Why is pi here? And why is it squared? A geometric answer to the Basel problem

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A most beautiful proof of the Basel problem, using light.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/basel-thanks
This video was sponsored by Brilliant: brilliant.org/3b1b
Brilliant's principles list that I referenced:
brilliant.org/principles/
Get early access and more through Patreon:
/ 3blue1brown
The content here was based on a paper by Johan Wästlund
www.math.chalmers.se/~wastlund...
Check out Mathologer's video on the many cousins of the Pythagorean theorem:
• Visualising Pythagoras...
On the topic of Mathologer, he also has a nice video about the Basel problem:
• Euler's real identity ...
A simple Geogebra to play around with the Inverse Pythagorean Theorem argument shown here.
ggbm.at/yPExUf7b
Some of you may be concerned about the final step here where we said the circle approaches a line. What about all the lighthouses on the far end? Well, a more careful calculation will show that the contributions from those lights become more negligible. In fact, the contributions from almost all lights become negligible. For the ambitious among you, see this paper for full details.
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
Music by Vincent Rubinetti:
vincerubinetti.bandcamp.com/a...
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with RU-vid, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
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Опубликовано:

1 мар 2018

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Комментарии : 4,4 тыс.

@battleclan 4 года назад

"In honor of Basel" or rather "We had to find something other to name it than 'Euler'"

@Goonercry 4 года назад

Too many Euler mathematical things 😂

@undeniablySomeGuy 4 года назад

@@Goonercry Euler's little theorem ( ͡° ͜ʖ ͡°)

@JackBarlowStudios 4 года назад

There’s a joke that mathematical discoveries are named after the second person who discovered them, because the first is always Euler.

@onradioactivewaves 3 года назад

@@JackBarlowStudios I thought that was more of a humerous fact than a joke.

@antanis 3 года назад

@@onradioactivewaves it's mostly a joke, but euler is incredibly influential nonetheless

@henryg.8762 5 лет назад

Other mathematicians: QED 3Blue1Brown: Badaboom badabing

@ViratKohli-jj3wj 5 лет назад

In India, it is *HENCE PROVED*

@shambhavisingh2981 4 года назад

@Tech Made Easy "quod erat demonstrandum" latin for ("that which was to be demonstrated")

@kingscross4233 4 года назад

@@ViratKohli-jj3wj why did you fail in semifinals?

@vinayvardhanyt2415 4 года назад

@@kingscross4233 😂😂😂

@ulti-mantis 4 года назад

I wonder if he got that from Beakman's World

@s_feles_2642 Год назад

I love the proof, but what I also find surprising is how the first four digits of π^2/6 are 1.644, like the year 1644 when the problem was first posed!

@ANTI_UTTP_FOR_REAL 6 месяцев назад

What???? Yo thats a sick coincidence Edit: I started a whole conversation just because of a mistake lol

@ducksinarow4958 5 месяцев назад

@@ANTI_UTTP_FOR_REALsick*

@mahdihasan6222 4 месяца назад

or is it?@@ANTI_UTTP_FOR_REAL

@tfg601 4 месяца назад

@@ANTI_UTTP_FOR_REALI love a suck coincidence!

@mrblakeboy1420 4 месяца назад

@@ANTI_UTTP_FOR_REAL i’ll suck something else

@user-qy3lz9wy4z 2 года назад

As a high school math teacher teaching calculus, this channel has provided wonderful intuitions about how to teach calculus to students in a wonderful way. The essence of calculus will be delivered to students in an interesting way thanks to all people who helped to make this video!

@sgs138 Месяц назад

I’m imagining you wheeling in a cart piled with 80 camping lanterns and placing them all over the ground while rambling through the proof and all your students just thinking their teacher is insane. 😄

@andrewthomas695 28 дней назад

What I find troubling is that how do you square a number that can't be precisely defined?

@SherinFunmes 4 года назад

*Pi is like an uninvited guest who shows up at every party where he isn't supposed to be*

@surfer855 4 года назад

Even Mister Bean hates Pi, for showing up at every party he goes to with his teddy bear!

@nathanwhitten8950 4 года назад

Maybe a surprise guest--always welcome, especially when not expected!

@tahsintarif6864 4 года назад

"e" also

@AdelaeR 4 года назад

Except ... he actually IS supposed to be there, he was simply uninvited.

@jamesperrin8619 4 года назад

pi is the party host.

@number-kv8px 3 года назад

As a Math major,I've read a great amount of solutions to this problem, but this physicly solution amazed me most.

@natanielmarquis6159 3 года назад

I'm wondering if this convergence can be easily formulated. I mean, the argument is really intuitive but saying "we let de radius tends to infinity" isn't quite enough. You must control the amount of light from the far away lighthouses in a precise way. Not so intuitive problem of summable families where the number of terms vary... However, I totally understand that the rigourous summable families aren't fit for such a video :p

@fangzhang9376 3 года назад

@@natanielmarquis6159 My idea is to ignore all lighthouses with a distance of more than R^0.9 away from the observer. The amount of light from each scales only as O(R^(-1.8)), while the number of them scales only as O(R). As R increases, the arc we are considering becomes longer and longer but also flatter and flatter.

@andreamarino95 3 года назад

@Nathaniel: you can go like this: Define a_i(n) to be the light emitted by the i-th bulb on the n-th circle, and if i is too big and there is no "i-th bulb" on the n-th circle, just set a_i(n) =0. You have that lim_n a_i(n) = (1/i)^2 , and by the above geometric argument you know that, for each n, sum_(i=-\infty) ^(+infty) a_i(n) = pi^2/4. The question is then if you can commute the limit and the infinite sum. The idea is to use Tannery's Theorem to conclude you can. Note that a_i(n) is decreasing in n: the i-th bulb has a fixed distance along the n-th circle from zero, but the circles keep flattening, thus the euclidean distance from zero increases. Also, note that when the i-th bulb appears is in the topside of the circle, so it has at least a distance of a radius from zero. At each step both the number of bulbs and the radius double, so that the radius of the circle when the i-th bulb appears is proportional to i. We conclude that |a_i(n) |

@integrando1847 2 года назад

the solution of euler is the most simple but very difficult to understand why?

