Winding numbers and domain coloring
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- Опубліковано 9 чер 2024
- An algorithm for numerically solving certain 2d equations.
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Special thanks to these supporters: 3b1b.co/winding-thanks
Writing and animations by Sridhar Ramesh, with editing and narration by Grant Sanderson.
Even though we described how winding numbers can be used to solve 2d equations at a high level, it's worth pointing out that there are a few details missing for if you wanted to actually implement this. For example, in order to determine how often to sample points, you'd want to have some bounds on the rate at which the direction of the output changes. We will perhaps discuss this more in a follow-on video!
Music by Vincent Rubinetti:
vincerubinetti.bandcamp.com/a...
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![ПОЛНАЯ ИСТОРИЯ ЭКЗОРЦИЗМА [Топ Сикрет]](http://i.ytimg.com/vi/qBySr6Bt0B8/mqdefault.jpg)
"Being good at math is not about being right the first time. It's about having the resilience to carefully look back and understand the mistake (and) understand how to fix them." ~3B1B
This applies to pretty much any field
Yes but it's very worth mentioning it in the context of math in my opinion.
This is true for how to live LIFE even.
@@juliusfucik4011 Studying Math and Physics has the added benefit of keeping most people open-minded and relatively humble--you can't fake a right answer, and no one is always right!
Yeah that's exactly what he said.
"To be continued", awwww. "Now", oh yay!
Nope no yes yes
He got me
I'm guessing he put an ad in there
I was about to be mad
Talking about colors and Stoke’s theorem was a perfect setup for *Green’s Theorem*
(Ba-Dum-Tss)
First of all, while reading the Ba-Dum-Tss, I actually imagined and heard the exact sound. Second of all, response to your comment, not quite as Green's theorem is used to derive the Stokes' theorem.
You're all wrong Greens theorem is a specialization of Stokes theorem badumtss
You can look at it in either direction. You can prove Green's Theorem and then use it to prove the more general Stoke's Theorem, or you can restrict Stoke's Theorem to 2 dimensions and then you have Green's Theorem.
There is a general version of Stoke's Theorem for differential manifolds with a boundary of any dimension. The 2D Stoke's Theorem as well as Green's Theorem are special cases of this theorem. You can look it up at e.g. Nakahara's Geometry, Topology and Physics or probably any other book containing differential geometry. It's really interesting stuff :)
Oh. I just realized the "Green's Theorem" was meant as a word pun *facepalm*. I bow my head before your punning skills.
green is not a creative color
To be continued.... Now
And was my loudest YEAAHHH BOI ever.
THIS ^^ :D
I was really annoyed, because i hate to be continued.
Like this it was ok though.
SAME I was like "how could you do this to meeeeee how can I sleep now"
And here we see a wild Meme-Matician, someone who likes memes and cares about math. They are endangered species, as many are overtaken by one interest, but some still remain.
I stopped to look for the next video
I really love how you are able to explain such concepts so clearly that even a highschooler who speaks English as a second language like me can understand them. Is anyone else here in the same position as me?
Tomas Roque
I'm a high school student (year 11) but English is my first language.
Not anymore, but I was a few years ago.
Yup this is pure beauty
Yeaahhhh Bwooiii !!!
IM NOT EVEN AT THE HIGHSCHOOL AND I UNDERSTAND WAWAWAWAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
19:11
3B1B: “And this time, I promise, there are no bugs.”
*ant chooses this exact moment to defiantly walk across my phone’s screen*
Perfect timing
Lmao
The ant knew.
Even though we described how winding numbers can be used to solve 2d equations at a high level, it's worth pointing out that there are a few details missing for if you wanted to actually implement this. For example, in order to determine how often to sample points, you'd want to have some bounds on the rate at which the direction of the output changes. We will perhaps discuss this more in a follow-on video.
Also, to answer what a few people have asked in the comments, it's certainly the case that you can have a zero that this algorithm doesn't find. For example, combining regions whose boundaries have winding numbers +1 and -1 will give a region whose boundary has winding number 0, and hence which wouldn't be searched. For that matter, this is very much true in the 1d case as well; you could have a region with many zeros where the function is positive at both ends. You thought you could get an algorithm guaranteed to find *all* zeros of *any* continuous function? We're only human!
13:30 I thought the video ends
"have some bounds on the rate at which the direction of the output changes. " do you mean like choosing an area that is closer to a part of the boundary whose winding number changes quickly without negating itself later? Create unevenly sized x and y and then emphasize on the smaller one that includes the rapidly changing winding number?
can you add fractional calculus to that list ?
