Researchers thought this was a bug (Borwein integrals)
Вставка
- Опубліковано 26 кві 2024
- A pattern of integrals that all equal pi...until they don't.
Next video on convolutions: • But what is a convolut...
John Baez has a really fun article about this: johncarlosbaez.wordpress.com/...
Help fund future projects: / 3blue1brown
Special thanks to these patrons: 3b1b.co/lessons/borwein#thanks
An equally valuable form of support is to simply share the videos.
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Hindi: Pragna1991
------------------
Original paper from David and Jonathan Borwein
carma.edu.au/resources/db90/p...
Other fun coverage of the topic:
schmid-werren.ch/hanspeter/pub...
johncarlosbaez.wordpress.com/...
Correction: 4:12 The top line should not be there, as that integral diverges
Timestamps
0:00 - The pattern
4:45 - Moving average analogy
10:41 - High-level overview of the connection
16:14 - What's coming up next
These animations are largely made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with UA-cam, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Reddit: / 3blue1brown
Instagram: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
For the very first time, the bug actually WAS a feature
imagine copying lol
Games have been doing this for _decades._
Lol the universe is like mojang
The developers of Real Life™ just left a bug in production and hoped no one would notice.
Math and physics seem to contain some undocumented easter eggs.
One of the main problems I have in making presentations is that I always try to make them like a story, avoiding spoilers so that everything leads up to the interesting take-home point, but you don't know what is coming until I get to it. This channel demonstrates why that's a flawed way of thinking for educational purposes. It's so much easier to follow along with these explanations knowing where they are going. The explanation at 4:22, while seeming like spoilers to me in the moment, was actually extremely helpful.
yes
If you want to guide someone to a destination, show them the whole map before giving individual instructions. That way if they make a wrong turn, they can have some sense that they’re going the wrong direction. Landmarks and reviewing the map partway through are important for humans learning how to get somewhere.
I don't think that's a spoiler, rather that's a hook.
Like movies doing "you must be wondering how I got here" type.
Hooks are really important in story telling as that builds the interest in the subject matter.
The actual Spoiler in this case is the relationship between the two graphs via Fourier Transform.
Don't worry. It is just two ways of making presentations. While Grant does claim his is superior (in some of his other videos), not everyone agrees. I suspect I would enjoy your storytelling style.
even in storytelling: foreshadowing or even straight up giving answers ahead of time to give a sense of dramatic irony is a useful tool for creating hitchcock-esque suspense in a situation where surprise is not sufficient for making the story good.
It's one thing to know *what* happens, another to see *how* it happens,
and sometimes knowing what happens makes you wonder how
I'm a retired electro-geek who last studied this stuff over 40 years ago. Having just discovered this channel, I wish I'd had this resource prior to slogging through the computational mechanisms available to us at that time. These verbal and graphical explanations are absolutely fabulous, and I foresee hours of enjoyable education in my future with a cup of coffee in one hand, these videos on my side screen, and a spreadsheet in front of me. Thank-you!
I remember turning my homework paper to 'landscape' to solve Fourier transforms 'by hand' in order to fit them on one line.
That's the most EE thing I've heard in a while, and I work as a plant electrician....
I thought exactly the same thing, I studied undergraduate electrical engineering 30 years ago which was very heavy on laplace and fourier transforms, and convolutions. This video would have helped me understand them infinity better back then!
Hey 3B1B team and especially Mr Sanderson,
I just wanted to say your videos never fail to enthrall and impress me. You have such a way of communicating high-level concepts that makes me feel exceptionally well-informed about the subject matter you cover. As of 3 days ago, I've finished my Bachelor of Mathematics degree, 4 years after having my love of mathematics reinforced by your popular video about 4 points on a sphere.
Your channel and its content are so important for young, mathematically-interested people and I cannot express how grateful I am for this content.
In so many words, thank you.
Congratulations on the Bachelors, that's outstanding!
And thanks for such kind words, it means a lot to me.
I think your space bar might be broken
@@TheOneAndOnlyZelenkaGuru utala pi toki ala (impossible, 100% fail)
@@TheOneAndOnlyZelenkaGuru it's probably just a moving averages problem 🤷🏽♂️
🤣😂🤣
@@TheOneAndOnlyZelenkaGuru i think it might come from using a foreign language keyboard on an iphone. when i was learning Mandarin, if i typed an English sentence the space bar would add extra large spaces for some words.
This is amazing.
I even love the way you visually explained moving averages.
hi
E
Hi, Destin.
i hate averages
Laminar transform
"a tiny positive number my computer couldn't compute in a reasonable amount of time"
You should do it the Matt Parker way, put it up on the internet and people will improve your code by a factor of millions in a matter of days!