@selfull5798 2 года назад

@number25 I was wondering what you think about my thoughts about Pi and Phi: Phi depends on .5, Division = Diameter Diameter = .5 = 1 diameter = 1 degree Radius is division of diameter (division of division) Phi is diameter x3 divided by/2, 3 halves, one and a half, 1.5 Pi is diameter x2 divided by /3, 2 third, 0.75, one and half of a half Phi x3/2 Pi ×2/3 Pi 1/2, 2/3, 3/4, 4/5, 5/6... Phi 2/1, 3/2, 4/3, 5/4, 6/5... Pi is multiplication of radius Phi is multiplication of diameter Basic principle of dividing/equalizing/sharing something in equal parts (itself) and multiplying something in equal parts (by itself) Calculating with circles, squares, triangles, bars, etc... it's very interesting if you think of Greek alphabet for example (Pi and Phi) and the first person to ever have to write "numbers" to explain mathematics and wrote it as such 1 2 3 4 5 6 7 8 9 0... and wonder why it was written precisely like that and if it is a "how to" calculate anything using geometry. It's easier to see if you put a set square of 360 degrees on a picture of old TVs test pattern for example. 360degrees being 1 circle You can calculate anything that way with degrees. You can calculate "nothing" precisely in the process, too as the outside of the circle. The TV and computer invention uses the same pattern and algorithm 4:4:4 of 1234 infinitely. Using both properly is an algorithm to multiply divisions or divide multiplications infinitely. The Greek alphabet letters are ways to calculate that way for specific functions, as well. But Pi and Phi are functions. Trying to give it a value would be like trying to give a value to +,×,÷,=,/, etc...

@sebastianbg5369 3 года назад

The first time you watch a 3b1b video you are puzzled by the new perspective it gives to the most common math problems. Then you incorporate that perspective into the way you solve problems (believing that you already understand everything). Then you watch the video again and new doors open, it's amazing how much ability you have to share knowledge!

@supimon9146 3 месяца назад

the part from 13:54 to the end of video really did stretch my grey matter. Here it is for slow guys like me 13:54 the fact that the lighthouses (factors) are aligned on a straight line on either side of the observer (origin) and are squared(so all negative factors are now positive), results in π²/4 = 2 (1/1² + 1/3² + 1/5² + ...) so 1/1² + 1/3² + 1/5² + ... = π²/8 15:27 the thing we want to find out is what this series is equal to : 1/1² + 1/2² + 1/3² + ... = ? in order to find that out, we need to figure out how much share each of these parts 1/1² + 1/3² + 1/5² + ... (lets call this O - for odds) and 1/2² + 1/4² + 1/6² + ... (E - for even) have in 1/1² + 1/2² + 1/3² + 1/4² + 1/5² + ... (lets call this full term as O+E) maybe 3/4 and 1/4 or 3/5 and 2/5 or whatever combination. we need to find it out. 15:40 is where you pay close attention to what he says: "now you can think of that missing series as a scaled copy of the total series that we want" implying E = some scaled copy of O+E since this is inverse Pythagoras, the denominator part in all the factors for e.g. the 2 in 1/2² or the 3 in 1/3² is nothing but the distance from the observer. if you double all denominators in O+E then you will get E. 1/(1x2)² + 1/(2x2)² + 1/(3x2)² + 1/(4x2)² + ... = 1/2² + 1/4² + 1/6² + ... proving E = some scaled copy of O+E so the earlier 1/2² = 1/4 become 1/(2x2)² = 1/16 . similarly 1/9 becomes 1/36 etc... so doubling the denominators, all factors in O+E become 1/4 of its original. therefore E has a share of 1/4 in O+E and therefore O must have a share of 3/4. or (3/4) of O+E = O But it is already known that O = π²/8 (3/4)(O+E) = π²/8 or O+E = (4/3)(π²/8) or the complete term O+E = π²/6 I must say that your idea of explaining the Basel problem using circles has indeed helped guy like me reason this answer perfectly - big thanks ! I have enjoyed all your videos - you are exceptionally brilliant

@joshuaharper372 15 дней назад

Ah, that helps me understand where the 4/3rds came from. I think the video was just a little too quick or succinct in explaining that bit, but you have filled in the implied information.

@joshuaharper372 15 дней назад

I just rewatched the last few minutes after reading this amplified explanation, and I finally actually understood and followed Grant's narration this time! Yay, finally!

@WilliamFord972 3 года назад

Math concept: [exists] Euler: “My name is involved in this.”

@maxwellsequation4887 3 года назад

Eulaaaaa

@kimba381 3 года назад

@@maxwellsequation4887 Even the Martians know him

@jneal1347 3 года назад

Soon may the Euler man come

@user-wm8xf3yv6i 2 года назад

JNeal134 to bring us sugar and tea and run

@aashsyed1277 2 года назад

@@maxwellsequation4887 HI I WATCH MATH ELITE TOO

@mariaceciliafp 5 лет назад

"I'm so tired of studying, guess I'll just watch some funny videos on youtube" Me 30 seconds later:

@moomin8470 4 года назад

Maria Cecília This is fun

@wisdom6458 4 года назад

Yeah, this is really fun if you know enough to understand :)

@xynyde0 4 года назад

Maria Cecília you’re not tired of studying. You’re just tired of studying the conventional stuff, the conventional way

@CapitalAvenuee 4 года назад

exactly definition jajajajaja

@sppss914 3 года назад

Wisdom not necessary. They just explained everything in such fine details, all that’s needed is just some imagination.

