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

A most beautiful proof of the Basel problem, using light.
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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
------------------
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[@battleclan]

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

[@Goonercry]

Too many Euler mathematical things 😂

[@undeniablySomeGuy]

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

[@JackBarlowStudios]

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

[@onradioactivewaves]

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

[@antanis]

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

[@SherinFunmes]

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

[@surfer855]

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

[@nathanwhitten8950]

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

[@tahsintarif6864]

"e" also

[@AdelaeR]

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

[@jamesperrin8619]

pi is the party host.

[@s_feles_2642]

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]

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

[@ducksinarow4958]

​@@ANTI_UTTP_FOR_REALsick*

[@mahdihasan6222]

or is it?@@ANTI_UTTP_FOR_REAL

[@tfg601]

​@@ANTI_UTTP_FOR_REALI love a suck coincidence!

[@mrblakeboy1420]

@@ANTI_UTTP_FOR_REAL i’ll suck something else

[@user-qy3lz9wy4z]

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]

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]

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

[@number-kv8px]

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

[@natanielmarquis6159]

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]

@@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]

@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]

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

[@selfull5798]

@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...

[@henryg.8762]

Other mathematicians: QED 3Blue1Brown: Badaboom badabing

[@ViratKohli-jj3wj]

In India, it is *HENCE PROVED*

[@shambhavisingh2981]

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

[@kingscross4233]

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

[@vinayvardhanyt2415]

@@kingscross4233 😂😂😂

[@ulti-mantis]

I wonder if he got that from Beakman's World

[@sebastianbg5369]

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!

[@yds6268]

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]

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]

its a limit.@@Taric25

[@WhoCares-ue5hk]

​@@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]

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

[@WilliamFord972]

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

[@maxwellsequation4887]

Eulaaaaa

[@kimba381]

@@maxwellsequation4887 Even the Martians know him

[@jneal1347]

Soon may the Euler man come

[@user-wm8xf3yv6i]

JNeal134 to bring us sugar and tea and run

[@aashsyed1277]

@@maxwellsequation4887 HI I WATCH MATH ELITE TOO

[@mariaceciliafp]

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

[@moomin8470]

Maria Cecília This is fun

[@wisdom6458]

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

[@xynyde0]

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

[@CapitalAvenuee]

exactly definition jajajajaja

[@sppss914]

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

[@gt6989b]

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.

[@SludgeFuZZ]

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]

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

[@pgbalagopalwarrier2298]

Ansh Shah All the best bro

[@srishtikdutta8946]

Ansh Shah same bro 😂😂

[@pgbalagopalwarrier2298]

Boards??

[@smithpereira459]

same

[@smithpereira459]

math paper tomorrow lol

[@TwoForFlinchin1]

this channel's quality is unmatched

[@martinelosudietz6795]

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]

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

[@mushtaqrasool9169]

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]

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

[@megablademe4930]

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

[@apollonmegara8220]

Im only a toddler and love watching this kind of videos

[@Rishabh_Joshi_]

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

[@corpeduhsmathsciteacher8402]

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]

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

[@krishivkothari5971]

@Timmie Collins Same

[@user-ss3ts5vm3q]

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

[@sirhasslich536]

Procrastination

[@facitenonvictimarum174]

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

[@facitenonvictimarum174]

@Tech Made Easy Thank you.

[@jacquelineliu2641]

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

[@AnushrutaMitra]

Exactly man... Here i am 1 year later

[@coltith7356]

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]

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... 🙂

[@EvilDudeLOL]

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]

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

[@sophmcamp]

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

[@gauravsingh3007]

My mind exploded so hard that my round skull became straight

[@hpsmash77]

@Federal Bureau of Investigation - FBI ...

[@doctorbones711]

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)

[@shahars3134]

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]

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

[@overpowered5919]

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

[@overpowered5919]

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]

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]

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

[@supimon9146]

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]

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]

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!

[@mattlambermon6583]

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

[@MartinMatten]

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!

[@apratimghosh109]

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]

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

[@SpaceyCortex]

This would be a problem yes.