Good point! If anyone out there has a deeper knowledge of computer algebra systems, I'm all ears.
@@3blue1brown Ask Matt Parker himself to do so 😂
The answer is π times 692235940415362523136988414491285998468620532382124599554066975879968202372479018941687306133557125141812009840009662733497578477395741589958741155007862285485649171111258286647871898412035813448185128487166238219335182872053769745063205146240398270221977832380760762866554366743397019522289256347615462644913261775369992728315584923236659323759817418582764754173499371387884058167010542953584434449476393697721676981883264752309900228411652423246081739021978704316749310333533596904537502580519003591630854375995694511316758712127072335981655643021189629703319518996608891858801563606731511756259150271536904664925444915995745598487882850973342179949112232261107451564475708164124601869338680457040736426834176357325238700023154772340405663484960868000544476177063934327405358840986142240740495891233632352852053087368646776262360895352822595554176491656178820976720387079767602962842304015276653872951276656719564661860009852322150747843167248021400524688931060413853949705429841350499311344844142812690878735649021359350878799892991941300536391836009746220081646980020619328507232729433224792490941993693654589654207336860144043824383616426523896328586666972201974975363869745131277430423497779482704923699635266814730743056122797451467295167944104959148848306171227734538923653674351260090426832081683750824578884795592847739029407231100114031692028834847718052811109661505435074338037197807509927683710026016782011198945921757041861903371723076024299552942686154078275262558274383125240246903963660244565495743790100779385689120612914314126748249032328644943967606168810945505133744051503793356677613465767506133403785838880077428117672171491305463631982985278470240463605873903007823368419732452411249428087806823995726037033029954428007284645945501678886834962638266386697203029172069359055069598160085557611071250819586513883262334808499877279404265739246453854314818930473784012514484630065265839504131463613716847280435096127167440453437234925013899740472517629852858007350702055473349166597916007035221009345839579731778913437555757452869569584725521765148238483120629334049015846611258643709781106104555540382601644817619693116271703781814763254333627569201647746337166728676209480537183667033744980348862985594703362234073685730010342405696049810927652018855284359782308568790335680407039194097771231043231125155243319999767116121609430970384357269423481232518264366416525210424503728896257581964154317685655227495297650147999531562443287526368243680227419154845562643905990854032891584723971919362819173221100539566110801807612543052603376782159573210538409554672405396295376724610561 / 466192705877572353389835231940601222951670101070134561239973049203355736797679164397624010041403280350403890172469013022072991611009891420460184365109844228257723354893359318552165960075193563963432495384489969443453032619043283947700320752061908507268402779707752507800584175761024396184554311926902174604278990716347817088215671830701294237737416084870577308225709433802172004179436883634224430186712907011255416169182951793876196020581124124871790191131026190817826295390668469484921153061628957182315532723627541561158527229962601975323545963581536503234088278520594697564268665056763039200953239157471488828427155355325822506814512931380692989201738194683671358027936731158164868428465160293810220914942856829006898677125278247051066719835366903281173005006414793603140852470513412338483125238870107400749975506910479167617492429188364161622090075380794841274064530078229588882401307373157838975174030799988659511414398333740739995186397781300193196193202757302833310640180721583180401458815210422678535674359307400703540019934939449079991546032209790021358416751689023180874723355242538833041051750335417189450271841435425842357705548951520412541807195011790313725815748442413653722842379292155259290874590674053279873682477462608416130078159361725049569136018863847769692711434304284648779994670136133745537730928927221014707549796285465899515205737181699779348683309919359212078797505708653203063359044435493651560670169020403706390659381796376355011534232215252329581727323206390983456393841689668653574596495767581620929318398055395068946540290994175198141078405809664168143560635261493106792900807310948239120108625189411525965104535724046516823181955565824022550341414576665251862252517262971555570824456334217133885793566352181827821799733460800562650876183474302696913558747118873383788058316498534666549582395993896633701984857946104957599858399846326694339088056360491291651580401291235916846039420291797011958951903525756596083400499833765015610160614682572190562581996175482897349716398200856207446583669457688398879025589194587622790093420738922616627264007078524712707438700538087126382407147484598708878421359007944754077319874892451013305146739264196759332202156279663574573700221915659282202010270420525069461223193977455730690186787162810767959480118208312050676001116374540365682589708346162179047550107133580021937933978889261168248964550472603798305071959551200598606625251939526329942481081494978482611041645840498095348015739875843098411344378730289107163926051684865403472243934262440308576997235702810712358963888128180945434447661046236283823959416606140230636235659383407828994007185098833311796358669296501389331388587322635430609807372093200683593750000000000000000000000000000000000000000000000000000
which has 137 decimals equal to 0.