@yds6268 2 года назад

This is amazing! I have a PhD in physics, and I've never seen this proof. It's probably the best intuitive proof for this theorem!

@Taric25 2 месяца назад

Oh, yeah, at 13:42 just expand a circle into a flat line and ignore all the geometry he just showed us to accept a handwaved answer...

@octs609 2 месяца назад

its a limit.@@Taric25

@WhoCares-ue5hk 11 дней назад

@@Taric25 Okay late response, but it's not exactly just expanding the circle. Notice also how the distance between each light house is preserved in expanding the circle. As the size of the circle is doubled, the number of light houses is also doubled, preserving the distance between each light house. Form a circle x^2 + (y-r)^2 = r^2. Intuitively, as you let the circle grow to infinity, more and more of the lighthouses basically touch the x-axis (or the real number line, in this case). Also notice how, in the limit, while the majority of the lighthouses will never actually be on the axis, any values that are elevated off the x-axis would already approach a distance infinitely far away. This would mean that the value of their light is already extremely low based on their distance from the origin alone (not counting vertical distance), as the value of light from that light house would be 1/(infinity)^2, which very quickly approaches zero. This means that the values on the real number line, in the limit, can again be seen as equal to the sum of all lighthouses. Hence, in the limit, it is safe to say that infinitely expanding the circle and keeping its bottom grounded to the origin would preserve the limit.

@Taric25 8 дней назад

@@WhoCares-ue5hk, extremely low does not mean zero, especially when considering an infinite sum.

@WhoCares-ue5hk 5 дней назад

@@Taric25 You're not wrong, but what I'm trying to say is that, as you increase the radius to infinity, more and more of the total length touches the axis, so we can say that as we increase the radius to an obscene amount, we can say that the x-axis is at least a good approximation, and becomes a better and better approximation as the radius increases. As you let the circle grow larger and larger, the part of the circle tangent to the line x = 0 becomes the only part of the circle that we can even "see," and so it becomes basically equivalent to x = 0 at every point.

@SludgeFuZZ 3 года назад

That was absolutely beautiful. I must admit that I would not have questioned why pi is squared, but I can honestly say that I really enjoyed the answer.

@funkycude57 6 лет назад

I've got a final exam to take in 10 hours and here i am watching 3B1B , best channel on RU-vid IMO

@pgbalagopalwarrier2298 6 лет назад

Ansh Shah All the best bro

@srishtikdutta8946 6 лет назад

Ansh Shah same bro 😂😂

@pgbalagopalwarrier2298 6 лет назад

Boards??

@smithpereira459 6 лет назад

same

@smithpereira459 6 лет назад

math paper tomorrow lol

@megablademe4930 5 лет назад

I am still in high school but love watching these videos,even tough I didn’t understand 95% of what he was saying.

@apollonmegara8220 5 лет назад

Im only a toddler and love watching this kind of videos

@Rishabh_Joshi_ 5 лет назад

I too am in 12 . Even though I can't understand

@corpeduhsmathsciteacher8402 5 лет назад

This is magnificent... your brains are building new neuronal connections as you watch and attempt to understand... And as you become ACTIVE in using new knowledge, (take notes and as you reproduce the ideas in your own words) you are building new neuron networks. Congratulations, you have just tapped into the process of becomming more intellegent!

@ddm1912 5 лет назад

@@ViratKohli-jj3wj abbe hindi nahi samjhega yaha pe kisiko 😂

@krishivkothari5971 5 лет назад

@Timmie Collins Same

@mushtaqrasool9169 Год назад

Professor gave us an insight of not only Mathematics but also Physics! Just shows how good of a teacher you are. Thanks for all of this.

@user-zm9wt8yh2l 8 месяцев назад

Excuse me, can we exchange math together, my friend?

@gt6989b 3 года назад

Unbelievably good :) I remember asking this same question in college, when I first saw this sum in a Fourier series class, and getting answers based on complex analysis :) This is so beautiful, thank you very much for posting this and providing fantastic insight.

@TwoForFlinchin1 6 лет назад

this channel's quality is unmatched

@user-ss3ts5vm3q 5 лет назад

why these subjects are so interesting only when i'm preparing midterm exam

@sirhasslich536 4 года назад

Procrastination

@facitenonvictimarum174 4 года назад

what term is at its mid point in May? just curious.

@facitenonvictimarum174 4 года назад

@Tech Made Easy Thank you.

@jacquelineliu2641 4 года назад

Tech Made Easy No, because a) the Chinese Spring term goes from Feb to Jun b) the OP's name is Korean

@godson200 4 года назад

Exactly man... Here i am 1 year later

@EvilDudeLOL Год назад

This is incredible. So intuitive that, as a 14 year old kid with not very wide knowledge of calculus, I could understand it all. Splendid explanation- such characteristics are very rare. Thanks a lot, 3b1b, for this absolute masterpiece.

@pinkserenade 10 месяцев назад

Dude, your future is bright! Keep going, keep getting curious

@martinelosudietz6795 3 года назад

The explanation you made at minute 2:00, is absolutely beautiful and huuuuugely intuitive... You don't get infinite bright at the origin by adding up more lights... I absolutely loved it.

@Owen_loves_Butters Год назад

I wouldn't say hugely intuitive, since in 2D, the brightness does indeed go to infinity.

@shahars3134 6 лет назад

Wow! This proof is so beautiful and not that complex. I was worried the channel will go down hill when I heard more people were going to join. But now I have no doubt in my mind that it's going to be GREAT! Good job Ben for the awesome video!

@3blue1brown 6 лет назад

There is no doubt in my mind that the new additions will make the channel better.

