Fourier Transform, Fourier Series, and frequency spectrum

[@EugeneKhutoryansky]

To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available). --To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable. --To turn the subtitles "on" or "off" altogether: Click the "CC" button under the video. --If you believe that the translation in the subtitles can be improved, please send me an email.

[@anonim5926]

What kind of program do you use for these animations?

[@نعلايتينعلايه-ر8ح]

What is the name of this music…?

[@EugeneKhutoryansky]

The music in this video is from the free KZbin audio library, and the names of the songs are the following. Hungarian_Rhapsody_No_2_by_Liszt Stale Mate

[@EugeneKhutoryansky]

Büşra Akyüz, I make all my 3D animations with the program "Poser." Thanks.

[@ishworshrestha3559]

Ok

[@exxodas]

When they showed Sine waves in 3D, that was one of the biggest aha moments I’ve ever had. These guys are amazing .

[@EugeneKhutoryansky]

Thanks.

[@Saleh19999]

i can relate to this

[@leondaz]

same

[@spectacularserenitytunes1533]

kzbin.info/www/bejne/qYvanIGjm7Wio5o

[@ydl6832]

This phaser diagram way of representing an sinusoidal oscillation is commonly used in physics.

[@EugeneKhutoryansky]

I recently created a Patreon account for people who want to help support my channel. The link is on my KZbin home page. Also, in case, you have not already seen them, I uploaded several other videos recently. As always, for each video that you like, you can help more people find it in their KZbin search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.

[@johnhall1700]

Yea...you have patreon. Mind if I ask whats your education in? Your videos inspire me to go into theoretical physics.

[@ronaldderooij1774]

+Jonathan Theodoulos If you google his name, it is easy to find out: Electrical Engineering.

[@afederdk]

How about a PayPal option? I would rather donate per video and can not be assed to create a Patreon account anyway.

[@EugeneKhutoryansky]

+Anders Feder, Patreon has the ability to accept payments through PayPal. As for donating per video, I wasn't sure whether to accept donations per video or per month, as both are options I could have selected in Patreon, but I decided on doing it per month as I thought that this would be less confusing to people. I am not sure I understand the last part of your sentence, but if you find that you can't create a Patreon account, please let me know. In any case, I really appreciate your interest in donating and in helping to support my videos.

[@afederdk]

Eugene Khutoryansky I meant to say that I am unlikely to go through the Patreon registration process, whereas I have already PayPal set up and ready to pay at the tap of a button.

[@ThePenguinTutorials]

This video literally made me cry. I wish our current educational institutions taught math with the same beauty that you, Euclid, and other great thinkers were able to pull out of the logic of the numbers. Seriously great job!

[@EugeneKhutoryansky]

Thanks for the compliment.

[@Thebossatmserfgsd]

🥺

[@nachtundnebel-up3nk]

in this country you must be stupid for democrats 2+2 = 5

[@emptykid]

🥺 🥺

[@Wallyisking]

I think this comment sums up why in 5, 10, 20 years, as the quality of free and inexpensive online education continues to improve, traditional brick-and-mortar education will lose its value significantly. If you can learn everything you would from obtaining a degree online at a fraction of the cost of attending a university, and more importantly can prove to employers that you know your stuff, how much will that degree be worth? From the employer's perspective it shouldn't matter as long as you can demonstrate your ability. Obviously there are exceptions (Doctors, for example), but AI, robotics and 3D printing could make short work of human doctors in our lifetimes - maybe we'll just train people with great bedside manners to supervise.

[@randyzeitman1354]

If I had that animation in college it would’ve changed my career path.

[@AA-gl1dr]

Never too late to try something new!

[@theoferreiramennabarreto1698]

I'm on my fourth year of Electrical Engineering, and the frequency spectrum has always been an abstract concept that I could never truly wrap my head around. Explaining it as a density of frequencies that, when inputed to sine functions, form a given function was a huge a-ha moment for me, thank you so much!

[@EugeneKhutoryansky]

I am glad my video was helpful. Thanks.