@@user-gc8cy5tx1w christ wow
@@user-gc8cy5tx1w holy mother of Gauss
I'm a retired machinist and I ran into this twice this while machining radii from for example 9.500" to 8.500" in decrements of .01". I called tech support and no one knew the answer to this. They had never heard of it. Now I know, 15 years later.
Could you please explain how Fourier transform comes into picture in your case ?
@@HemantKumar-xn8mn The radius I was machining decremented by .01" from 9.500" - 8.500". When the control got down to , for example, 9.130, 9.130-.01=9.12. Not so. the variable read 9.119999999... . Then when the control got down to say, 8.729999999..., instead it read 8.7299999999...8. This happened on every piece I machined. No one could explain why.
@@SUNRA131 this is probably actually due to floating point imprecision and not due to the problem featured in this video. basically, theres only a finite amount of floating point values that can be represented with a certain number of bits, and since some numbers dont perfectly translate to a corresponding floating point value, itll choose the nearest one instead. most of the time this works fine, but sometimes it doesnt. a good example of this is if you try doing 0.1 + 0.2 in many programming languages, itll compute to 0.30000000000000004. entering 9.13-0.01 into python returns 9.120000000000001
@@EinsteinsBarber Makes sense. Thanks.
@@SUNRA131 Going into it a little bit more: Computers represent everything as binary, including floating point numbers. The way floating point works is you divide your 32 bits (or 64, or 128... you get the picture) into 2 distinct parts. The exponent, and the mantissa. This is analogous to scientific notation in decimal: 1.2*10^5 has 2 parts: the exponent (10^5) and the mantissa (1.2). What ends up happening is that since we have a limited number of binary bits for both the exponent and mantissa, we end up with gaps where certain numbers cannot be represented exactly (without using more bits). In addition to that, it seems that nobody uses the error handling defined in the relevant standard to detect when a number is not representable. This can lead to compounding errors when an inaccurate representation happens in the middle of a multi-stage computation.
If you're really curious, look up IEEE 754 on wikipedia :)
As someone who worked extensively with convolutions and Fourier Transforms in physics and engineering: This is a beautiful video and I’m excited to see where it leads us.
Once he showed that moving average it made click in my head and all the lost knowledge about fourier and convolutions from my university came back to me.
@@Herdatec I flashed back instantly to second year college and getting a B- in Signals and Systems. Heard him say sinc(x) and everything repressed came back
The next video on convolutions and their relationship to FFTs is out! ua-cam.com/video/KuXjwB4LzSA/v-deo.html
What a time to be alive! Thank you. Also what a good timing! The last episode of Veritasium was also about the fast Fourier transform and there, Derek mentioned you! :-)
Bless your soul! Your videos are the only thing that bring me sanity
I like the hint about multiplying large numbers being related to convolution. It took me until well after grad school to realize that the long multiplication I was taught in second grade, was actually a convolution.
@@stevenspencer306 Really? Seems interesting!
@@Math4e Are you holding on to your papers?
I've been trying to wrap my head around convolutions forever, so seeing that you're going to be doing a video about them has just made my day :)
Anything specific you're hoping to learn? Or any specific contexts where you saw them and were confused?
Me too! I was always confused how convolutions seems to be meaning different things at once, like folding and multiplying functions and doing f(g(x)) ..
@@3blue1brown As a chemical engineer, the only context I've learned them in is just for how to use them to take inverse integral transforms (basically just using the definition of convolution). I'd love to see more about the motivation and intuition behind that definition
@@ollerich32 f(g(x)) is composition, not convolution.
@@3blue1brown I find this topic in my statistics classes..i would like if u cover this in context of convolution of probability distributions.
Is it weird if I'm not studying or doing anything remotely to do with this kind of math, but absolutely loved it? It's strangely soothing and entertaining.
it's not weird. it's nice to find out that there are things you don't understand that will work themselves out in a very elegant way.
Not at all.. it's pretty much the story of my life :D Downside is I really have to put in some discipline to not be binging on interesting content too much :P
(or maybe it is, but in that case I love to be weird)
I find these so calming and beautiful, despite never really being good at maths. There’s such a sense of elegance and awe to these big concepts, and they always make me feel like I’m experiencing something beautiful.