@overpowered5919 6 лет назад

+3Blue1brown It already is! Awesome video as always. Can you make a video about the honeycomb conjecture?

@overpowered5919 6 лет назад

arxiv.org/abs/math/9906042 You can download the proof and I think the way you guys depict concepts is incredible so please consider it

@intuited9754 6 лет назад

The only thing that would make this channel worse is if the current fans start gatekeeping. Very happy to see you explicitly subverting that!

@jayasri6764 6 лет назад

michael einhorn sadly,I believe sum of any other higher powers is impossible for a human to compute,since the extension would need higher dimensions than 3,which we are unable to properly imagine,on our own

@sophmcamp 6 лет назад

15:11 "The number line is kind of like a limit of ever-growing circles" MY MIND IS BLOWN

@gauravsingh3007 4 года назад

My mind exploded so hard that my round skull became straight

@hpsmash77 3 года назад

@Federal Bureau of Investigation - FBI ...

@doctorbones711 Месяц назад

Believe it or not, this perspective becomes very concrete in what is called projective geometry - and it is just as useful there as it was in this proof! (Sorry for dropping in 6 years after your comment)

@coltith7356 Год назад

I think this is the fourth proof I see of this, and this is certainly my top or second favourite.The other proofs I know involve Fourier series, the residue theorem for infinite sums or a Lebesgue integral. The first two weren't that easy to understand when I was studying them because I hadn't quite yet understood everything that we were using to prove this, and the Lebesgue integral was actually quite cool because even though the function used came out of nowhere, the theorems used were very explicit on what they do and then the basic integral we get didn't require much more understanding. But I learnt these 3 proofs in Uni, and they would have seemed like total garbage if I had seen them before, whereas this one is actually understandable for most people out there who are willing to listen carefully and pause the video to think about it from time to time. This is what makes this channel so great and useful. It offers new persepectives and gives everyone intuitive and clear explanations, that only require a little of motivation from the viewers. Most videos are almost self sufficient, you don't need to watch an entire series to understand the video that caught your attention, they give you a better understanding of where everything comes from but the explanations are clear enough that you can do without those additional previous videos. Truly amazing.

@ronald3836 3 месяца назад

I'd say the most intuitive one is the one by Euler. I discovered it myself as a student learning about the Taylor series of sin(x) and cos(x). From olympiad problem solving I knew that quantities such as the sum of the inverse roots of the zeros of a polynomial could be expressed in terms of the coefficients of the polynomial. Then I wondered what if I do this with sin(x)/x as an "infinite" polynomial. Lo and behold, out comes sum 1/n^2 = pi^2/6! I was aware that I could not formally justify these manipulations, but then I found out to my surprise that this was how Euler had "solved" it. If it's good enough for Euler... 🙂

@mattlambermon6583 2 года назад

I needed this alooonnngggggg time ago... Love it, maths like this just has some kind of purity to it, I can never describe the feeling when your mind come to that moment of realisation and clarity, wish I had kept up with mathmatic skills and practice, and not have to start right back at the beginning lol

@cesiupro123 4 года назад

15:11 "the number line is kind of like a limit of ever growing circles" - i've been thinking of a number line like this since forever, i thought i was insane, but it makes sense now

@ibrahimmahmoud8592 4 года назад

DUDE SAME this video blew my mind with that statement

@isaakvandaalen3899 4 года назад

It gets worse. You can think of the complex plane as the surface of an infinitely large sphere. Lines on the surface of an infinitely large sphere wouldn't just approach being parallel, they would become parallel. Now if you plot the graph Y=1/X where X approaches 0, you might think it shows that as X nears 0, Y approaches infinity. Now I know a lot of people don't like it when you say that N/0=Infinity, but screw them. I do what I want. The real interesting thing here is if you take this exact same equation, Y=1/X, but after you plot it you run it through a Circle Inversion, you get to see what happens as the Y axis approaches infinity. Now I haven't actually done this, but it seems to me that the line Y=1/X would map right through the point of ''Infinity" and come out no worse for wear on the other side. Although this would be hard to see, as the plotted line would be hugging the Y axis pretty tight as it approaches that point. So to me this seems to pretty definitively answer the question that N/0=Infinity. Some people make the argument that this can't be true, because -N/0= minus Infinity. To which I say they are the same thing. The number line's an infinite circle, travel an infinite distance and you loop right back round again. Some people think that this can't be true, because if 1/0= Infinity, and 2/0= infinity, then does 1=2? To that I say no, 2/0 = 2*(1/0) = 2*Infinity, which is twice as big as the infinity we got before. It's different. This might not seem to make sense, but if you've ever heard the solution to the problem, How do you free up infinite rooms in an infinite hotel where every room is occupied? The answer is to move every guest to an odd numbered room, leaving an infinite number of rooms now unoccupied. Some infinities are bigger than others, this is not a contradiction.

@josephzeltsan1350 4 года назад

This is not how you compare infinities. 1/0 is not infinity, because division by 0 is not allowed, and infinity is not a number. 2*infinity is not a "bigger" infinity. "Infinity" is in no way shape or form a real or complex number. What you can do is write that the limit of 1/x as x approaches "is infinity", which really means that as x approaches 0 1/x grows without bound. Analysis is good, treating infinity like a real number is unacceptable.

@andrewzhang8512 4 года назад

@@isaakvandaalen3899 Your fourth paragraph contradicts your statement.

@aculasabacca 4 года назад

@@isaakvandaalen3899 I've always felt like the infinity of space is a circle that comes around to a singularity. Not sure why I think this but psychedelic drugs may have played a part. :)

@MartinMatten 4 года назад

Well, this approach to the Basel problem is amazing! It combines physics, geometry, and maths in the same run! The inverse Pythagoras theorem is something new to me. Will check this out further. Thanks so much for this discovery on pi day!

