The most beautiful function in Math: Sinc (3B1B Summer of Math Exposition
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- čas přidán 22. 08. 2024
- Massive shoutouts to 3B1B, Khan Academy, Numberphile, and all mathematicians spreading the good word.
Sinc function Wiki article (containing other cool properties): en.wikipedia.o...
Dirichlet Integral (containing the hairy details of how to do these integrals/sums): en.wikipedia.o...
I recommend looking at the Feynman Method, which doesn't require any knowledge of Fourier/Lagrange transforms.
Flammable Maths doing the sin(x)/x integral: • DESTROYING THE DIRICHL...
Flammable Maths doing the (sin(x)/x)^2 integral: • Doing double Feynman I...
Derivation of sinc's Fourier Transform: math.stackexch... (for our purposes, set omega_0 = 1).
For a great video about sinc's applications to signal processing (how many signal can be reconstructed from samples using sinc), check out this other video from #SoME2 : • Shannon's Paradox: An ...
Music is "Prime #19" from Echochrome.
The programs used were Powerpoint, Mathematica (for the simple plots), and Kdenlive (to add the music).
#some2 #anime #maths #math #funny #edutainment
Respond to this comment with your credit card number, and I'll guess your height and weight OwO
Edit: 3B1B stole my video idea, we've made it big boys :3. For real though, if you want more sinc goodness, check out his video czcams.com/video/851U557j6HE/video.html .
Alright, it is... 124
I will give you my height and weight and you give me my credit card number (but your credit card number will have more use). Deal?
Weight: 80kg
Height: 1.92 m
@@TheElderBot Your weight is equal to Earths gravitational force acting on you and it is measured in newtons. Mass is measured in kilograms.
less anime pls
4546 5229 9001 2593
I am 6'9" and weigh 420 lbs. You'll never guess my credit card number, unless you use the sinc function.
heccin chonkerino
bro you are tall
1337?
bro was born and decided to get 🅱️ig
@@ishaanlohani r/woooosh
Funny note: because of the Lindemann-Weierstrass theorem we can safely say that the sin of any algebraic number is transcendental, and thus so is the sinc. It's mind boggling how the infinite sum is pi tho.
Funny edit: as pointed out in the replies, 0 is the only exception.
"the sin of any algebraic number is transcendental, and thus so is the sinc." Nit-picky Nick here. 0 is algebraic, and sin(0)=0 is not transcendental. Furthermore, since sinc(0) is usually defined to be 1, that's another counterexample. The Lindemann-Weierstrass theorem only applies to non-zero algebraic numbers! Otherwise, yes.
@@hOREP245 Whoopsie :P
@MorTobXD No need, I can live knowing I made a small mistake :P
@@davidebiclol
@@davidebic gigachad mindset
I love the "if I add my height and my weight out pops my credit card number" explanation because it pretty much sums up my mathematical experience of college calculus
Edit: I passed both of my analysis classes and I still don't understand how they came up with some of this sh*t
Wow, great video! I can confirm, my therapist is in fact now on the next Forbes cover.
This was pretty funny, but I wish there was more explanation besides "I'll leave some articles in the description, also check out these Wikipedia articles". I still don't feel that feeling of internal understanding that 3b1b videos give me. 6 minutes into the video, I was thinking, "so are they going to explain _why_ all the crazy results are true?"
I wanted this video to be as accessible as possible. Unfortunately, going further requires knowledge of deeper concepts, like convolution, and a bunch of things need to be proved to justify it, all of which are hairy mathematics. But at 5:55 , I definitely impart the important big picture of why: sinc is super nice and simple in the frequency domain, and all the crazy sums/squaring we were doing have analogous operations that can be performed in the frequency domain. But because things are so nice and simple in the frequency domain, you ultimately get nice answers overall.
Definitely check out the links if you want to see the hairy integrals done outright, but it won't really provide any insight. The main insight is what I said about being nice and simple in the frequency domain. Cheers!
Sometimes, a good video leaves you wanting more.
I don't think you'll be surprised by the explanation.
Pretty sure it's just number crunching and some neat theorems here and there
you mean you want the bad part of math divulgation, that false feeling of having actually understood something.
also this video is really neat, presented with pure experimentation just shows how crazy apparently theoric shit has actual real and useful application, a subject matter presented many times in 3b1b
@@yuw8410 It's not at all the bad part, I have no idea what you're talking about. Both absolutely have their time and place.
I've watched CZcams math videos for probably 10 years now, and oh boy there is a myriad of them that try to be funny and quirky in their methods, but yours seriously made me laugh. not only that, but it was educational, and I appreciated the part about "plugging in arbitrary units into F=m*a". good job, you've earned a sub!