[@coningham7195]

@@SpaceyCortex Would it be Euler's problem?

[@alexwang982]

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

[@user-vj2sh1kc1x]

Stop Eulering

[@vaibhavgupta627]

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

[@KIMARO1423]

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

[@user-cd4bx6uq1y]

Yes

[@LaTortuePGM]

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]

The universe welcomes you; enjoy your stay.

[@1996Pinocchio]

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

[@callofdutymuhammad]

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]

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

[@protocol6]

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

[@matanlevi5873]

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]

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

[@ilprediletto]

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]

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!

[@seannee3896]

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.

[@whogashaga666]

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

[@cesiupro123]

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]

DUDE SAME this video blew my mind with that statement

[@isaakvandaalen3899]

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]

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]

@@isaakvandaalen3899 Your fourth paragraph contradicts your statement.

[@aculasabacca]

@@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. :)

[@glaucophane]

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

[@pgbalagopalwarrier2298]

drkscpe No its not

[@glaucophane]

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]

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

[@MA-bm9jz]

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

[@nerdy5999]

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.

[@hyprk5590]

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.

[@JayJay-qs8nd]

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]

Thank you. Really helped to clear things out

[@fchenhku]

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.

[@elliottsampson1454]

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

[@tankizoltan1752]

Just wow.

[@DivyanshMMMUT]

Nice observation man

[@Macion-sm2ui]

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]

1.644*9* so actually yeah

[@RoselineJerryA]

nice

[@IExistSometimes]

The beauty of these animations is beyond anything else on youtube

[@IExistSometimes]

Also it clearly shows how to summon math satan

[@geniusfollower]

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.

[@kabirsethi2608]

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 👏.

[@MKWKezer]

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!

[@rs-tarxvfz]

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

[@user-lj8mr6fk6s]

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

[@neeleshbansal3299]

but for mathematics FIELDS MEDAL

[@FiXioNxd]

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

[@Phobos_Anomaly]

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

[@FiXioNxd]

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

[@aderinsolajoshua1186]

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

[@ShivamShukla-nw6pu]

So elegantly explained. Just amazing. Thanks a lot!

[@balajisriram6363]

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

[@Copperbotte]

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]

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]

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]

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

[@twilightknight123]

@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]

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.

[@sadmansakib3902]

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.

[@TheMarkEH]

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

[@vari1535]

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

[@rewrose2838]

😂

[@b.clarenc9517]

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

[@trickytreyperfected1482]

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

[@Tactix_se]

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

[@marwanaljohary7615]

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

[@tedalper1464]

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.

[@FESchon]

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.

[@kushkumar7389]

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

[@TranSylvainie]

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 !

[@ugursoydan8187]

OHH ONE OF THE BEST VIDEOS ON KZbin!!!! THANK YOU SO MUCH EXCELLENT VISUALIZED!!!!

[@ericbell1137]

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]

How is it going?

[@youssefwassef2231]

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

[@johanwastlund8422]

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]

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]

@Johan Wästlund you rock!!

[@amineaboutalib]

can you share ur paper? :)

[@thetherorist9244]

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

[@breadandbutter644]

@@thetherorist9244 well i cant find it

[@peterfarhat5767]

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

[@epicmorphism2240]

Give up guys! GIVE UP

[@Emperor_Shao_Kahn]

i know this and im only 5 its so easy

[@why-ue1tu]

@@Emperor_Shao_Kahn joined 11 years ago hmm...

[@tombratcher6938]

I'm betting on old age tbh

[@user-ot3vb6xs4u]

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

[@marksvendensen3830]

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.