@geniusfollower Год назад

What a beautiful video. Kudos to all the animators and of course to you for explaining the beautiful proof. The best kind of math is the kind of math that makes you tear up when you discover the truth. And this one did.

@seannee3896 3 года назад

Fantastic stuff! I am relearning applied mathematics from this excellent approach. This is much more in keeping with the way Archimededes and Newton thought about mathematical thinking in science. Feynman would love all this, I think.

@matanlevi5873 6 лет назад

You say that wherever pi is showing up there's a circle hiding, one circle which I would die to find is the one hiding in the fact that the integral of e^-x^2 from minus infinity to plus infinity is the square root of pi. One of your best videos in my opinion by the way.

@monsterstein 5 лет назад

integrate over two dimensions and take the square root at the end. since you integrated the square of what you should have...

@ilprediletto 5 лет назад

If you integrate exp(-x^2-y^2) in the Real plane, you can evaluate the integral via substitution polar coordinates, and dx dy=rho drho dtheta, then you integrate with theta from 0 to 2 pi, because 2 pi is the length of the circle with unit radius. Then here's to you pi!

@JoaoBapt 5 лет назад

The usual proof is to think of this integral as the square root of the double integral exp(-x²-y²) over the plane. To evaluate this integral, switch to polar coordinates - here's your circle!

@glaucophane 6 лет назад

the fact you can take a summation to infinity and turn it into a circle is absolutely stunning.

@pgbalagopalwarrier2298 6 лет назад

drkscpe No its not

@glaucophane 6 лет назад

P G Balagopal Warrier I don't mean the way we all learned it... dividing circles up yeah I get that but what he's done with that is astonishing.

@pgbalagopalwarrier2298 6 лет назад

Sorry dude i was randomly spreading negatitivity on random comments. I dont even get half of what this guy is preachin

@MA-bm9jz 6 лет назад

sorry you are not summing to infinity, if you want a shorter proof check fourier series

@nerdy5999 6 лет назад

It absolutely is. If you think about it, is a circle just an summation of infinity small points? If you took a zero dimensional point, could that infinite summation of points make any mathematical dimension? These are the questions we all should be asking.

@whogashaga666 6 месяцев назад

Such an explicit explanation and high quality video! Can't believe I missed this video for five years.

@vaibhavgupta627 Год назад

The way you explained this is just awesome. This will remain in my brain forever.

@marwanaljohary7615 4 года назад

That was the most exciting math lesson I've ever been to. Thank you for making math so fun.

@Copperbotte 6 лет назад

You can also use Gauss's law to approach the same solution, rather than a geometric approach. Gauss's law works since a radially symmetric field that's magnitude weakens via the inverse square law has its radius term fall out in a surface integral. This means no matter where the lighthouses are within a sphere of radius R, they can be represented by a single lighthouse of combined magnitude in its center. This also means that same combined lighthouse can be represented by equally spaced, equally lit lighthouses along its boundary. By using this law within a cylinder, and holding the "lighthouse surface density" to be 1/2, you find the surface integral to equal to π^2, and a quarter of the cylinder is π^2/4, the same result as using the geometric method. (The circle is quartered to eliminate lighthouses on the negative side of the number line, and double counting when the number line curves upwards to form the circle)

@3blue1brown 6 лет назад

Hmm, this seems super clever, but I'm not quite sure I follow the connection between the continuous "lighthouse surface density" and the discretized case.

@Copperbotte 6 лет назад

This surface integral works by taking the sum of the areas as the areas approach zero. However, if you hold this distance to be constant, the radius must increase to give the same result. Similar to zooming in to see individual differentials, which at that scale, would be discrete. Since I'm using a cylinder, and putting the lighthouses only on the circular boundary, the circular endcaps can be ignored.

@TheAgamemnon911 6 лет назад

That is indeed very clever. But I don't think it is as elegant, because you need more advanced theorems for the proof.

@twilightknight123 6 лет назад

@Copperbotte, I'm not quite sure if I'm understanding what you're saying correctly, but I don't think Gauss' law works the way you think it does. If you have a random assortment of charges in a Gaussian surface, you can calculate the flux through the surface by assuming a lump of charge at the center, but this does NOT tell you anything about the field produced. I'm also not following your math or your explanations.

@anywallsocket 6 лет назад

Yeah no. Gauss's Law will tell you E and V for electrostatics. But there's no way you're gonna get a pi^2 term out of a cylindrical integral unless you make the length pi or something specific like that.

@ericbell1137 2 года назад

I' ve recently gained a passion for mathematics at the age of 27. Now that there's no pressure its lovely. Polynomials make me smile and I'm excited to be on this journey.

@austinscott4695 Год назад

How is it going?

@hyprk5590 2 года назад

I didn’t know this approach about series even though I learned it when I was a college student. Thank you for your explantion. I always enjoy watching your video.

@kushkumar7389 4 года назад

I have seen its proof by Fourier series but the way your team animated and gave physical proof is simply awesome... great work, cheers.

@elliottsampson1454 4 года назад

0:40 challenge posed in 1644 first 4 digits of awnser 1.644 coincidence I think not!

@tankizoltan1752 4 года назад

Just wow.

@DivyanshMMMUT 4 года назад

Nice observation man

@Macion-sm2ui 4 года назад

Next digit is 9 (for 90 years when problem was unsolved) and 34 (for 1734, year before Euler solve this problem). It can't be coincidence

@Potato2017 4 года назад

1.644*9* so actually yeah

@RoselineJerryA 4 года назад

nice

@kabirsethi2608 2 года назад

Marvelous proof probably the most interesting one I have seen of the Basel problem. Used the euler product formula to prove this beautiful result but this geometric proof is brilliant. Congratulations 👏.