Yeah, I watched all kinds of youtube videos all day and yours is the only one that made me laugh.
Yeah that Flammable Math try to act funny but every time comes out as cringe.
Fahrenheit = meters * amperes
This was incredible! The math, the humor, pls do more
3:48 Kansan here, I’m a figment of my own imaginary imagination I guess.
I clicked on the video because I've never seen anime and math together in a video, and it was totally worth it. xD
Well done, lad.
Someone must make this into an academic branch, Weebonometric Analysis! I'd totally sign myself to a full research career xD
You should watch Congruent Angle's Thesis then.
@@Felipe-sw8wp oh, that's the opening theme from Evanglelion, a classic. Another classic without that _complex_ story is Nth-root Shippuden, and if you wanna have some laughs, you should see Seven Deadly Sines
@@filiperodrigues97 LMFAO I admit I'm impressed you could come up with those. But I wasn't being creative there's an actual video if you search Congruent Angle's thesis on youtube (the name turns out to be Cruel Angle's Thesis, but the result will show up anyway).
@@Felipe-sw8wp LOL my bad, I assumed that was some kind of pun session, not an actual video. I guess I'm not enough _integrated_ with people over the internet, so I'm not good with all the references xD
I saw an example interview in the University of Cambridge for Engineering where the task was to sketch the graph of "Sin(x)/x" and was wondering why they would ask such a weird question with an equation that has no way of being used for anything but after watching your video It's truly fascinating to see such a seemingly random function to have such nice properties and even widely be used in the Engineering field.
Fun fact: It is possible to integrate sin(x)/x from 0 to inf with the Feynman trick of differentiating under the integral sign, but to do it you have to have the crazy idea to introduce the parameter as F(a) = integral of sin(x)/x e^(-ax) dx. Taking F'(a) removes the x in the denominator and you're left with something you can do with integration by parts. Then notice that F(inf) = 0 and from that you can deduce the answer, F(0).
exp(-sx) makes me think Laplace Transform, so maybe not too bonkers -- not that I remember the uni class about why one would ever use a Laplace Transform
i like the idea of replacing pi creatures with exaggerated anime girl expressions
You missed the opportunity to say: "let that *sink* in".
"I literally squared EVERY SINGLE TERM, and got the SAME DAMN ANSWER."
I love this XD
I actually was thinking of leaving, but the anime memes are good and then the sinc function blew my mind. Good job on predicting all my reactions.
Truly is a lovely function :)
Leaving the questions of convergence to the nerds, u can see why the sums equal what they do by using Euler’s formula to break the sine function into a difference of exponentials and then using the power series of the logarithm to see that the solution is just the logarithm of some number divided by i.
The pi comes from the fact that log(-1)=pi*i so dividing that by i just gives us regular old pi.
I’m not sure how strict that proof is but that’s the way I’ve always evaluated it.
For the series its quite easy. However, idk how they're called in English since I studied calc in french so names will be in french.
It's quite easy, there's Abel's theorem that states: SUM(Un.Vn) will converge if:
1) Vn is positive and limVn=0
2) there exists M so that SUM(Un)≤M.
We have our sum : SUM(sin(n)/n) which is basically SUM(Un.Vn) where Un=sin(n) and Vn=1/n.
1/n -> 0 and it's positive so 1) is satisfied.
SUM(sin(n))
Holy shit, this is legit. Super cool insight :3
"sin(1) has nothing to do with π or e"
[e^i - e^(-i)]/[2e^(iπ/2)] :
"Am I a joke to you?"
7:50 Ah yes, summer of math *explosion*, my favourite maths contest
normal protagonist : has the power of God and anime on his side
gigachad protagonist : has the power of math and anime on his side, defeats God
I absolutely love how the video tries to be serious but at the same time funny, and oh boy does it do a good job with it
Great video
Sounds like the sink function is really good at washing off all the random noise from your signals
But actually,it is good at slicing off the frequency range that you don't want to have from your signals.
This is such a fantastic video!
An interesting and humble way to approach sinc(x) without calculus using only elementary college algebra AND another useful point based off what you mentioned about the series sum of sinc(n) converging to PI, is that if you perform the series of normalized sinc(n) for all values n, it equals the constant 1 polynomial. If unnormalized, it equals PI.
The reason is that it sums to 1 can be shown that Lagrange interpolating polynomial basis of degree n always sums to the constant 1 polynomial for every degree, which is proven with The Fundamental Theorem of Algebra.
