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

3 жыл бұрын

What kind of program do you use for these animations?

@user-sh4wp3lo6b

3 жыл бұрын

What is the name of this music…?

@EugeneKhutoryansky

3 жыл бұрын

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

@EugeneKhutoryansky

3 жыл бұрын

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

@ishworshrestha3559

3 жыл бұрын

Ok

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

@EugeneKhutoryansky

3 жыл бұрын

Thanks.

@Saleh19999

3 жыл бұрын

i can relate to this

@leondaz7380

3 жыл бұрын

same

@spectacularserenitytunes1533

3 жыл бұрын

kzread.info/dash/bejne/pY6rzbCmlrPVnZs.html

@ydl6832

3 жыл бұрын

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

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

@EugeneKhutoryansky

5 ай бұрын

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

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

@AA-gl1dr

2 жыл бұрын

Never too late to try something new!

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

2 жыл бұрын

I am glad my video was helpful. Thanks.

@chriswalsh5925

Жыл бұрын

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

@user-di4bt7qu2i

Жыл бұрын

@@chriswalsh5925 lol, awesome!

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

6 жыл бұрын

Thanks for the compliment.

@Thebossatmserfgsd

Жыл бұрын

🥺

@nachtundnebel-up3nk

Жыл бұрын

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

@emptykid

Жыл бұрын

🥺 🥺

@Wallyisking

10 ай бұрын

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.

this feels like one of these "wow enough internet for today" videos you randomly end up with at 3 AM in the morning but it's actually 2 PM and you're studying for uni

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.

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

Жыл бұрын

Thanks!

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

6 жыл бұрын

Thanks. I am glad you liked my video.

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

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

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

@igorjosue8957

3 жыл бұрын

We can add together INFINITE ^ 10 sine waves

@Operational117

3 жыл бұрын

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

@__8120

2 жыл бұрын

"there's more" "No"

@gustavocortico1681

2 жыл бұрын

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

@rhonaldcastiglionemcbriard2023

2 жыл бұрын

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

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

@EugeneKhutoryansky

7 жыл бұрын

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

5 жыл бұрын

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

@ogunsadebenjaminadeiyin2729

4 жыл бұрын

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

@ruprectmonkeyboy1856

4 жыл бұрын

@@tommushrom5929 different application.

@vector8310

3 жыл бұрын

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

Hungarian Rhapsody in the background is distractingly beautiful

@jankiel3284

4 жыл бұрын

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

4 жыл бұрын

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

3 жыл бұрын

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

@lucaferlisi2486

3 жыл бұрын

Agree

@Ozzz77118

3 жыл бұрын

ahahah you are right. I felt same way

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

@EugeneKhutoryansky

2 жыл бұрын

Thanks.

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

@MrPabloguida

3 жыл бұрын

Hehehehehe....

I recently created a Patreon account for people who want to help support my channel. The link is on my KZread 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 KZread search engine by clicking the like button, and writing a comment. Lots more videos are coming very soon. Thanks.

@johnhall1700

8 жыл бұрын

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

@ronaldderooij1774

8 жыл бұрын

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

@afederdk

8 жыл бұрын

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

@EugeneKhutoryansky

8 жыл бұрын

+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

8 жыл бұрын

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.

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

@Star3

8 жыл бұрын

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

@Annihilator49

8 жыл бұрын

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

@alboraek

8 жыл бұрын

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

@chillingmeko

8 жыл бұрын

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

7 жыл бұрын

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

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

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!

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

@YamilSG

7 жыл бұрын

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."

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

@EugeneKhutoryansky

8 жыл бұрын

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

@flitzgerald7984

6 жыл бұрын

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

6 жыл бұрын

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.

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

19 күн бұрын

I am glad you enjoy my video.

@jamesday1295

18 күн бұрын

@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.

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

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

@marzipug5439

3 жыл бұрын

You think people just make this stuff up? lol

@vijaybabu4082

2 жыл бұрын

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

@scp-yearsago

2 жыл бұрын

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

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

2 жыл бұрын

Learning styles are a myth

@junogoose

2 жыл бұрын

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

@MonkOrMan

2 жыл бұрын

@@junogoose Yes

@ryanmitchell6721

2 жыл бұрын

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

@junogoose

2 жыл бұрын

@@ryanmitchell6721 I was being facetious.

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

@EugeneKhutoryansky

Жыл бұрын

Thanks. I am glad you liked my video.

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

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.

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

Жыл бұрын

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!

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!

This was the first time I truly understood the importance of sine waves...thank you so much

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

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

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

What a trip! Thanks for providing the 3d intuition of fourier and for taking time out of your life to make these vids.

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

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

@EugeneKhutoryansky

8 жыл бұрын

Thanks.

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

Ай бұрын

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

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

3 жыл бұрын

Glad my video was helpful. Thanks.

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.

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

7 жыл бұрын

Glad to hear my video was helpful. Thanks.

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

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

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

Thank you for making the time to make these- the animations definitely help

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

@EugeneKhutoryansky

Жыл бұрын

Thanks for the compliment.

