank you!
Wow that was absolutely amazing! I haven't seen something on UA-cam that caught my interest so well in a long time. I subscribed and I hit the Bell.
This actually reminds me of something I've recently started wondering about Adobe Illustrator. The whole objects orbiting other objects, which is what he's doing here. Or, more specifically, vertexes orbiting other vertexes.
4:22 "it's actually gif" Well, the g stands for 'graphics' (Graphics Interchange Format) so the g should be pronounced as in graphics... the Turkish guy pronounced it correctly.
but you pronounce PC as "pee-see" not "pee-kee" right? oh btw I'm not a fan of calling them "jeefs" either.
The U in SCUBA stands for 'Underwater' but you don't pronounce it 'Sc-uh-ba'
he is saying like "graphics g" but in turkish language its pronounced also "graphics g" and i guess he is call it "graphics g " because of that
It's an acronym so the creator decides how it should be pronounced and Steve Wilhite called it as Destin said. ;)
In the words of the format’s creator, “choosey programmers choose gif”. It’s pronounced like the peanut butter brand.
That is fantastic! I really appreciate what you do! I am not well educated and I am not a mathematician, but I thoroughly enjoy the way you take complex ideas and break them down. Keep doing what you are doing!
Love this ... just finished a master's degree and had some of this math in a "controls course" ... this is your best video ever. greetings from Switzerland. Hats off to Doga and your video channel. Daniel
Watching this instead of studying for the diff eq final tomorrow haha. Great visualization and I'm so glad I could kind of understand it after this semester! The SmarterEveryDay drawing function reminds of some kind of CNC laser cutting/milling program. Are fourier series, or something similar, used for vectoring?
This will revolutionalize the way Fourier series are explained in classes!
WoW. I'm not a mathematician and have been trying to understand and visualise the Fourier series. Got it now! Thanks! KeepSmiling 😊🌺 I like the kids kit too. Will try it.
As an EEE student, I totally loved your video
Keep up the good work 😊
We apply this on vibration analysis, the time waveform, which also can be transformed into frequency spectrum via fast fourier transform (FFT)
Wow, those thousands circles moving like crazy and drawing a perfect face was beautiful. I'd love to put them in 3D and see them layered in VR 😄
I went to undergrad school in archaic times when the lab "desktop" computer had a thick cable to a big hot box under the table. We did Fourier on this machine but glowing 'nixie' tube readout did not give you a solid feeling about it. Therefore, we were required to make a 1st + 3rd + 5th harmonic approximation of a square wave using free running sine signal generators stacked on top of each other with a signal combining network output to an oscilloscope. It was maddening to hold them stable enough at low frequencies so we could take a picture for the lab report.
Fourier series has some limitations representing impulses but can be extended using the integral form (infinite number of frequencies in any interval which is more useful than it sounds) and windowing...multiplying the function to be approximated by a function 'window' like an exponential before doing Fourier analysis. I don't think there are many instructors who can seat the concept firmly in a students mind so that, for example, the student is not flabbergasted to find LaPlace transforms can solve physical system time behavior but also reveal the frequency response at the same time. They don't see this method as a form of Fourier analysis (and maybe missed the superposition idea). Anyway, it's easier to use SPICE computer network analysis...and I do.
Very nice video indeed!
Very easy to visualise!!. It took me lot of effort when I was student to understand what here seems so simple. Thanks and congratulations
oh man ... somebody explained it to me finally in simple terms. Thank you!
As a synthesizer nerd, seeing this video pop up made me so happy.
EDIT: and as a Georgia Tech grad, so did watching it!
Ahaah dare I say it if I'm guessing right FM synthesis.🤓
Loved it. Please continue working with doga. ❤️❤️❤️
@0:48. Ahhhh the green grid paper. While I was visiting a book store for my son's college tour, I *had* to pick up a pack. It's been thirty years, but boy did it bring back memories!!!!! Everyone else thought it was weird, but I bet every engineering can commiserate and understand the nostalgia.
"We can approximate anything as long you have enough terms." That right there is what makes mathematics so beautiful!!
A 78-years old SWISS-boy says you: Fan-tas-tic! Thank you for this video!
"sagt dir" nur einen Hinweis auf Englisch würde man "tell" stattdessen "say" da nutzten
You are not alone, Hans. This 72 year-old had the same reaction!
This 52 year old Aboriginal Australian had the same reaction too. Hello from Australia : )
This spoiled 15 year old punk was blown away too.........
Best non-slo-mo video you've made!
This is the best posdible explanation for fourier series.
Thanks a million.
I was smiling in amazement the whole time: I can watch this video for eternity!