[@chriswalsh5925]

@@EugeneKhutoryansky Yes, he was freqing out there for a bit

[@X19-x5f]

@@chriswalsh5925 lol, awesome!

[@flamurbedrolli802]

@@EugeneKhutoryansky and it is . I have seen many many other videos , but to be honest , today i saw this and i can tell this is the fundamental video , Period

[@SkaldRPG]

This blew my mind! What an amazing example of how much more accessible even complex math becomes when visualized like this! Hats of my good sir!

[@EugeneKhutoryansky]

Thanks. I am glad you liked my video.

[@jamesday1295]

I've watched a few of these vids back to back now. I did my engineering apprenticeship back in '97 - '01. I could perform the math, answer the questions, but never really visualized what was occurring until now. A set of really great resources, showing what are beautifully simple ideas to grasp when taught in the right way.

[@esmashahin2167]

This is the most beautiful thing I've ever seen on KZbin, the only visual I've found thus far that truly captures the magic of sine functions. Thank you

[@EugeneKhutoryansky]

Thanks for that really great compliment about my video. If you have not already done so, you may want to also check out some of my other videos too.

[@tommushrom5929]

No. They should introduce triangles. Then it would be considered beautiful and comprehend to me.

[@ogunsadebenjaminadeiyin2729]

Beautiful comment. Thumbs up to Eugene for the awesome animation.

[@ruprectmonkeyboy1856]

@@tommushrom5929 different application.

[@vector8310]

@@tommushrom5929 are you a competitor? Or just envious by nature?

[@utkarsh-wl8sg]

I cried watching this. Lots of thanks to you from India.

[@EugeneKhutoryansky]

I am glad you liked my video that much. Thanks.

[@DaveWasThereMan]

kzbin.info/www/bejne/pqjEha2mg9RneKMsi=gLyR-K53BdwAppJA

[@chanfarishafi6467]

Bro ! I can feel ur emotion !! From india 😂🫡🤝

[@Ste-q9n]

It is the best lesson on the topic, cheers a million ​@@EugeneKhutoryansky

[@Vexxter09]

The moment I saw the sine wave in 3D my mind was blown it was like my brain had unlocked a level it didn't know existed, seriously huge props to yiu my dude you're genuinely making people passionate about math

[@EugeneKhutoryansky]

Thanks!

[@MistressGlowWorm]

My God, the Fourier Series is one of the most beautiful and poetic equations ever created.

[@nrazz9985]

Hungarian Rhapsody in the background is distractingly beautiful

[@jankiel3284]

I also was impressed by beauty of the music in background, and I wanted to ask what score it is. Thank you mister N Razz

[@DerpMuse]

Whenever I hear the Rhapsody move into the loud main variation of the song, it always reminds me of Looney Tunes! Good memories, great music.

[@RedAllStars2010]

I was just looking through the comments to see if the name of the music was mentioned. It is very good.

[@lucaferlisi2486]

Agree

[@Ozzz77118]

ahahah you are right. I felt same way

[@Christina.N.]

I have been taught this in lectures and I just did not "get it" . Until now. I cannot thank you enough for your hard labour. You are truly a pioneer.

[@EugeneKhutoryansky]

Glad my video was helpful. Thanks.

[@mfaraday4044]

When ever I came to this channel I feel two things either I am in future or I am in 19-20 century . The power of physics

[@devbanthia6154]

the animation that you've used in this video is truly mind boggling. I wish my college professor taught stuff in the same intuitive way that you do

[@musicaddikt7444]

This is genius....plus you managed to overlay the right kind of music that constructively interferes with learning. Good work, I love your videos, being a highly audio-visual learner.

[@MonkOrMan]

Learning styles are a myth

[@junogoose]

@@MonkOrMan So, teaching modalities are also a myth? Good to know.

[@MonkOrMan]

@@junogoose Yes

[@CommunityGuidelinez]

@@junogoose Where are they a myth? The dishwasher station at Dairy Queen?

[@junogoose]

@@CommunityGuidelinez I was being facetious.

[@bobbyrp9881]

I never expected this to be this deep and simple at the same time.