3Blue1Brown i feel like this is describing a flag with wind currents on the earth. Its possible to have the wond change 360 degrees without a lull in the wind.
If all the sattilites in the world went down we wo7ld still be able to locate low pressure systems by wind stations evenly distributed around earth.
11:24 plot-twist of the century
13:36 plot-twist of the millennia
in the same century???
Liam Swick yes
"plot-twist"
Pun intended?
Jazz music stops playing
EmissaryOfSmeagol a
To be continued... now!
*gets goosebumps*
also when the theory was wrong/incomplete.. it's like.. math on-the-fly XP
Shreyansh Darshan
I almost had a heart attack when I thought it was a cliff hanger
I was about to ragequit math at that cliffhanger
Stop playing with our emotions 3blue1brown!
true
That "to be continued" had me checking the length of the video on the edge of my seat. Never wanted to hear more about math in my life. Inspiring stuff and fabulous animations. Thank you
i just became a patreon of yours.
your videos just tickle every cell in my brain and i love it.
keep up the good work :)
I'm so glad that I live in the time when 3Blue1Brown exists
Yeah 👍😎
The real reason this is worth 20 minutes is because 3 Blue 1 Brown uploaded it.
Word!
Excel!
LaTeX!
Vim!
Internet Explorer!
"To be continued"
Me: Whoa whoa whoa whoa whoa
"Now"
Me: Phew
This was a great video. Probably the best I've seen on this channel. You show very beautiful visualizations of complex functions, which are often considered "unvisualizable" even in upper level Complex Analysis classes. Your fluency with modern technology to provide these "impossible visualizations" is a huge contribution to mathematics education. You also told a compelling story including a twist, which allowed you to remark about the "culture of mathematics" which was a great bonus. Even though there are a couple technical issues (as you addressed in a comment), the sheer storytelling and animations make this video an excellent accomplishment.
High praise indeed for Sridhar's debut video . . . and I agree
I do not know why, but watching your videos really helps calm me down when I'm stressed. This stuff is fascinating.
Wow! You covered path integrals, snuck in (hidden) taus everywhere, and gave my brain a whole new way to visualize 2D functions. This video was fantastic!
TFW colorblind…
We've got mild deuteranopia and the gradient wasn't so smooth, although it was still a continuous function. Are there any types of colorblindness that would make the function noncontinuous?
No, because the Luminance of the color wheel is continuous, no type of possible color blindness can ever make the color wheel look discontinuous. However some types can make it look symmetric.
nope, it just make you unable/hard to say if a square has every color in its borders
Ah yeah that makes sense. Thanks :)
it's not obvious to see it, but it's not really relevent since he show the results for every loops.
5 years ago? This video is so technological and professional! Incredible!
There are many mathematicians explaining things in a neat and super understandable way on UA-cam. I love watching those. And then there's this guy. Holy shit. I can't count the times my mind has been blown beyond recognition. Thank you so much.
That REALLY did not feel like a 24 minute video. Thank you 3Blue1Brown for making me fall in love with Math. ☺
When the "To be continued" came up at 13:32, I was sad that the video was over, but satisfied with what I had seen so far. Cheeky surprise!
Amazing video as always, Grant.
... and Sridhar :-)
Sir!... believe me YOU ARE THE BEST TEACHER IN THE WORLD!!!!!!.. YOU ARE PERFECT..your concepts,graphics,teaching skills .. YOU ARE DOING A GREAT JOB .. YOU EARN OUR LIKES AND SUBS
22:45 The most blatant advertisement of a channel I've ever seen. I'm gonna watch them all anyway because these videos are hecka good!
This is the best channel on UA-cam. So we'll made and thoughtfully executed. You and your team inspire curiosity, thank you!
This is interesting. I do hope you do more "Essence Of" series, though. I often send students to those when they have issues with calc or linear algebra. It would be great to have more to use.
In his Patreon page interview introducing the two new team members, Sridhar and Ben, Grant throws in right at the very end that Sridhar will be picking up the baton on the Essence of Probability series, and Ben will tackle Essence of Set Theory (I think). Grant himself will do an entire follow-on Essence of Linear Algebra series to add to/enhance what topics covered in Essence of L.A. Part I. With now three people creating content, the flow should increase commensurately.
this sounds great! i also would love to see an essence of multivariable calculus. should lend itself well to visualization. ...and complex calculus....and tensors! keep the good work going!