@TheMarkEH 2 года назад

Fabulous. I wish that I had been taught mathematics by someone with these kind of insights. Thank you.

@rs-tarxvfz 4 года назад

I want to nominate 3Blue1Brown the noble peace prize for year 2020. Thanks.

@user-lj8mr6fk6s 4 года назад

because of his wife having cheated on him it can not be))))

@neeleshbansal3299 4 года назад

but for mathematics FIELDS MEDAL

@FiXioNxd 4 года назад

Абдаллах Муслим wow im ruski look Im making cringy jokes using bad English))))))) so funny right?))))

@Phobos_Anomaly 4 года назад

@@FiXioNxd Firstly, you don't know that that person is a Russian, secondly, what does that have to do with bad jokes?

@FiXioNxd 4 года назад

Phobos Anomaly I think his intention was to make a joke, second, I guessed by his name hes Russian.

@TranSylvainie 6 лет назад

Best math channel ever. Clever, original, beautiful, soothing / motivating voice... Just perfect. I've been following it since the very beginning. Every new release feels likes christmas. Please keep it on !

@tedalper1464 2 года назад

this is a beautiful, clever approach! [though it sweeps under the rug a few questions of *convergence* -- when you say the larger and larger circles converge to the sum on the integers there is a potential pitfall: for any finite, very large circle, MOST of the circumference is *not* near the x-axis and there are many many lighthouses on that part of the circle. But this is easily fixable, you can show that the total sum contributed by them becomes negligible as they are all far from the origin and their individual contribution to the sum may be bounded by a quantity that shrinks proportionally to the square of the radius, but the quantity of them grows only linearly with the radius. This is a much slicker and accessible proof than the one I've previously shared with high school students that requires some polynomials involving trig functions and vieta's formulas.

@KIMARO1423 2 года назад

평소에도 수학을 좋아하는데 이렇게 재미있게 설명해 주시고 한글 자막까지 달아주셔서 감사해요.

@user-cd4bx6uq1y 2 года назад

Yes

@apratimghosh109 6 лет назад

There are already so many mathematical results named after "Euler", that if they had called this "Euler's Problem" or something, it would start getting confusing...

@alephnull4044 5 лет назад

Proof: en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler QED

@SpaceyCortex 5 лет назад

This would be a problem yes.

@coningham7195 5 лет назад

@@SpaceyCortex Would it be Euler's problem?

@alexwang982 4 года назад

The other Euler formula The other other Euler formula The other other other Euler formula

@user-vj2sh1kc1x 4 года назад

Stop Eulering

@balajisriram6363 6 лет назад

whats stunning here is not just the geometry behind the problem, but the effort and intellect of the 3Blue1brown Team

@JayJay-qs8nd 3 года назад

At 16:40 you could state that since 1/4 of the TOTAL apparent brightness (B) is contributed by the sum of the inverse square of the even integers, then 3/4 of B comes from the sum of the inverse square of the odd integers, which has just been shown to = pi^2/8. Thus 3/4 of B = 3/4 x pi^2/8. So B = pi^2/6.

@marvin.marciano Год назад

Thank you. Really helped to clear things out

@fchenhku Месяц назад

Excellent explanation! The video is brilliant but the explanation around this part is puzzling to me. How I convinced myself of the final answer was via this: pi^2 / 8 * (1+1/4 + 1/4^2 + 1/4^3 +...)=pi^2 / 8 * 1/(1- 1/4) = pi^2 / 6.

@sadmansakib3902 2 года назад

It's very beautiful. I saw that for mathematical proof only the inverse pythagorian theorem suffice. But, using lighthouses made the video more beautiful. Thank you for the video.

@MKWKezer 6 лет назад

That was simply fantastic. You really show what it's like to love math. Combining light and geometry to reveal the circle in 1 + 1/4 + 1/9 + ... = pi²/6? How great is that? You sir are amazing!

@johanwastlund8422 6 лет назад

This is wonderful! As I said in my paper, it's based on proofs by Yaglom & Yaglom, Hofbauer, and others, and I added some of my own ideas. I thought of the light sources as stars revolving around a common center of gravity, but light-houses are arguably easier to move around! :) I hope the "light-house proof" now becomes folklore, and I'm happy to have contributed to that!

@thetherorist9244 5 лет назад

again just another distraction from the truth about pi and the information contained within its code and sequences...I find it strange that after the last decade and 9000 pages of text i have written on pi i haven't had ONE single person interested in it.....lets play a game folks...lets see who knows anything about pi that isn't common knowledge.....

@anushreesabnis5856 5 лет назад

@Johan Wästlund you rock!!

@amineaboutalib 5 лет назад

can you share ur paper? :)

@thetherorist9244 5 лет назад

sure...let me just give you all my work@@amineaboutalib

@breadandbutter644 5 лет назад

@@thetherorist9244 well i cant find it

@aderinsolajoshua1186 Год назад

I keep watching this video over and over again, so rich in information and good animation

@ShivamShukla-nw6pu 3 года назад

So elegantly explained. Just amazing. Thanks a lot!

@Selicre 6 лет назад

This was amazing! I love how geometry and algebra, while being based on completely different axioms, can represent the same concepts, and they way you switch between them is astounding.

@LaTortuePGM 6 лет назад

it's fascinating and frustrating at the same time to see how super-abstract concepts can be linked to some weird geometrical ones, like honestly wtf ?!

@n_x1891 6 лет назад

The universe welcomes you; enjoy your stay.

@1996Pinocchio 6 лет назад

La Tortue PGM What do you think where these concepts are coming from?

@callofdutymuhammad 6 лет назад

La Tortue PGM This problem isn't very abstract(for a physics student) it all depends on your understanding and where you're coming from.

@LaTortuePGM 6 лет назад

tbh i prefer abstract stuff, so i kinda struggle when it comes to more concrete, geometrical structures. still in high school though lol.