Not at all! Looking "under the hood" & getting an explanation of How Stuff Works is fun for the Curious, whether they're going into math/manufacturing/car repair/etc or not
yes
As a Hungarian-German, the name Borwein is pretty funny:
Bor in Hungarian translates to wine, and so does Wein in German. So their name is basically wine-wine
Born to be a sommelier.
And as just a German, you think to yourself why they're putting boron in that wine.
@@deaconmaldonado7947BORn to be a sommelier
Arnold Blackback approves.
@@mortenbund1219All the other elements argon
As an electrical engineer student as soon as I saw sinc(x) I immediately thought: Ah yes, definitely something with Fourier Transformation later in this video. Here we go again!
Yep! This is the foundation of all signal processing! Takes me back to my analog systems and signals class!
😒
Im also an electrical engineer student and we see this next semester.
afterso many classes I realize the entire world can be broken down into vectors and sin() cos().
@@sebagomez4647 It's even cooler than that. Using the trigonometric functions is convenient because you're familiar with them already, and they're easy to generate with analog circuits. The Laplace transform and Z transform generalize this further to also take complex arguments (instead of a real number x). And in digital signal processing, all hell breaks loose -- Why not transform any function using a rectangular wave? Why not transform them using quantized waves? Look up leaflet transforms [correction: WAVElet transform].
Systems and signals is the class that makes you appreciate Fourier and Laplace Transforms, and math in frequency domain / complex numbers as litteral magic.
The trick is that litterally any real world function, and many "mathland" function can always have their Fourier transform taken and be expressed as an infinite sum of sinusoids or complex exponentials (which are easier to work with), and then you just do regular multiplication and perform the inverse transform and you have the answer.
One of our jokes is that "laplace is god" because its just that much easier for solving differential equations. (And most high level physics equations are differential equations in their most generalized form)
I was taught by both Borwein brothers (Johnathan and Peter) at Simon Fraser University in math undergraduate here in British Columbia, Canada. Peter was a joy to take complex analysis with. Jonathan's 4th year real analysis course was... less joyful. Brilliant man, we as his students weren't ready to hold the volumous and requisite knowledge in our brains at all times. Still, I greatly appreciate the experience and am glad I passed his course!
The very best hour of my undergraduate was a day where Peter Borwein, 10 minutes into our scheduled hour long analysis class on a hot summer day, chatting about anything but the course material, said "I didn't want to teach today anyways", and we spend an hour just talking about mathematics and science.
I would pay good money for a recording of that hour.
Woah nice! Was it analysis in R^n and general metric spaces or more like measure theory and functional analysis?
Had no idea they were professors at SFU! I've just started my first year at SFU as an undergraduate majoring in data science.
Very cool that they are teaching at SFU. SFU really deserves more credits than it gets. Despite all the trash talk from UBC, SFU seems to be quite strong in several departments.
@@MaximBod123 I am doing DS at Michigan. Seems like the SFU program has quite a bit of business focus that is absent in our program. Goes to show how underdefined the discipline is ig
So if we alter the series with 1, 1/2, 1/4, 1/8, 1/16… the integral will always be pi since the sum of this series will always be less than 2
But we would have to multiply by 2cosx
How can we check this if it is true or not
@@kaanetsu1623 I could probably do it rn but I'm busy; but just use a calculator/desmos no? If not desmos use a graphing engine and input the function
That's not the same as the series from before, because the numbers were all decreasing by -1/2, your suggesting to do 1/2^n
@@kaanetsu1623 one could study the convergence of the series towards PI
I’m currently studying maths at undergrad level, and the difference between 3B1B and the teaching I am receiving is day and night. You do so much to motivate and illuminate with these videos. I know that to learn the detail will involve a lot of hard work, and then I’ll have to develop my understanding by exercises and problem solving. However, now that I am fascinated and have a picture, this is a joy, not a chore. Thank you so much and keep doing this sort of thing.
Yes and Grant has made what, about 5-10 hours total of videos in this manner for his channel? While in your math classes, you get 40 hours or so of content for every course. UA-cam will always win out for ‘most interesting’ content. A good in-person educator will take the best of what is online though and bake that into the daily teaching.
Maths pronouns: they,them lol
i’m currently studying electronic engineering and i’m pretty familiar with all of this frequency domain stuff, but the sudden “aha” moment I had at the end was really something else. 3B1B really knows how to neatly wrap together seemingly disparate pieces of information
Its so nice when you know enough math that you can figure out the problem yourself midway through the video
Yeah, god bless learning filter design many years ago...
Yeah must be, but that's not me lol
It's also not really nice when you don't know or better yet understand anything in the video from start to finish. That's me.