It can then be shown that the Lagrange basis polynomials of infinite degree and of equally spaced points converge to shifts of the normalized sinc function. This means the limit of interpolating polynomials of n for equally spaced points converges to the normalized sinc function. This means that the series of normalized sinc converges to 1 at integer points, and PI, for unnormalized sinc. (IE it is Lagrangian)
The Lagrange polynomial of infinite degree for equally spaced points is the normalized sinc function. This fact also means it forms the upper limit of interpolating polynomials (IE: It interpolates data most accurately)
In the real world this beautiful fact is the motivation behind sinc interpolation (The Shannon-Whittaker formula) in engineering and computer science, and is a motivating reason why engineers use sinc as a sample function: At 0, it is defined to be 1, (since it has a limit at 0 the singularity can be ignored and it is assumed 1) but it is zero for all other integer points, which makes it a useful discrete delta function for the integers and it has the sifting property under multiply-accumulate allowing it to sift through discrete collections of points.
As a non-mathematician I can't say it's honestly the most beautiful function in math, since the majority of beautiful functions in math haven't been discovered by humans yet, only a small amount, but it's still really incredible, especially in information theory and signal processing. Engineers are very familiar with this function as its a tool they use every day in their lives.
This is definitely one of the funniest math videos I've seen in a while lol. And I can see you're a man of culture too
This perfectly integrates the merits of a de-stressing shitpost and an educational documentary
bro these videos are literally fire :)
math + weeb = happiness.
This is my summer break
Why am i watching a maths lecture
And why am i enjoying it
2:35 If you expand sinc(x) into its Taylor series: with sin(x) = x - 1/3! x^3 + 1/5! x^5 - ...
There is no 0/0 for x=0 for sinc(x) = sin(x)/x
Finally, everyone says it’s undefined when sinc(0) should be 1.
@@pepitogrillo3368 It's still undefined for x=0 if you simply plug in x=0. Just like x/x, it's undefined for x=0, although x/x = 1 as rational function.
Sinc: I am the most beautiful function!
- pathetic
Sinc: who said that?!
y=x: pathetic. I'm the most beautiful and useful function. If you give me 1, I give you... 1. If you give me your credit card number - I give you a FUCKING CREDIT CARD NUMBER. TRY THAT! You can't even calculate someone's height or weight! If someone give you credit card number you just answer with some random bullshit!
Raised Cosine: Are those bad words here because you're LINEAR!? You're that pathetic muggle indeed! One who don't know about magic should not consider magic as crap! That's why you get few gifts on Christmas,but Sinc gets a lot!
I didn't believe a single bit when I started this video. Now I do.
I love you for making this video. The sinc function came up in my work & once I found out it had its own name, I tried to find out more about it without much success - thank you for making this!
I had managed to discover the bit about it being the FT of the Box function, but didn't know about all those wonderful formulae!
Wow, this is the most beautiful video I could possibly have randomly stumbled across at 4am, and I love it.
However, after liking, subscribing, and sharing it with a couple of frens, I can only say that after finding out you don’t have more such videos, my disappointment is immeasurable and my day is ruined. Thank you.
I wish I had teachers/professors like you, mate!
Keep it up!!
Just to add to how amazing this function is, the mentioned Fourier transform is proof of the Heisenberg principle
what in the f?
@@nanostrafes It is true - the Heisenberg uncertainty principle for position and momentum is based on the relationship between the position domain and the momentum (frequency) domain. Read the "Kennard Inequalities" here: en.wikipedia.org/wiki/Uncertainty_principle#Wave_mechanics_interpretation.
However, I'm not sure why sinc is important in this case. If any function is to be remarkable in this context, it is the gaussian: It makes the inequality in the Heisenberg principle become an equality.
I did not understand anything, my brain hurts, and my back, but I learned one thing: there is a difference between something not making sense and understanding something doesn't make any sense. Which doesn't make sense. I'll go back to bed now.
very good function, it just looks so nice and im always happy to see it whenever youre calculating the intensity of light wenn it hits one or multiple slits
Convolution with the sinc function gives you a sharp, steep falloff in the frequency domain. It is considered an "ideal" filter due to it's absolutely steep roloff - a brickwall filter, a filter that leaves no frequency escape above it's cutoff frequency.
However, in practice, convolution with a sinc filter means either applying a windowing function to the sinc filter, so that it ends in zero on both extremes, case in which the filter is no longer yielding a perfectly steep roloff at it's cutoff fq *or* you don't apply any windowing and end up convolution with a signal that doesn't start / end in 0, case in which you'll get the steep roloff, but have ripples (or ringing) artifacts in the frequency domain, also know as the Gibbs phenomenon.