[@robrobinette]

A truly eye opening visualization of a complex subject. Thank you Dr.Khutoryansky for creating and sharing it with us.

[@davydlima5539]

I'm speechless with the level of didactics in this video. Congratulations

[@EugeneKhutoryansky]

Thanks. I am glad you liked my video.

[@EugeneKhutoryansky]

You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link: kzbin.info_video?v=r18Gi8lSkfM&ref=share You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit button for both the title and for the subtitles, as they are submitted separately. Details about adding translations is available at support.google.com/youtube/answer/6054623?hl=en Thanks.

[@impossiblephysix2633]

How you animated this

[@brauliomamani6581]

Physics Videos by Eugene Khutoryansky por favor como se llama el programa que grafica todas estas maravillas de la matematicas

[@Nando79958]

Obrigado

[@saddam8783]

Which software is you to make this awesome video??

[@ruiojack4078]

science career road.~~~kzbin.info/www/bejne/jJSwZppofMljo6s

[@pc300000]

This is a work of art ! I am a big fan of simplicity and visualization. I think the biggest service to humanity is when you open someone's eyes in awe by simple explanation of a complex subject with brilliant visuals. You have done exactly that Dr.Khutoryansky. Please keep up the good work. May I ask which software did you use to produce these amazing 3D animations?:

[@EugeneKhutoryansky]

Thanks for the compliment. I make all my 3D animations with the software "Poser."

[@scottliu9555]

r

[@ankitaaarya]

Truee agreed

[@someonlinevideos]

Chandran Palanisamy the true sign of genius is I. The simplicity with which one can explain complex important ideas 💡

[@Zooofa0610]

@@EugeneKhutoryansky 😍

[@divyakhandelwal2400]

Before this, I never knew that what I was studying had any meaning.... thanks a lot!

[@marzipug5439]

You think people just make this stuff up? lol

[@vijaysingh4080]

i can feel it....but am little late.

[@scp-yearsago]

I am about to fail this semester in college and now I am realising Math can be this beautiful.

[@chillingmeko]

That moment you have an ahh moment as an Electrical Engineering student. XD feels good bro.

[@Star3]

+Ohmeko Ocampo i just had the same thing happen, Finals for Discrete has me searching, lol

[@Annihilator49]

+Murdered Ink I had a teacher who explained Discrete math topics the same way as Eugene explains these topics.

[@ahmedmborik]

+Ohmeko Ocampo Indeed! I was just screaming ahhh like an idiot now

[@chillingmeko]

Peter Bayley I wasn't so fortunate, but thank goodness for the internet that there are other out there who can explain these topics well. :)

[@YamilSG]

HAHAHHAHA Aeronautical engineer here. Same for me. In my case i wasnt that fortunate, but its amazing to know were things come from. Gives you a whole nother level of understanding

[@franciscojavierizkierdoitu5966]

I never would have believed that the Fourier representation could look even better. Congratulations, Eugene!!

[@EugeneKhutoryansky]

Thanks.

[@THE_ONLY_GOD]

That sinusoid as an upward spiral moment was really wonderful. Thanks!

[@EugeneKhutoryansky]

Thanks.

[@wolfgangbenfattoum8137]

aahhhhh....it makes so much sense when in 3D. Wonderful job.

[@YamilSG]

hahaha I love how most of the engineers and people that use this kind of things are like "Damn. I thought i knew what i was doing but i actually had no idea."

[@davidwright8432]

Seeing what's going on in equations (such as sums of sine waves) is the same as the difference between looking at a musical score, and hearing the music. The music is the point - the score, just a way of capturing change in a static form (as equations are, to what you show here). Thanks! You're a fantastic resource for anyone wanting to learn this stuff. Me, for instance!

[@chriswalsh5925]

Great analogy...yes, it's the difference between understanding the notation of something and understanding the thing itself at a deep level... I sometimes think the language of mathSSS obscures more than it reveals :D eg imagine trying to describe the colour red to a blind person. You can use the most complex language in the world but it wouldn't work. I think there needs to be some way of describing motion of things separate from language and maths, maybe. What that is I don't know, but it would be pretty handy!