I don't know if anyone will ever see this comment, but as an Electrical Engineering student, I guarantee that Fourrier and Convolution are very powerful tools. We can analyze an entire circuit through equations modeled using fourrier and laplace. Note: I was taken by surprise, I wasn't even looking for videos on this subject.
These video's are so incredibly well made that, not only is the math beautiful and well-explained, but the scripts 3Blue1Brown uses in these videos is just as beautiful and meticulously constructed. This is one of those subtle things I love about science and math - that it teaches you to speak carefully such that what you say has exactly one meaning. It's a truly difficult art to master but if achieved, the speaker is effortlessly satisfying to listen to.
In a 20 minute video, 3b1b teaches what my school takes 1 month to teach
I've always thought your visualizations are among the best I've ever seen. Thank you 3Blue1Brown for getting me back into Mathematics after graduating from university!
He uses a Python library called Manim to make them.
@@seneca983 He created Manim! :)
I'm not at all a math student, but I come to this channel every time I want to relive that feeling of "wow everything is connected, this is so beautiful"
For some reason watching this video the though of DNA telomeres jumped into my mind. The fact that they shorten but remain relatively functional all the way until that critical threshold after which they fail to produce coding sequence protection. It’s just fascinating how our world’s laws just mesh and meld into one another from math to biology to space-time geometry
Well, as one science communicator on youtube puts it: "Physics is everything"...
Only that there is absolutely no relation whatsoever between telomeres and this. It's an artificial relation that only exists in your head, sorry.
Thats not how telomeres work. They aren’t “used up” they are just buffers.
this person seems to know what they are doing, I know this is 3 months old, but hoe do you think they work
I am French chemist...very far from math in general... but your way of explaining and showing interesting mathematical things made me read my old book of mathematical analysis :D Thank you and please continue!
I get you. I study molecular immunology, far from the math lands too, but these videos help me grasp the wonderful elegance of mathmatical problemsolving. Fascinating stuff!
A professor in college had this on his door along with a warning about assumptions and patterns. It's been in the back of my head for years to look into this and understand it!
me watching these videos to feel smart, knowing full well that i don’t understand a word he’s saying
As a math enthusiast that became engineer 25y ago 3B1B makes me feel I can still understand complex & fun stuff like this 😍 definitely the best youtube chanel ever, there was nothing like this before youtube
Fun to watch after finishing an electrical engineering degree. Feels like the second you found moving averages, I could see the convolution and Fourier transform. Made me feel like I learned something in the past 4 years
If this didn't immediately trigger you Fourier transform reflex as an electrical engineer you would have grounds to sue whatever school gave you the degree. That sort of negligence would be unheard of.
Amazing to see the convolution and Fourier relationship conspire to create this interesting pattern!
fun fact: Bor means wine in Hungarian, and Wein means wine in German, so if you translate it, it's the winewine integral.
This is fantastic! I have actually used the relation between the convolutions of rect functions and the multiplied sinc functions in my work. The convolution of rect functions is actually one way to express a jerk-limited motion curve. Separating it into the sinc functions in frequency space can help tremendously to understand the impact that such a motion curve has on a control loop. Really cool to see this here! 🙂
Fun fact (also somewhat connected to the Fourier transform): You can actually integrate sin(x)/x using the Feynman trick of introducing a new parameter and then differentiating under the integral sign, but to do this you needed to somehow come up with the crazy idea of setting F(a) = integral of sin(x)/x * e^(-ax) dx from -infinity to infinity. (The rest is a routine calculation of finding F'(a), and integrating it back to get F(a), and substituting a = 0.)
For what it is worth, what you are doing is essentially the Laplace transform. You just first note that the integral is even, so you only worry about the positive half of the axis.
It's not too crazy once you realize you're just trying to eliminate the pesky x in the denominator
Guessing exp(-ax) is not crazy necessarily. It is done often in physics, because 1/x diverges when integrated, so one uses a strongly decaying function like exp(-ax) to "help" make it converge faster, then you remove the "help" at the end. A similar trick is used in quantum electrodynamics where the Coulomb force has a potential V(x) ~ 1/x. The exp(-ax) factor corresponds to if the photon actually had mass a, and then at the end of the calculation we set a = 0 because photons are actually massless.
Yep, you're definitely describing the laplace transform and it is actually a generalization of the fourier transform
I'm only just beginning to actually intuitively grasp the Fourier transform over the last year or so of some really excellent videos coming out, but now I got the Laplace Transform to try and figure out! lol
This video takes me back to my senior-year signal processing class back in college and learning about Laplace transforms and convolutions. I knew that the term "convolutions" sounded familiar and it seems like Fourier transforms are just a special case of Laplace transforms! This is why I love your channel - it brings back memories of learning (and the trickiness of these topics) from the past and it's sending me into a deep rabbit hole of trying to remember much of this topic. I've never commented on any of your videos before but thank you for this great video and all the others you have done over the years.