I wrote "ideal" in quotes because the filter it's only ideal if the convolution is done with a sinc filter of an infinite length, obviously not possible when working with discreet signals as one would do in DSP.
But there's a nice formula that can compute what is the resulting roloff of the filter for a certain filter length, so that you get to design the filter as accurate as detecting a variation of 0.001 mV in a signal. Really useful when ridiculously high roloff is required (like an insane 180dB / octave - usually 12 to 24 dB per octave is considered "normal" for audio processing).
For those interested in finding more about the sinc filter, look for "Digital Signal Processing: A Practical Guide for Engineers and Scientists" by Steven W. Smith, chapter 16 - "Windowed-Sinc Filters".
Thanks, willing to learn!
Fun note, if you take the binomial coeficients m!/k!(m-k)! and turn it into a function at k, such that the nth x gives you the coeficient of the nth therm, namely f(x)=m!/x!(m-k)!
And set m = 0, (as if it was a degree 0 polynomial) the function behaves oddly similar to a sinc(kx), and if you decide to check this out, namely, if you try to evaluate what f(x)=1/x!(-x)! would look like, you would end up ( after using the gamma function so that f(x)= 1/Γ(x+1)Γ(1-x) ) exactly into sinc(πx), and yes, you can try to work out the other cases where n is different to zero, i'll be kinda messy but you'll see that it will be basically sinc(πx) times some inverse functions.
I came for the anime + math in the thumbnail and stayed for the quality content, this is great, keep it up. I just read your page (from your bio), damn your talent at writing is admirable, you make very complicated topics interesting and you are able to relate abstract ideas to the everyday life of the general public, while showing why they are important, not only that but as seen in this video you can also make them funny. Your ability to understand "high" concepts and also transmit all of your passion is simply amazing. And you are also good at music! I'm speachless.
I think this might be the first math video I’ve watched that made me actually laugh rather than just smile and blow air out of my nose
Everybody gangsta until they learn that the Fourier transform of the sinc function gives you a rect function
Seriously, this actually blew my mind
sin(2) actually can be defined with e and a bit of Pi. So there is something to do with it. Just use euler identity for the job and you're done. A few I's e's and pi
Not just sin(2). Any trigonometric function can be defined with e, at least when not fully simplified. For example, the sin(x) function is equal to (e^ix-e^-ix)/2i.
@@grifogrifoo true
Math and anime girls is a wonderful combo, well done sir.
I clicked for the love of anime girl, and I left with the love of the sinc function.
Tes trop cringe freee
It looks like a damping wave, so ill use it for generating water ripples. Ive been using sin(x) the whole time and decreasing its amplitude by the time passed. Thx
This should win for simply all the elegant memes
This is the first #SoME2 video I've watched and I already had a blast. Well done!
How did you combine anime and math so perfectly? Amazing video!
Also, yes, this function absolutely blew my mind
That’s what ruined the video.
@@ophello Disagree
@@ophelloON GOD BRO
Huh, my classmate learned to draw all of this mess in geogebra, and it looked like real
Finally math explained with anime girls
Summer of Math EXPLOSION!
Megumin approves
This mans is having too much fun with this video. What a bunch of weird fascinating behavior of sin(x)/x
So that's why sinco de mayo is a holiday
3:04 It really isn't all that bonkers. The sum of sincs of negative integers is equal to the sum for the positive integers, and sinc(0) is defined to be 1.
And swapping out an infinite sum with an integral and getting a similar answer isn't all that surprising, especially considering sinc(x) pretty quickly tends toward 0.
But getting the exact same answer is surprising. I wouldn't expect it for anything that isn't a step function.
@@Anonymous-df8it Same. I wouldn't downplay this cool behavior at all.
@@Anonymous-df8it Step function, I'm stuck
came for laala manaka on the thumbnail, stayed for the math
Agred I am in love with this function and nothing makes sense other than this feeling.
This functions shows up in both amplitude modulation and digital sampling, and mathematically shows that they are the same. Noted that in two different engineering classes back in school, back in the 1980s.
This is the BEST video of the all internet !!!
I love it !
Thank you for this video !
„Go back to procrastinating on my phd thesis” as if the video was something else 💀
You're really good at keeping viewers' attention! This was really engaging and eye-opening, one of the best videos I've seen, coming from a content creator myself. Liked and subscribed 💛
Glad I didn't skip the video during the first few mins
This is the first time I've actually burst out laughing watching a math related video. There's so many people that try and fail miserably, but you my friend, are something else.