[@Contra1828]

1:45 pi creature: OH MY GOD, you put me in front of an INFINITE line of lighthouses, I'm TWO DIMENSIONAL, light falls off as 1/x, the harmonic series diverges, YOU ASSHOLE I'M BLIND NOW

[@pikcube]

This comment is under appreciated

[@satyapalsingh7849]

Really a nice one...need more attention

[@granthamilton1918]

This comment gets the good comment award

[@nickpro8116]

Grant approximated the infinite series only with a finite number of lighthouses, so the pi creature is fine I guess

[@ifroad33]

The animation is so amazingly well done I am just stunned

[@CrannBethadh]

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]

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

[@aj76257]

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

[@gabrielpardo1]

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

[@luijonas675]

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

[@ilyrican]

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

[@dovi_bun]

12:57 that circular right angle kills me to this day

[@novalerdsuwanrut7241]

Tech Made Easy the right angle symbol is drawn with a square, not a circular arc

[@zke.ac6p2y58]

Maybe he just forgot the dot inside... ? ^.^

[@napalmnathan9163]

my first clue into I'm not buying this.

[@ishworshrestha3559]

Hh

[@divasv]

90° is written there, square angle is used but it is not a rule neither it will affect in any way

[@KaliFissure]

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. :)

[@mjorozco3786]

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

[@Selicre]

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.

[@harrystuart7455]

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

[@eduardkuegler3655]

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.

[@robperkins6023]

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

[@ofermagen895]

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]

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]

@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]

Ofer Magen that's me!

[@BoB-Dobbs_leaning-left]

" 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]

Steve Barnes hahaha so funny. Get a life.

[@boeingaviator9503]

2 videos 1 week is this christmas all over again ?

[@ManojP847]

It's Holi, my dear friend! Happy Holi! Happy Mathleting!

[@PrimusProductions]

Is that what it's snowing?

[@plaguedoct0r]

Yes christmas is all over. Christmas is being in december time.

[@BarryLeeahtam]

You are the best and you could not be happy

[@chinisa.innukshopa]

We can't wait another day-please, Solstice, don't be late.... [Apologies to R Bagdasarian]

[@probablyapproximatelyok8146]

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.

[@yfshen3785]

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

[@alvarojimenez1252]

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!

[@harshit.k071]

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.

[@FidgetyGamer]

Engineering student- never seen such a great visual of geometry and loved it!

[@mrzold]

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!

[@MG30001]

Mind blown. Multiple times. And I'm only half way through it.

[@echonnet]

Wish you had it now

[@RazorM97]

we should have expected this, if you look at the earth from any of the points like now, it is straight, if you look from above is huge and round at many points because is a sphere not really but oval like, you can have a sphere if you cut some parts correctly, our eyes narrow it. if you look through a microscope and so on, the width of a single hair wire is like a million atoms or probably more source: kzbin.info/www/bejne/f3eun6RvhrJplaM

[@RazorM97]

just think how we are bending over along with the surface of the earth, seen from space our heads shoulders and so do not surpass the circular size and shape of the earth in fact they are the same and fitting with any part of it

[@kummer45]

The guy got creative using the superposition principle. Using light? He's dealing with "waves". That man studied mathematics, physics and of course, computer science. Expect combined ideas of this sort in all of his videos.

[@andreamarino95]

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.

[@tr911tr]

Amazing work! Thank you so much for sharing!

[@pe1900]

13:51 i swear if the middle of the transition there makes a cardioid from the lighthouses that is so beautiful

[@surbhibhanot8635]

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

[@saikiran2310]

2 videos in a week!

[@zokalyx]

this is making me so happy

[@gregoryfenn1462]

You have 314 likes for this comment. NO ONE MUST SPOIL THIS.

[@milanstevic8424]

if and only if...

[@anirbanmookherjee938]

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

[@Meic909]

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.

[@vatsaltrivedi2755]

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...

[@aenygma]

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

[@dribrahimaldhaify4469]

Thank for all of you great team...

[@carlosalbertoandradesilva9442]

very clever of you getting pi involved in all this mess, it all started with that small circle 2/pi diameter and the exchange of 2 lamps for one!

[@vidyanandwagh6277]

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!

[@midodoku]

SO impressed from physical anaysis of basel problem. Nice Work!

[@petersinger5459]

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 !

[@ilangubernick2753]

This is brilliant. Im really amazed. thanks for sharing.

[@YatriTrivedi]

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]

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

[@YatriTrivedi]

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]

cool, thanks for sharing !

[@TheTobyrobot]

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

[@YatriTrivedi]

bibop224 anytime! Definitely check them out!