Excellent ! That may also be an example of why proofs by induction are required : observing the first terms of a sequence never tells you for sure what happens next ...
👏👏👏👍👍👍 this is *the* best channel of its kind, the team never compromises the rigor while maintaining uncluttered vivid visualization! Extreme quality of their work, the modesty of the presentation, the simple fact the text and the formulae are correct and proof-read to near perfection (in contrast with their ubiquitous competition) , all these features make the channel uniquely useful in their contribution to noosphere :) 👍👏🏆
As an electrical engineering student, convolution and Fourier transform are very useful and interesting concepts. I loved this video.
he makes such niche and complex subjects seem so simple! very nice
And he's so cultured that he didn't use the word niche, he said "esoteric".
They aren’t that niche and complex but yeah! Master work. ⚙🕰
@@05degrees they absolutely are lol. Most people never even go past solving a triangle. For most people even basic differential calculus is completely foreign
These topics are certainly ... complex
Fourier transform and convolutions are not exactly niche ... It is bread an butter in electronics, to name a few: control theory, communications, signal processing.
Absolutely amazing. The problem itself and the quality of this video.
Your videos are, by far, the BEST videos on whole YT...
Explain these concepts, with the simplicity and naturalness you use... How it can be even possible? Out of this universe...
Thank you 3B1B.
These are the kind of awesome videos that I wish I had back in my undergrad in Physics. So helpful and intuitive!
Great timing! Just this week I took a dive into signal processing and I learnt about fourier transforms and convolutions, you chose a very interesting aspect of this area of math. Awesome video, I'm looking forward to the rest of the series!
These videos are just gorgeous. You make seemingly complex problems almost unnecessarily intuitive. It's a thing of beauty.
This type of stuff takes me back 20 years to my college days in the best possible way. Thanks for helping keep that feeling of wonder and amazement alive.
Just when I was looking for a good maths problem to ponder, you swoop in to save the day! Thank you Grant for everything you do.
I just finished a signal processing course and this is what we did all semester. So satisfying to have it explained here!!
Wow those visualisations are magical, your channel never stops to give me goosebumps. Thank you so much
Wow, I've never even taken a calculous class and somehow I was still able to fully understand why the pattern doesn't hold because of your explanation and animations! This is such a well put together video!
As soon as you started talking about rectangular pulses and the value of f(0) I immediately realized it was going to be Fourier frequency analysis and the DC offset, amazing video!
Wow, im so amazed! I had some lectures on Fourier transform and it is AMAZING to see these integrals be so wonderfully explained! Thank you so much ❤
Mathematicians: Math isn't mathing
This was thrilling and magical the whole way through. This is one of the few times I've been able to see what was coming at every turn - I just spent yesterday working on a problem involving convolutions, and when that little hint popped up about the relationship between the Fourier transform and the integral definition of a convolution, it was an exhilarating feeling!
Signals and Systems 101... Laplace transforms Heaviside step functions and convolution are our bread and butter as electrical engineers. I loved this video more than you could know!
Electrical engineering is all about abusing complex numbers to cheat math and save time.
The Fourier transform is probably my favorite function in all of math because of how powerful it is. (Along with the rest of the time->frequency domain transforms, we had a joke that laplace is god in college because it made differential equations so much easier, not to mention how it sinplifies every linear circuit into a simple transfer function multiplication and convert back to time domain, not that it was always easy to do so.)
I knew a lot about that, and still I learned a lot. Such a magical video. Keep this up. Can't wait for the next in the series.
This left me breathless! Wow!
What looked like an insane thing not only got explained, but even gave me the feeling that I understood (most of?) it!
You are an amazing explainer of math!!!
Currently studying 'Signals and Systems' so a video about convolution is absolutely godsent.
Great video as always!
that little drawing sequence caught me by surprise, its really fun and i love how seamlessly it mixes into the video! didn't even feel like its a new thing, nice :)
Oh man, can't wait for the next video! Even though I know convolutions very well and have used them quite often, I never was able to really wrap my head around how they work, and I'm looking forward to changing that with the follow up video :)
Love this video!
As an electrical engineering student, I immediately link everything together as soon as I see the rect(x) and the moving average!
slowly but surely i'm learning that sinc() is the heart and soul of digital audio. your graphics of how the Fourier transform of a pulse relates directly to the wavy sinc() just makes me feel warm and fuzzy. add in the superposition principle, and digital audio just makes total sense to me - many thanks for showing me another way to think about this
The fact that 57k other people watched this with me within the first 2 hours of its posting makes me happy for humanity.