Although, I hope this doesn't pressure you into making funny math videos (if you do make more videos in the future). I'll be just as happy to watch math videos that are cold hard math with no jokes.
Edit: are you of Indian/Pakistani origin? Your accent sounds like a mix of American and Indian English.
This is definitely the funniest SOME out there
Summer of Math Explosion, I love it. The crowning jewel on a hilarious summarization of some incredible mathematics.
Sum of sinc function gave most bizarre result ln(1-e^i)/2*i+ln(1-e^-i)/2*i=(pi-1)/2
I actually thought for just a moment that sin(1)+sin(2)=½π
LOL, nagatoro in thumbnail brought me into this video.
Wow, my friend sent me this yesterday, and I thought it would have way more views. This vid was funny af man, thanks so much :))
This was hilarious and got me hooked really well, but it feels like there could've been _more_
Where did this come from, and can you give a real-world example in signal processing?
Wonderful question
Easy! Probably you know Nyquist sampling theorem saying you can perfectly reconstruct a signal from it's discrete points (samples) if frequency is limited. But how do you do that? You replace each sample with sinc function multiplied by the sample value, then sum everything together, and you'll get the perfect reconstruction.
a real world example would be ideal interpolation kernels or ideal low pass filters
"Sorry Dorothy, but this is cold hard math, and it's more real than Kansas will ever be"
01:20 I'm no expert in physics but man I wheezed
that was a fun ride i'd love it if you made more such videos on your channel
Yo. Diagnosed as an adult with ADHD, apparently my parents and teachers framed my misbehavior as a character fault which led to a lifetime of believing I'm stupid, bad at learning, and not meant for anything logical. They are not to be blamed for their ignorance of my condition, but the work left for me to do is immense and at times, overwhelming. What is a young 29 year old with no bachelor's to do to find motivation to not just make it through but dominate and smash school records?
The answer that serves me to this purpose is to find the people who love the topic in they are sharing. If my hard work to learn the topic doesn't produce an appreciation of it, then I must be learning from a teacher who is failing the transfer of the topic. Treating it like a common, irritating task to just be done with.
You sir, in this short video, have helped me appreciate a new mathematical phenomena with the use of multiple logical and topical perspectives and a nice garnish of memes. I barely had to do any work to listen, it was a very enjoyable experience and you deserve to know that your persuit to make and share this video has successfully transfered a sense of awe of this topic to another person.
Even crazier is that integral of windowed sinc function is also pi.
i had a bizarre experience while watching it
instant subscribe
Their is no difference in discrete and integral solutions. And as you've added the negative terms so, we can expect pi. It's not a big deal... Sine is already related to circles, which in turn relates to pi. In engineering Sinc is great function with a lot of applications, and the reason I think is because of that pi, a constant that we can found everywhere in universe - as circles and triangles are everywhere.
The video is funny though. 😁
I thought I was the funniest math guy, but you do a great job
"It's all π?"
"Always has been"
this video just made me love maths even more
Time to reset the day without using trigonometry
Fantastic video about my favorite function, nice.
"more real than Kansas will ever be" :D
The anime gifs definitely helped me to calm down from this video
Very good content, I never knew I needed math videos that don't take itself that seriously until now
my brain just melted
Maths and anime did collide on that fateful day
Sinc is pretty cool indeed
Time for your channel to blow up 😂
I'm too stoned for this.
bro the function is a cutie pi
5:05 e^πi : "am I a joke to you"
I love complex math videos with anime girls as a background XD
But to be honest it works so good dam well
Amazing interactive math
"Maybe this is all a dream, and you will wake up back in Kansas. But I am sorry Dorothy, this is cold hard maths and it's more real then Kansas will ever be"
badass
RIP Kansas
I seem to be hearing poor residents screaming and guns firing hard in Kansas... Though I'm sitting thousands of kilometers away from it...
...and in a different dimension.
"It's like if I added my height and my weight and got my credit card number" Holy shit how big are you
10 times more satisfying than e^i*pi business
Madlad just woke up one day and decided to combine Math and anime, and it fucking works. Blew my mind. At this point I wouldn't be surprised if some guy decided to do an anime rocket science video tomorrow
Actually, sin(1) = (e^i-e^(-i))/(2i), so it has something to do with e.
Why you gotta do me like this ;-;?
I think he means that x has something to do with a certain number if x is in Q adjoined e
"explosion contest" lol
0:46 Trig identities! sin(1) + sin(2)
= 2 * sin(3/2) * cos(1/2)
You sure did explode that math
One of the best math videos I've seen lmao