Actually, Grant does multivariable calculus by Khan academy! ua-cam.com/video/TrcCbdWwCBc/v-deo.html
@@piman7319 well but 4 years later theres only 2 prob video, no set video, and the calc and LA kinda stopped?
Οkay. I can not be the only one who is so calm when I hear his voice. Like.. I use these videos to help myself fall asleep.
Enlighten the world about the thinking process that mathematics is, is a noble and difficult endeavor. Love your videos, they reinforce why I fell in love with math.
Watching the end bit and seeing the top of the "infinite list of all topics," my reaction was, "MAKE VIDEOS ON EVERY ONE OF THESE!"
And the Gödel will start to complain that .... Never mind.
I am always impressed that you manage to make such incredible videos that they actually have constitently less than 0.5~1% dislikes, are there any other channels as overwhelmingly awesome as this one?
That bit around 4:10 gave me some intuition about vector fields, thank you so much. That has been one of the most difficult concepts for me to grasp while reading up for my physics studies that start later this year.
I'm gonna watch this video, sleep, then watch again first thing in the morning tomorrow. Because your videos are so educational watching them twice is the least I must do.
Excellent as always. I feel like this needs a library implementation - C++ or even ansi c. Thanks for the video. And you are absolutely right about getting some backing.
I just watched it and kept looking at the animations so I paid no attention to what you said. Time to rewatch it!
I god damn love this channel, I do honestly believe this is one of the greatest, if not the greatest, maths-related channel on youtube. It's entertaining to watch, your animations are extremely good and the videos overall are just great to watch. I don't think that I have ever watched a video of yours and not learned something new even if I knew a lot about the topic already. I've spent so much time on this channel, just binge watching videos over and over again. I do believe that I have learned more than I've learned in the school.
Entertainment, knowledge, explanation, everything is here, in one single place.
Fifth year medical student here. I enjoy every single video you upload. Thank you
I share these videos with other engineering majors at university. When there are confusing proofs in the lectures, your videos explain everything we need. Not to mention the animations make even the most confusing subjects easy to grasp (specifically Fourier transforms). Keep up the amazing work!
One of your most engaging videos ever. Thank you for continuing to provide this excellent service!
When the music stopped after the initial algorithm failed: 3b1b.exe has stopped
*record scratch*
Oh, you scared me SO MUCH with that "to be continued" thing, my heartrate at least doubled
I think it's really cool that you did science with it. Making a hypothesis and find out it's wrong. It may get people more interested but it shows the process of coming up with rules and algorithms like this.
Loved the video! You are my favourite mathematics teacher ever. You bring a new perception of mathematics.
My heart broke when I saw the "To be continued..", but then I was filled with immense joy when the ..now popped up. Grant, you sneaky troll!
As a Control Engineer and also (future) teacher, it has always been a little bit difficult to teach about Nyquist Theorem to Stability on Control Systems. I did the complex function course as student, but not all students in engineer classes do. So, explaining the theorem and the loops and winding counts is not the difficult task, but, explaining the reasons behind the theorem has never been an easy task. Thank you so much for such amazing and clarifying video. I've always told people about 3Blue1Brown Channel, but this video has been the perfect visual aid I've seen in this Channel so far (and all of them are very good indeed). A masterpiece I must say. Muito obrigado, Portuguese for "Thank you very much".
Amazingly insightful! This video should be used by the teachers around the globe!
This is an ingenious and massive step in the right direction. Love your work 🤗
I really hope you get paid enough for the amazing work that you're doing. Thank you for this. You put in a lot of effort and hardwork and it is visible in your videos. I wish you loads of success in life.
As someone who fell in love with math a long time ago, I have to say that this is one of the most beautiful and enjoyable videos I have ever seen!
Another accurately and clearly explained 3blue1yellow video.
Hey man you put so much work into these videos. 20 minutes is the least I can do right? Keep up the great work I love learning math from you.
the best mathematical visualizations possible, thank you
3BlueOneBrown, the only math UA-cam channel that I actually get excited when a new video comes out.
Another wonderful video. 3Blue1Brown is perhaps the best educational channel on UA-cam.
this was ..... amazing. please more!!!! i literally love everything you post
just hopped onto patreon too, keep it up!!
You're amazing! I have studied lots of algebraic geometry & alg. topology and am aware of the proof of the fundamental theorem of algebra using winding numbers and you still managed to give me a completely new view of it.
This channel is an absolute gem and higher education should be looking to this as a paragon of math and science education.