[@naimulhaq9626]

As a student of electrical engineering I had to struggle visualizing Fourier series, but to visualize your beautiful representation was refreshing and spellbinding. Thank you. How easily all shapes of wave forms can be constructed from sine waves.

[@charithjeewantha]

I think this is the best use of 3D animations I have ever seen in the internet!

[@EugeneKhutoryansky]

Thanks for the compliment.

[@maciodb]

I'm impressed with how good this video is. This is how every tutorial and teaching video should look like.

[@EugeneKhutoryansky]

Glad you liked my video. Thanks.

[@afederdk]

Though I was familiar with the Fourier transform, I had no idea its mathematics could be visualized in this intuitive way.

[@EugeneKhutoryansky]

+Anders Feder, I am glad to hear that you found this to be an intuitive visualization. Thanks.

[@flitzgerald7984]

Physics Videos by Eugene Khutoryansky You should be super proud of yourself for imparting knowledge in such a lucid way, that too as non profit service.

[@defenestrator9119]

I just found your videos, and they are amazing. You take the most advanced concepts and visualize them in such understandable and intuitive ways... well, as intuitive as you can make these concepts to us humans. Every one of your videos I've watched so far has helped me connect things in a way I had not before.

[@eslamsaeed622]

I would like to thank you so much for all these beautiful videos in your channel. Actually, these videos show how physics and science are beautiful and enjoyable not complicated and boring like what exists in books and theoretical resources.

[@EugeneKhutoryansky]

Thanks for the compliment about my videos.

[@brendawilliams8062]

I agree

[@timmy3211]

15:00 Hitting me with an existential crisis there. Great video as always!

[@MrPabloguida]

Hehehehehe....

[@nicos1097]

I am very fortunate to have these videos as I begin my engineering degree. I will always refer classmates who are struggling to understand concepts like the Fourier transform to this channel!

[@deadpool1729]

Mr. Eugene, your videos are extremely good. You have incorporated visual presentations using technology at a terrific degree. I entered college and studied electronics engineering on my 1st year. Now that I'm at 4th year, I seriously recommend your channel to the freshmen and future engineers.

[@EugeneKhutoryansky]

Thanks!!!

[@germancampos1498]

What else can we say?... this is such a piece of art, taking into account the effort we had to have to abstract such math back in engineering college... great job!

[@gisforgirard]

I feel like this video answers all of quantum mechanics and in fact all of reality to me like nothing ever has before. Fantastic.

[@The_Black_Vegeta]

Holy SHIT! It finally makes sense! It all finally makes sense! It’s beautiful! 😭

[@miltonderezende7906]

I am an electrical engineer here in Brazil. Congratulations on this graphic animation where we can clearly see the result of the sum of the different harmonics of the Fourier series

[@EugeneKhutoryansky]

Thanks.

[@jayakrishna2154]

After watching this video I learned a ton. Even though I studied this in my four years of engineering I didn't realized this is what happening. THANKS a ton

[@EugeneKhutoryansky]

Glad to hear my video was helpful. Thanks.

[@KcKc-bh6lu]

It's so beautiful explaination. Many of my coworkers were some of the best engieers and scientists. But none of them could explain the Fourier Transformer in such simple and clarity manner of yours. Thank you. Most engineers know that complex wave forms were the sum of sine waves. But it's visually difficult for me to grasp the concept.

[@EugeneKhutoryansky]

Thanks. I am glad you liked my explanation.

[@SonnyIC]

'We can add together FIVE sine waves' Me: Goddammit you gotta stop

[@igorjosue8957]

We can add together INFINITE ^ 10 sine waves

[@Operational117]

@@igorjosue8957 Limits? Broken! Reality? Rejected! Hotel? Trivago!

[@__8120]

"there's more" "No"

[@gustavocortico1681]

"add more waves" "but sir we are close to critical-" "ADD MORE WAVES!"