@protocol6 6 лет назад

Roverse You can check out anytime you like, but you can never leave.

@crweewrc1388 2 года назад

I wish more math teachers follow in your footsteps. The fact that you actually inspire your students by trying to show off the true beauty of mathematics is far more helpful and amazing than just letting students do problems. Most of my classmates, although smart, only sought to do math in order to past entrance exams for university. But they failed to see the hidden beautiful world this subject offers. I've been researching and analysing different aspects of math all my life. And I really love what I've seen. I wish I had friends who loves math as much as me.

@Meic909 7 месяцев назад

I was lucky enough to have a great mathematics and geometry teacher. Many questions in algebra are most fruitfully investigated when they are given a geometric interpretation.

@IExistSometimes 6 лет назад

The beauty of these animations is beyond anything else on youtube

@IExistSometimes 6 лет назад

Also it clearly shows how to summon math satan

@marksvendensen3830 6 лет назад

This is the best math video I've ever seen! You and Mathologer have inspired me on a consistent basis for a while now, but this video is my favorite so far.

@FESchon 2 года назад

What a beautiful video. Thank you so much for sharing these extraordinary insights and your deep love of mathematics in the form of these precious videos, they are truly artistic masterpieces.

@schizoframia4874 2 года назад

Why are you red.

@jan_Majeken 12 дней назад

lmao

@methodzero 8 дней назад

He's brown

@practicemodebutton7559 7 дней назад

how racism started:

@schizoframia4874 7 дней назад

@@practicemodebutton7559 😂

@muqq1q12ruiohammadrayyanmu6 5 дней назад

Why are you stupid @schizofraia4847

@youssefwassef2231 5 лет назад

This is beautiful man!! I wish the whole world can see and appreciate how amazing your explanations and representations in your videos are. You're showing the true beauty of maths

@vari1535 4 года назад

Even with a straight line as "a circle", I still can't draw a perfect circle.

@rewrose2838 3 года назад

😂

@b.clarenc9517 3 года назад

Just pretend that you drew an accurate representation of the projection of a sphere in a higher-dimension curved manifold.

@trickytreyperfected1482 3 года назад

@@b.clarenc9517 I'm going to pretend I know what that means.

@Tactix_se 2 года назад

I mean if you think about it, it doesn’t really make sense because it would have to mean -infinity and +infinity lead into each other at the top of the circle

@ugursoydan8187 2 года назад

OHH ONE OF THE BEST VIDEOS ON RU-vid!!!! THANK YOU SO MUCH EXCELLENT VISUALIZED!!!!

@mjorozco3786 Год назад

I was shocked, even one member of your Patreon animated this AND made the geogebra! thank him!

@CrannBethadh 5 лет назад

I've been wondering how this equation related to Geometry for more than 20 years since I first saw it in college. THANK YOU!

@VinayAggarwal 3 года назад

This reminds me of Earth. It's spherical but still feels pretty flat, even though it's size is finite.

@alvarojimenez1252 3 года назад

Amazing! The first thing I thought of when I saw the animations was that it kind of looked like De Moivre's formula with lightbulbs and now I wonder whether there is a hidden proof in the video. Tremendous work!

@user-ot3vb6xs4u Год назад

stunning animation, so amazing,straightforward to explain it. astonishing beautiful

@eduardkuegler3655 3 года назад

Thank`s a lot for this new inspiration. Your technique might bring new insights, whether we have to improve the invese square laws for gravitation and electromagnetic forces. You go from odd numbers and extrapolate to all. But with odd numbers we have the problem, that after the prime fermat number 65537=2^16+1 , mathematician´s have not yet found a new one- and only the prime ones are constructable. I think 3*5*17*257*65537=2^32-1 is the last possible odd number you can use in physics.

@harshit.k071 6 лет назад

I am currently pursuing UG in Mech Engg and the video gives the answers to most of my expansion related problems I've been facing since my high school days. Wow!!!!! it's beautiful.

@andreamarino95 3 года назад

I have been doing math for a few years, and I still think that this is one of the clever stuff I have seen in a while. Feels great.

@gabrielpardo1 3 месяца назад

You can get emotional with the result of such care, passion and commitment. It is a work of art.

@robperkins6023 2 года назад

I've lived in Basel. Grant, you make transcendent videos about math and you say the word "Basel" very differently than I learned. It kept my attention. :-D

@surbhibhanot8635 4 года назад

Sir, your concepts are so crystal clear...please don't stop making these types of videos.

@vatsaltrivedi2755 6 лет назад

Watching your video for the first time and feeling myself so unfortunate that I didn't watch it until now..... Awesome work... I'm gonna recommend it to all my friends...

@TheDigiWorld 7 месяцев назад

I wasn't necessarily able to understand the proof but still i appreciate how a crazy-looking, complicated infinite product can be explained using not just brute-force math but a combination of math and intuition

@aenygma 6 лет назад

wow. I have no idea why you do these exegeses, but i'm eternally grateful for it. such eloquent and elegant explanations tap into a deep sense of beauty. Thanks Grant

@ofermagen895 6 лет назад

An amazing proof. Perhaps the best proof I've seen in a while. I really like that even a high school student could follow along with it

@jeremydyar7566 5 лет назад

It makes you think that your understanding is finely tuned in until you pick up a pencil and paper and try some on your own

@jeremydyar7566 5 лет назад

@Ayush Dugar I'm literally catching up on my infinite series and sequences as we speak. I'll try this after I'm done

@fluent_styles6720 5 лет назад

Ofer Magen that's me!

@BoB-Dobbs_leaning-left 5 лет назад

" I really like that even a high school student could follow along with it" If only we had a President that could do that.

@deanmarshall1011 5 лет назад

Steve Barnes hahaha so funny. Get a life.