My background is in optics rather than maths; as soon as you switched the discussion to rect functions, I could see the whole remainder of the discussion laid out. Very satisfying, and a great discussion!
I always love your videos, great visual and oral explanations that make difficult phenomenon clear (this one in particular)! Thank you so much for your work!
Not only visuals are amazing, the wording script of what you say is insanely accurate and well-thought
4:53 Oh shit. Grant is a gamer
I'm familiar with all the concepts of this video except for convolutions, so I'm looking forward to the next one. Excellent, well thought out explanations and graphical representations.
This is fascinating! I happen to be looking into Fourier transforms right now, so this was a timely video indeed. I love the insight about the interval shrinking by what is effectively a subset of the Harmonic Series (which diverges). Interesting to see a real world example where what initially appears to be a weird arbitrary cutoff point turns out to have a rather elegant explanation. I look forward to the next video!
Your visuals are top notch as always! Such great explanations here too!
This takes me back to my childhood. Sitting in the Oakland Public Library reading a book about the unknown formula of an egg. You have such an amazing way of presenting!
Please elaborate on the formula of the egg 🎉
@@caseyj1144 If I remember correctly, the formula didn't exist. There were close approximations. The book as a whole explored lots of mathematical mysteries both solved an unsolved.
You gave me so much nostalgia with that comment! I loved trying hopelessly to understand math and physics problems as a kid but still getting exposed to some interesting ideas.
Convolutions and Fourier Transforms were my favorite parts of math. Great introduction here.
Excellent presentation. The way you have started the video, it compelled me to watch it, and I am happy to see that you have made the connection between Fourier transform and convolution process. The most interesting thing is that you have given clear justification why the integral value deviates from pi after the stage of 1/15. I am Professor and researcher in signal processing (more specifically Adaptive Signal Processing) and I tried to make extensive treatment to Fourier Series, Fourier Transform (FT), DFT, convolution in my channel. I hope if anyone is interested to learn more about these topics, he/she may follow the channel. I really like your animation. It is so beautiful and interesting. Thanks for your great effort.
Please create a full calculus curriculum.
I struggle to put in words what it is that you are doing that’s so captivating when presenting mathematical beauty. It’s exploration, a lot of passion and curiosity, and part mystery. I want to say it’s like you take a complex mathematical concept and make it concrete and less abstract, but that’s not it. It’s more like you present an abstract concept in a way that makes me care about knowing despite the abstractness.
Your passion and ability to search and uncover, then explain mathematical concepts in a visually stimulating manner is what is missing in the generally abstract presentation that is a math course.
Gosh. Convolutions were always a difficult topic to learn as it is tricky to wrap my head around computing them. The representation of the moving average to explain convulsions is quite elegant and cool. It is always great to understand the intuition and the deeper meaning behind math concepts. Can't wait for the next video!
I went through all this in college, but that was *many* years ago. It’s nice to see the beauty of it laid out again (and without having to worry about reducing it to practice on a test next week 😁)
I agree. Having to learn something for a test takes the fun out of the thing that I'm trying to learn.
I as an undergraduate engineering major, had digital signal classes and never could I think of these visualizations in any way. The moment that convolution was explained as a moving average something just clicked in my brain as I went “Ooooh. That’s why I had to do compute all of those stuff”
Your illustration at 12:40 showing how the initial wave form and the fourier transform can be calculated from it as a continous integral of the corresponding constituent waves is absolutely genious.
For a person that thinks much more visualy than most this was a perfect for me
Thank you
Just finished one of my EE semesters and we learned convolution and Fourier transforms/series. This video would of been so nice to see a few months ago. Still great to see though.
Always makes my day better when I see that 3B1B uploads
Why do you manage to blow my mind with every single video you make?
Whenever I want my mind blown I came to this channel, works every time .
As an electrical engineer, I was nodding along the whole way. Love your videos as fun reviews on key concepts! Also thanks for tackling these more "convoluted" topics ;)
We use them in dealing with the waves produced by an earthquake in geotechnical engineering.
I've been aching for a high-level Fourier video for *so long* that I almost considered making one for SOME1, but holy cow this is amazing! Kudos
12:35-12:50 I have never seen a more beautiful and succinct depiction of WHAT Fourier transforms are, and how they work. Brilliant.
Wow this is absolutely extraordinary. I mean the way you explain it and how you visualize it. Really well done!