[@rhonaldcastiglionemcbriard2023]

For a sec I was fully expecting that the rest of the video from that point would be a count upwards to the end, with 0 discussion of the fourier transform lol

[@jamesday1295]

3yrs since I last praised this video, apparently. The beauty of seeing familiar 2d waveforms in 3D with waggling sticks still makes me smile.

[@EugeneKhutoryansky]

I am glad you enjoy my video.

[@jamesday1295]

@EugeneKhutoryansky really good :) I'm not sure why, but visualising a mathematical concept seems to come much less naturally than something abstract, which is a bit odd. It took me back to my apprenticeship, watching the oscilloscope as i plugged in / unplugged various modules, watching the waveform change, clipping, filters etc. I understood an 2 inputs give a certain output, but just couldn't visualise the process.

[@FireFox2382]

This is one of the most beautiful videos I have seen! I cannot even imagine the minds of the scientists who discover these maths hundreds of years ago before modern technologies 🤯

[@EugeneKhutoryansky]

Thanks for the compliment about my video.

[@kentvandervelden]

I always enjoy the clever ways you explain concepts. In this one, I was expecting an explanation of how to perform Fourier decomposition and the FFT. I hope you'll consider that topic for a future video.

[@EugeneKhutoryansky]

+Kent A. Vander Velden, yes that is a topic for a future video. Thanks.

[@MrPabloguida]

The future of leaning will depend on our abilities to create video/animation/graphs aiding materials like this. Well done, guys!

[@fortoday04]

That was absolutely brilliant. I think I also watched one of your video about entropy. You're doing amazing work. Keep it up!

[@EugeneKhutoryansky]

Thanks.

[@Daniel-lc6ol]

probably the most outstanding explanation of FFT i've ever seen so far. This is brilliant. Thank you so much.

[@EugeneKhutoryansky]

Thanks for the compliment.

[@wernervannuffel2608]

For all frequenty modulation synthesis sound designers this video is a goldmine !

[@EugeneKhutoryansky]

Thanks.

[@aarevalo49]

Beautiful. Really enjoyed it. I hope this will help young students want to pursue a deeper understanding of signal processing. The more general form of this idea introduces imaginary numbers (i.e., complex math) creating a very powerful tool. The Fourier transform is just one, among other types of transform, that allow us to manipulate the way we see mathematical functions. Just like logarithms allows us to do multiplication by adding and then transforming back to get the desired product, Fourier transforms allow us to study the characteristics of signals in time by looking at the signal in the frequency domain.

[@1st_ProCactus]

No such this as an imaginary number. Its not imaginary if it means something.

[@nunyabisnass1141]

ProCactus imaginary numbers are numbers that not represent real things. You play with them all of the time in basic algebra, where the number doesn't correspond to any real thing. In the classical equasion y=mx^2+b. You only input numbers as you need to solve for various values of one type or another, but these numbers do not represent a real thing.

[@kamalbahadur007]

Mathematics + Computer Science = Best thing that can happen to you. After seeing this video, I couldn't resist myself from subscribing.

[@EugeneKhutoryansky]

Glad to have you as a subscriber, and I am glad that you liked my video.

[@rikushimo8413]

Thank you for helping me visualise mathematics.. I was searching for something like this since my childhood.. I knew maths is beautiful but was not able to visualise it.. This video made my day.. Heartfelt thanks to the creator of these videos.

[@mingteagathering9591]

This is so far the best video about the Fourier Series and Transform I've seen...Thank you!

[@EugeneKhutoryansky]

Thanks for the compliment.

[@northantsgarage2444]

Complex topic made incredibly simple and clear. 10/10. Absolutely brilliant.

[@EugeneKhutoryansky]

Thanks for the compliment.

[@SafeAndEffectiveTheySaid]

Amazing! You made complex math like Fourier Series simple and straighforward

[@EugeneKhutoryansky]

+Adriano Me, thanks.

[@geetmalhotra6460]

wow this is the best explanation with animation that i've ever seen. "AWESOME"👌👌👌👌👍

[@D3tyHuff]

You forgot to mention that we can add together 6 sine waves!

[@PaulSmith-gi5bf]

what about 7?