13:33 To be continued...
-Ohhh maaaan!
Realizing there is 10 minutes more and the word "now." comes up = Absolute happiness.
"Being good at math is not about being right the first time, its about having the resilience to carefully look back and understand the mistakes and understand how to fix them" -3Blue1Brown 2018
There is so much beautiful topology and complex analysis in this video. I love it. I'm a grad student in mathematical/theoretical physics and I really appreciate what you're doing to educate people.
Can you please tell me why any small circle near the zero needs to include all the colours? Thanks.
Correction: when I said the loop contains a zero I meant to say the zero is on the interior of the loop.
That... that was beautiful ! Thank you. Thank you for the being that clear in your explanations, for always choosing so interesting content, for the neat animation well appreciated, for the colors in this video and for the kind message at the end.
I'm a maths teacher and your channel blew me away quite a few times ! And I love that. :)
So thank you again !
Hey you are uploading more frequently these days. Keep doing that. I just love you guys. You always make me proud for loving math. I particularly love those videos where you solve tough looking question by applying simple concepts or some seemingly unrelated concept like mostly with pi related things or that n points on circle are joined to each other then in how many regions they will divide. I love those kind of things.
Complex graphs are the prettiest thing.
This was a joy to watch. I took all of complex calculus without understanding what I was doing... now *this* is what math is all about!!
I just love how he explains these concepts that are so complicated, but you still know it can’t be explained better
The cliffhanger in the middle was going to make me verbally attack you. Good thing you didn't hype up the stakes and leave us hanging..
thanks for continuing your awesomeness! i can't think of a video by you which I don't absolutely love (and learn such surprising things from); you totally rock!
The explanation and quality are simply amazing, great work!
You are one of the best youtuber! I have always liked math and found it interesting... but you have made it more interesting!
I like your pick of the background music! It blends so well with whatever you try to explain and make it more interesting and mysterious! Keep up the good work!
I love these videos! You impress me every time.
17:49 Gave me Goosebumps!
i often feel like skipping throught videos that are longer than 20 mins but i never skipped throght a 3b1b video once its more like this could go on for 20 more mins :) keep up the amazing work
I love the twist showing the narrow algorithm SEEMS to work but isn't an actually robust solution. A surprise like that really grabs your attention.
Wow... It's beautiful!!! Another amazing video 3B1B
11:20 the timing was very well done
These color maps are beautiful visualising of functions!
Thank you for continuing to make these awesome videos, your work has helped bring more than a few topics into focus for me!
Well first of all 100 salutes to you all .
Well you all are genius.
Right now I am college students.
I don't have that money to support you all.
But in future I will pay.
Keep awesome and good work going...
They are a team. Not just one guy.
̷ ̴ ̸ɢ̸ʟ̶ɪ̸ᴛ̶ᴄ̶ʜ̷ ̶ ̶
Well I didn't realise that.
Thanks for pointing out.
FWIW, the last two videos have been the debut videos of those two new team members, Ben and Sridhar. Up until those it was indeed just Grant.
This is a bit of a broader question that also applies to any dimensional algorithm like this, but you can't search every point. So how do you make sure you don't accidentally miss something when running the algorithm?
Also interested to know this.
I would guess you would split the boundary into lines and integrate over those lines, or something like that.
Taking integrals along the paths is the way if I remember university's math correctly
I suspect that to apply this perfectly a path integral may be necessary, but I'm no expert and this would in general be too difficult. Of course there are always numerical methods - look at closer and closer points on the perimeter until the change in arguement is less than some threshold to be pretty sure nothing happens there, or the magnitude is sufficiently small to call that point a zero and move on to another region.
Azazeo Ainamart I believe you are right
Best chanel in the game!
Love your visuals. Very engaging exposition.
The video was beautiful and enlightening! So many of the concepts from multi variable calculus make more sense now. Thank you!
"Being wrong is a regular part of doing math"
Sneaking growth mindset into a video? Nice to see all that discussion on Ben, Ben and Blue has paid off.
I'm not sure if this is possible but I would really like to see a video that explains the code behind these beautiful animations. As you pointed out, they are mesmerizing.
And thank you 3blue1brown for introducing us to the beautiful side of mathematics. I can't describe how grateful and happy I am to be a part of this awesome community. By the way, that moment in the video when algorithm didn't work out as planned really motivated me to keep working hard and never to give up. Keep doing a great job!
This was one of my favourite videos from you!!! So entertaining!!! Congrats to the Ben and Sridhar! They're doing great!!!