As someone who never heard of Convolution and all that stuff. Things got confusing rather fast after like minute 10. But thanks to visuals and simplified explaination i could follow somehow. Top vid.
I graduated in electrical engineering munching transforms and convolutions for breakfast, but never really understood what I was doing and where it all came from. These videos have been full of a-ha moments so far. Looking forward to the next one!
It's especially nice to watch this video after your convolutions video!
Yet another amazing video. Your animations and approach to explaining the problem are just great and help a lot in understanding.
as someone in college right now with a focus in signal processing, I immediately knew where this was going within the first 3 minutes, and it was immensely satisfying to see that confirmed :) great video as always.
I highly appreciate the math behind your videos. A video on Taylor's remainder theorem would complement his existing videos on Taylor series and enhance our intuition in calculus.
Thank you for the videos and the work you are putting in to explain those concepts!
This is great!
Absolutely amazing! Every video you make is pure gold
Imagine testing a computer program for bugs and you find a bug in math
I really like you continuing on Fourier transformation and convolution. Especially cause in my field of work it's not just an abstract math concept, but a real physical phenomenon - if you shoot a crystal with x-ray out the other side comes the Fourier transform of the crystal structure - inanimate crystals can perform Fourier analysis, not just computers and mathematicians, which I personally find really cool
This is one of the coolest things I've read today. Could you elaborate on how that happens by any chance, or point me to a source?
@@rampadmanabhan4258 Ok, basically it goes like this: Atoms in a crystal want to have the lowest amount of energy in their positioning to each other. So they will situate themselves a certain distance apart, everywhere. So we have the same distances, over and over, in a periodic structure everywhere in the crystal. If we now shoot an X-ray beam on that, the electrons of the atoms will to move in sync with the waves of the X-ray beam. This input in energy means, that they also produce X-ray waves of the same frequency, spherical in all directions. Because the crystal is periodic, this will all add up. But because they are a certain distance apart, the electrons are not in sync to each other, but to the highs and lows of the incoming X-ray wave. So if one wants to add them together, one has to account for this phase shift of 2pi * distance between any tow atoms / wavelength of the X-ray beam. So to get all outgoing X-rays one has to Sum over all atoms, or in the limit, take the integral over the electron density in the whole crystal, multiplied by e^(2*pi*i*distance/wavelength), which is the exact same formulation as the fourier transform. The distances between atoms get transformed into what we call the reciprocal space, similar as time resolved things get transformed in 1/time or frequency space. If you google "Laue method" you will see pictures of such experiments, where on the other side of the crystal you get the primary beam, and many different dots on the plate, going out of the crystal at a certain angle. Due to what is know as Braggs law, the reciprocal space is directly connected to the sine of the angle of the outgoing beam, and the intensity of each dot is proportional to the phase-accounted summation of the number of electrons in each so called diffraction plane. Because remember, the crystal is periodic? So the distances between atoms are too! And so each in plane of atoms each atom will have the same distance the next atom in the next identical plane, and so the amount of distances are finite. So these finite amount of distances mean that one gets a dot for each plane, and not the full plate full of varying intensity X-rays. So basically the crystal produces a fourier transform of itself. Or the X-rays do. Or both togehter. But one then has to do the step back, perform ones own Fourier transform to get the original atom positions, which is a whole other story. For more information I want to direct you to Kevin Cowtan, who has excellent resources on his Website of the University of York, especially "THe Fourier Picture Book" and the interactive tutorials for Structure Factors and Ewald Spheres
A awesome result with a great visual explanation. Keep up the good work. Thank you.
Fascinating. Looking forward to the convolution video!
oh man I'm reminded of the tons of exercises on convolutions i had to do for our waves courses in university. I really did like the science behind it, but the extensive math problems were a pain. Still, when we learn this in applied science and see what can be done with it in regards of waves, its fascinating.
Convolution is when all the other electrical engineers in my class revealed that they had truly abysmal geometry and spacial reasoning skills, its fundamentally sliding graphs across eachother to set up a very ugly integral almost exactly like how he did the sliding average animation.
Fortunately we quickly moved on to learning that you just transform to frequency domain, use normal multiplication, then transform the result back to time domain. (And make a computer do the actual arithmetic of it, systems and signals is the class you are told to make matlab do all of your integration for you)
How can someone not fall in love with the beauty of mathematics !
You are brilliant in explanation 😘
love from india🇮🇳
Wow! You've broken this down so well!! At first glace I thought there was no way I was going to understand why, but the answer became obvious in your simple example. Thank you!
This is the first time I’ve felt genuinely excited for the next release of a heavy esoteric math / cs video series. Bravo!