[@seansong2808]

Thank you sir! Your videos are way better than UC Berkeley electrical engineering classes!

[@EugeneKhutoryansky]

Thanks.

[@navidjabarimani3678]

this video was just amazing in every sense, music, animation, voice-over, and information were just perfect

[@EugeneKhutoryansky]

Thanks for the compliments.

[@RoboBoddicker]

Fourier transforms and Liszt. Like peanut butter and chocolate ; )

[@EugeneKhutoryansky]

Glad you liked it. Thanks.

[@TheSacknasty]

they don't mention the word fourier transform even one time?!?!

[@maxzhao]

Haha I thought the same :D

[@martinzitter4551]

Does he serve Supper with the video?

[@DavidSaintloth]

That's the beauty of the visual approach .... you SEE the transform emerge by the simple shifts of perspective. The transform emerges from recognizing that a combination of sin waves of various frequency , amplitude and phase can be summed to emerge over sample time a particular output wave form ...this is exactly what the transform does...the video doesn't actually provide the equation but one should be able to see how the terms of it emerge from simple visual examination of this video...which is what makes it so awesome. If I'd had this video as a guide when I was learning the FT back in the mid 90's as an undergrad it would have cemented the mental imagery I was creating to try and understand it myself.

[@BehzadVaziri]

it is like "your are in the jungle and complain about trees which dont let you see the jungle!!!". This whole thing is fourier transform! You better have an idea about that then watch this wonderful video.

[@ravitiwari4707]

FYI,, fourier representation of any signal is a way in which signal can be represented as sum of orthogonal signals(complex exponential signals which is a sine wave by a simple formula) ....

[@nitishmishra9365]

A lot of gratitude to you for this video,clearing the concept in crystal clear way

[@AA-gl1dr]

You should win an award for these videos they’re so incredibly helpful

[@EugeneKhutoryansky]

Thanks for the compliment. I am glad my videos are helpful.

[@ehudv9276]

Holy crap, theses videos are so good! that "ahh" moment mentioned below, I had the same thing, seeing the video from 00:00 to 00:40 Now I understand how cosin/sin wave functions relate to circular motion. simply amazing thank you

[@fahmylina123]

Much more than excellent video

[@selamawithailu4796]

I 6 yum uiiuii 8 km

[@seandafny]

Stamp

[@matteopisanello7787]

My eyes are seeing sine waves, my mind is seeing tom who's bashing jerry on the piano

[@ff-ti7nj]

lmao

[@chengyanpan4138]

couldn't agree more lol

[@kipropkigz1285]

the last part was really deep... thank you for this amazing work

[@EugeneKhutoryansky]

Thanks.

[@michaelzap8528]

I show my grade 7 son this video and give him a little bit explanation, he instantly know and master the Fourier things, polar system, vector concept and imagine number including Euler formula. This video just game changer.

[@EugeneKhutoryansky]

I am glad my video was helpful for your son. Thanks.

[@TheWindsofWonder]

Me during the entire video: “YOOOOOOOOOOOOOOOOOOO”

[@JuanLopez-vu7oo]

Lol same

[@MrZzfitzz]

Came here because I was instructed by my college professor, will return again when on psychedelics

[@om1gee]

Wow, this is the most relaxing thing I've seen on KZbin! So beautifully made with the music in the background ! The ending is amazing, too :) You think you're searching a rather dry and boring topic on KZbin and than you find art like this :-D

[@oracledsp]

I am just starting to learn the mathematics of audio signal processing using Max 8, and this was one of the most helpful and informative videos I have seen!

[@EugeneKhutoryansky]

Glad my video was helpful. Thanks.

[@Lily-tr9pn]

Read about sin, cos,tan theta in high school and wondered what was it all about...then said goodbye to mathematics forever...wish someone had taught us these concepts like this...hats off to your beautiful video👍👍👍

[@EugeneKhutoryansky]

Thanks.

[@xBoomshock]

“can you feel it?...the force...its always been there...”