I must say it here, Grant you inspire me not only with your soothing voice and video content but also by the thought how much hard you must be toiling to make these videos.
You know, all of this reminds me of cauchy's residue theorem. The zeroes are like poles and computing the winding number is the same as performing a contour integration.
If the contour integral is an integer multiple of 2πi, then the area contains a pole (or multiple).
Though the similarity is really nice.
Can anyone please elaborate on how it is related?
Why is 3B1B consistently high art
This is seriously the most beautiful video I've ever seen. First video of yours I've seen, instant subscriber
This video had my mind blown. I could never had imagined that mathematics could be so beautiful (and colourful)! I started watching your videos from a year ago, and I must say that my perspective on math has drastically changed, thanks to you. Being a junior college student, comprehending your videos is difficult for me, but nonetheless, I enjoy learning the ways in which mathematics can surprise me and make me think harder than I had ever imagined. *Hats Off for you* Keep enlightening us!
I was like noo thats not right, it isn't always like that
few minutes later:
*top 10 anime plot twists*
edit : why don't you put an ad after the 'clicky stuff'
Pausing at 19 min to think out loud: is it not possible for a 0 winding number loop to contain a 0 spot? I'm imagining two 0 spots next to each other, that share the blueish region in between them. So a loop that surrounded both of them would go through red and yellow and green, and then back to red, then back to green, then back to red, because the loop never passed through the region in between the two 0 spots. As an example, at 14:09, look at the two dots to the right. You can imagine a loop that goes around the two of them, which goes from red -> yellow -> green -> yellow -> red and completely avoids the blue, yet it clearly contains two spots inside of it. Is this rule not exhaustive? ie, non-zero winding number implies 0-spot, but zero winding number does not imply lack of 0 spot?
I guess it's like how in 1D equations, there can be two spots with a positive value (or 2 spots with a negative value) and it's still possible that there are 0 spots between them.
Yes, that's correct! In essence, we like to think of each of the zeros of the function as having a winding number equal to the winding number of a sufficiently small loop around it (think about why we can do this by looking at Grant's discussion of small loops around zeros). It can be shown that the winding number of any zero is either 1 or -1 ( again, why is this?) Then, if you draw a loop around multiple zeros whose winding numbers sum to zero, the winding number of the loop is zero, because we know that winding numbers add together. Therefore, we know that any region enclosed by a look with winding number zero contains an even number of zeros (could be none).
Hmm, if that's true, and it seems to me that it is, why does the algorithm work correctly? In the example in the video, we're saved from quitting too early because there are an uneven number of zeroes. But for instance, for a polynomial of degree 6, wouldn't we gray out the entire region enclosed by a loop that contains all 6 zeroes right away, and so we wouldn't find any zeroes? I'm sure I'm just missing something, but I'm not sure what it is.
I think that was why he said that "We don't know if these cut off areas contains a zero or not" at one point, or, not guaranteed to not contain a solution.
We only knows that those that do have an integer winding numbers definitely contains a zero.
That said, this does bring me to think that this rule might just not work when there is, said, 2 points of which you can draw an area which boundary avoid certain colors. and the algebra ignores this area completely and tries to round off other areas, which would render it useless.
XD
Star the Triple Devil Yes, and here also we have the condition that there are an even number of zeros in the region in question
Man you are taking this animations to another level! Ever improving. Just amazing.
This is so fcking beautiful, the voice, the explanation, the visuals, the background music. Dude this is art. This is perfection.
I actually cried out in pain when it said "to be continued"
13:32 My heart litterally stoped for a few seconds...
F & Yu stopped*
Whenever I want to sleep. I play your video. It's so soothing.
can this get any better !! you explain things in the best possible way
Could you do a video in residue theorem and contour integration? I took a class in complex analysis but I think seeing one of your videos would really help solidy my understanding, much like your linear algebra videos did. Physics grad student here, so I'm interested in the aspect of applied math/physics.
One of most happiness moments: see 3b1b uploaded a new video 😆
チュimoc see him say to be continued ... NOW !
its amazing, you somehow find a way to keep making better and better videos!! the brilliance of high level math concepts naturally and simply tied together through beautiful and ever evolving illustration is... unparalleled :’o)
I never understand what he's talking about but damn I love these videos. It makes math real. Something that you can really see. Beautiful
7:30 "[...] a more theorhetical STANDpoint"
3b1b mentioned 「stands」, confirmed for JJBA fan