@KaliFissure 3 года назад

This also beautifully shows how an apparent point charge can be distributed over a spherical surface harmonically. The incident angle always compensating for the distance. If all our physics models of point like particles instead distributed the charge over measured actual spherical size the fuzziness is the uncertainty principle. The closer you get in proximity the less of a sense you have of size. At the surface the charge takes up half the screen so although you can accurately measure the charge let’s say, the light from the original lighthouse, You can now only say that the charge lies somewhere in THAT direction, 50% of your view. Or it’s morning and I’m confused. :)

@peterfarhat5767 3 года назад

If you don’t fully understand these videos Never give up.....Soon you will realize the intuition this channel give you

@epicmorphism2240 3 года назад

Give up guys! GIVE UP

@Emperor_Shao_Kahn 3 года назад

i know this and im only 5 its so easy

@why-ue1tu 3 года назад

@@Emperor_Shao_Kahn joined 11 years ago hmm...

@tombratcher6938 Год назад

I'm betting on old age tbh

@vidyanandwagh6277 5 лет назад

At the beginning you said, "You've never had the experience of your heart rate increasing in excitement, while you were imagining an infinitely large lake with lighthouses around it. Well, if you feel anything like I do about math, That is gonna change by the end of this video." And that does. I literally had goosebumps at the end!

@elardenbergsousa3836 6 лет назад

This is by far the most mindblowing demonstration of whatever you call math or related to math. Thank you so much for this.

@probablyapproximatelyok8146 3 года назад

I just realized and worked out that if you use pretty much the same argument, but the starting lighthouse is θ of the way around the circle instead of π of the way around the circle, you can show that the sum of (x-πn)^-2 over all integers n is equal to csc^2(x). This essentially provides a geometric way to show that the sum of over all integers n of (x-πn)^-1 = cot(x) if you can justify interchanging the sum and the integral. This cotangent identity can be very useful when trying to find other sums as well. Great video.

@mrzold 2 года назад

ok i'm a little past 4 and a half minutes in and i am very excited about learning how the inverse square law, which i was taught in astronomy class in college, makes TOTAL SENSE now, decades later. Jeez why didn't they explain it that way?! Thanks!

@YatriTrivedi 6 лет назад

Your videos are in my top 5 most-looked-forward-to. I'm really happy you were able to increase your team! I think it's about time I join your patreon campaign.

@bibop224 6 лет назад

what else do you have in your top 5 then ? Just curious

@YatriTrivedi 6 лет назад

bibop224 top 4: Welch labs, mathologer, standupmaths, and Mark brown. Here's a few extra: artifexian, nativlang, alliterative(the endless knot), PBS spacetime, PBS infinity series, scishow, nerdwriter1, every frame a painting, holy fucking science, I like to make stuff, and Chris salomone

@bibop224 6 лет назад

cool, thanks for sharing !

@TheTobyrobot 6 лет назад

Yatri Trivedi wintergatan is great aswell. Not so much science, but more engineering and music!

@YatriTrivedi 6 лет назад

bibop224 anytime! Definitely check them out!

@ArthurSeijiNishikawa 6 лет назад

I've been sporadically watching your videos, always impressed by the quality. But this one blew my mind. You've gained a new subscriber!

@ifroad33 4 месяца назад

The animation is so amazingly well done I am just stunned

@anirbanmookherjee938 9 месяцев назад

I got lost at the multiplication by 1/4 step, guess I have to brush up my algebra... but as always new ways to see things to common series and more visually demostrated to grasp why rather than its like that.... Truely as one person said below A great piece of art

@aj76257 6 лет назад

Two 3blue1brown videos in one week? It’s a dream come true!

@HHH21 3 года назад

Till now I was just learning principles, theroms, formulas given in my book without proof I was really got angry to learn without proof but after seeing your videos I got ideas that how the these are derived and how it actually works. Now I am feeling better now.

@moumnyxt3337 11 месяцев назад

Excellent presentation of a very beautiful proof. A minor complaint: the proof is incomplete, as, in the final step, it is assumed that only the points (lighthouses) close to the x-axis contribute. At every step of the doubling-the-circle process, there are points, eg, directly above the observer, far in the y direction - these never disappear. To complete the proof, it must be shown that, in the large radius limit, the contribution of those points tends to zero - it shouldn't be difficult but, as is, the proof is incomplete.

@yfshen3785 Год назад

what a masterpiece,the best video about math I have ever seen,I have to be shocked by the calculation pfrocess though I had already leanrned this point

@Someone-cr8cj 6 лет назад

"Alright, buckle up"

@SomeoneCommenting 6 лет назад

Good thing he didn't say "Buckle up, Buckaroos!"

@harrystuart7455 6 лет назад

Absolutely incredible, your videos never fail to blow my mind, keep up the good work!

@luijonas675 2 года назад

I love this channel so much, it makes math visible!

@petersinger5459 8 дней назад

Thanks for this. While there are several series that give a value for pi the Basel series converges realky quite quickly. I was learning to program in python and wondered how easy it would be to calculate the first 50 or so digits of pi. using a fairly simple program and the Basel sequence you can churn out the first 50 or 100 digits quite quickly and easily - a lot of fun !

@johnchessant3012 6 лет назад

Badaboom Badabing!!

@ldavilla8971 6 лет назад

Badaboom badabing

@Hyuts 5 лет назад

16:55

@guptahaha 3 года назад

Badaboom Badabing

@ilyrican 6 лет назад

I just had an increasing heart rate excitement when I clicked on the new video notification!!

@markborz7000 3 дня назад

As a physicist I love this solution. Very intuitive! Mathematics and Physics complememt each other in a wonderful way. The greatest mathematicians were also excellent physicist: Newton, Riemann, Poincare, Hilbert, Weyl etc....