[@失われたフグリ]

教育界はこういう分かりやすい説明をもっと取り入れるべき。

[@krakenmetzger]

I feel like I'm doing math on the moon

[@LL-ue3ek]

FINALLY! a unique video that's as brilliant as it is simple, and as educating as it is inspiring, has been found in the KZbin sea of Fourier Transform videos. I've been struggling to understand how frequency circles relate to a real-time waveform in a Fourier context until I watched this video. It made such a seemingly complex geometrical bundle of mess look as easy as 1+1=2. Thank you for clarifying the confusion that acted as a roadblock to a deep understanding of Fourier Transforms. ABSOLUTELY SPECTACULAR !! It's so inspiring because it shows the immense and unimaginable intelligence and power of the creator behind his creation of nature.

[@EugeneKhutoryansky]

Thanks for the compliments about my video.

[@jayelms2706]

I can't believe there are ones who don't like such a super professional and clear presentation !!!!!!!!!!!!!!!! Might it be envy? Btw, such a neat voice can only come out from someone who understands how Fourier Transform works inside her throat.

[@EugeneKhutoryansky]

Thanks.

[@barbequelove]

Wow, this is freaking amazing, this really helps solidify my understanding!

[@photon2724]

Now it's not every day that higher level mathematics blows your mind!

[@fifaham]

Amazing presentation - I wish we had similar computer applications that can do those graphics presentations when I was in the university

[@OjashShrestha95]

This video invoked the feeling of pure ecstasy. God Bless you!

[@EugeneKhutoryansky]

Glad you liked my video. Thanks.

[@z00ey]

beautiful - even the orchestral swells are perfectly timed to the revelations of the changes in perspective. You've given us all at the next evolution of learning maths

[@EugeneKhutoryansky]

Thanks for the compliments.

[@johnaybier418]

Mindblowing representation. My first thought: DNA

[@alikhan81]

you mean RNA?

[@ndv2212]

each time I watch this video, several ahh moments happen haha. Thank you so much for this helpful vid with beautiful graphical representation!

[@NityanandaHazarika]

omg! I never imagnined sine waves in 3D ever. Thanks for giving the feeling.

[@garymitchell4719]

Staggering graphics that perfectly demonstrate what is so hard to get across verbally. Been a maths student for 50 years yet I learned something today that was staring me in the face all that time but I never spotted it before - the sine wave formation on the 'z' axis. Also a great way to demonstrate why mathematicians use radians and not degrees in complex situations. Astounding video..... many thanks.

[@EugeneKhutoryansky]

Glad you liked my video. Thanks.

[@sandhyasandy9269]

aree yaarrr u came up with such a amazing video on youtube .really very very happy after watching this video 👏👏👏👏👏👏👏

[@sanjiblamichhane]

25 years in college and now this video to understand a circle. Welcome to engineering baby :D

[@Prometheus4096]

Couldn't get over the random Liszt.

[@Eloss69]

The best music for the best subject

[@828burke]

The sphere and line model REALLY makes it make more sense to me than the typical circles connected at midpoints that you see in videos. It lets me see that its essentially adding a vector instead of just being random. I could have comd to thst intuition eventually but this helped a lot.

[@EugeneKhutoryansky]

I am glad my video was helpful. Thanks.

[@ElvenSkywalker]

Your videos are truly a gift to humanity. Please keep up your marvellous work!

[@EugeneKhutoryansky]

Thanks for the compliment.

[@vex_ahlia7]

you had me hooked at 0:43 with that change in perspective.

[@krzem3532]

OMG this is an amazing explanation! Thank you so much!

[@EugeneKhutoryansky]

Glad you liked my explanation. Thanks.

[@ImTheMargo]

The guy at 14:06 is brilliant. Thanks for adding him)

[@aname6142]

+Maral Gurbanzadeh lol a little something for the ladies for a change ! (:

[@rajnikanthdsp4368]

what a concentration

[@Aemilindore]

The most beautifull video I watched. SO grateful to creators.

[@nokian9005]

This video is SO close to being a 10/10 perfect video. I don't know what it's missing, it's just really really good.

[@nictai186]

work of genius. i wish i have seen this 5 years ago. lol