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

@ff-ti7nj

3 жыл бұрын

lmao

@chengyanpan4138

2 жыл бұрын

couldn't agree more lol

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

7 жыл бұрын

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

@scottliu9555

5 жыл бұрын

r

@ankitaaarya

4 жыл бұрын

Truee agreed

@someonlinevideos

4 жыл бұрын

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

@Zooofa0610

2 жыл бұрын

@@EugeneKhutoryansky 😍

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

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

@EugeneKhutoryansky

5 жыл бұрын

Thanks for the compliment.

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

Жыл бұрын

Thanks!!!

Me during the entire video: “YOOOOOOOOOOOOOOOOOOO”

@JuanLopez-vu7oo

2 ай бұрын

Lol same

Wow, this is the most relaxing thing I've seen on KZread! 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 KZread and than you find art like this :-D

Uno de los mejores canales, y sin duda el mejor que encontré en cuanto interacción gráfica, amor eterno a los creadores!

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

Жыл бұрын

Thanks for the compliments.

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

@EugeneKhutoryansky

8 жыл бұрын

+Adriano Me, thanks.

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

5 жыл бұрын

Thanks for the compliment about my videos.

@brendawilliams8062

2 жыл бұрын

I agree

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

@EugeneKhutoryansky

Жыл бұрын

Thanks for the compliments.

Changed the whole concept, the way of thinking I was having about the wave form. Superb video.

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

The best music for the best subject

I'm making this comment in 2019, also, although I comprehend the language spoken I appreciate the narrator's explanations. Such a great post for those of us who want to learn more

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

Жыл бұрын

Thanks for the compliment about my video.

Much more than excellent video

@selamawithailu4796

7 жыл бұрын

I 6 yum uiiuii 8 km

@seandafny

7 жыл бұрын

Stamp

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

4 жыл бұрын

Thanks. I am glad you liked my explanation.

I wish I would have seen this video during PDE/ODE course. The visuals really help with understanding FTs.

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

3 жыл бұрын

I am glad my video was helpful. Thanks.

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.

@procactus9109

7 жыл бұрын

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

@nunyabisnass1141

7 жыл бұрын

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.

Excelente Video, obrigado por ele, sou Estudante de Engenharia de Controle e Automação e este vídeo me ajudou bastante a entender o comportamento da frequencia no dominio do tempo.

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

@EugeneKhutoryansky

5 жыл бұрын

Glad you liked my video. Thanks.

Wow your visualization of trigonometries are just admirable!!!!

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

8 жыл бұрын

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

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

@EugeneKhutoryansky

7 жыл бұрын

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

Thanks for the great video. I wish these videos were available back in 1994 when I was taking electrical engineering in college.

Eugene Khutoryansky, Great Content. The animation and explanation were perfectly in sync. Thanks for uploading!

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

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

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

@EugeneKhutoryansky

4 жыл бұрын

Thanks for the compliment.

FINALLY! a unique video that's as brilliant as it is simple, and as educating as it is inspiring, has been found in the KZread 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

Жыл бұрын

Thanks for the compliments about my video.

I feel like I'm doing math on the moon

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

@EugeneKhutoryansky

6 жыл бұрын

Glad you liked my explanation. Thanks.

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

@EugeneKhutoryansky

5 жыл бұрын

Thanks.

I only know the formula defining Fourier transform and Laplace transform and used them for analyzing linear systems. I loved how Laplace transform method simplified the way for solving linear 2nd order ODE which was commonly found in RLC circuits. Found their application also in Control Engineering both in conventional and modern methods using state space. But now, having been working as software developer for years, I forget many things about them, except the formulas. :)

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。

THis video is so visually beautiful!

@EugeneKhutoryansky

8 жыл бұрын

+vineeth venugopal, thanks.

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

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

@EugeneKhutoryansky

2 жыл бұрын

Thanks for the compliment.

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

I wish I had this video back when I was a student!

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

4 жыл бұрын

Thanks.

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

@EugeneKhutoryansky

3 жыл бұрын

Glad you liked my video. Thanks.

Hungarian rhapsody playing on background, love it XD

I really liked this video, especially the use of a multi-hinged arm to draw waveforms. It's very well done. I would, however, like to propose one correction. If a wave's amplitude were exactly zero, then its frequency must also be zero, because a wave with an amplitude of zero would be a flat line and not a wave. I think you intended to say, "infinitesimally small," instead of, "infinitely small." Real waves must have non-zero amplitudes and non-zero frequencies. Other than that one caveat, this video is great and I would recommend it to anyone who wants to see how waves stack on each other. Thanks for making these videos.

@deiz92

8 жыл бұрын

Maybe the 'Dirac Delta Function' answers your concern.

@WeShredForBeer

6 жыл бұрын

i thought the same about the sheet of paper example. The papers have a Volume, infinitesimally small however NOT 0. Nevertheless all theses videos are probably the best way to learn about advanced math and physic concepts.

Thank you so much for explaining Fourie series with an awesome illustrative graphics

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.

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

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

@PaulSmith-gi5bf

3 жыл бұрын

what about 7?

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

5 жыл бұрын

Glad you liked my video. Thanks.