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

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

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

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!

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!

Other mathematicians: QED 3Blue1Brown: Badaboom badabing

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

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!

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!

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

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

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

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.

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

this channel's quality is unmatched

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.

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.

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

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!

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.

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.