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

What people said

Strong rec user357980 · 2018-03 · Mathematics Stack Exchange · on “Is there any geometry behind the Basel problem?”

Funny you should ask this today. A great video video [this video] by the YouTuber 3Blue1Brown was just posted today. (Aside: I recommend all his videos.) The proof is based on the result mentioned by "3 revs" in the MO thread mentioned by user296602 above.

Strong rec Isti115 · 2018-07 · Mathematics Stack Exchange · on “Why does the sum of inverse squares equal $\pi^2/6$?”

Regarding the involvement of $\pi$ I would recommend watching this beautiful explanation from 3blue1brown! [this video]

Strong rec For the love of maths · 2018-10 · Mathematics Stack Exchange · on “Understanding the Maclaurin Series”

There is an excellent YouTuber named 3Blue1Brown [LINK] who you may have seen on advertisements in this very site. His goal is for "explanations to be driven by animations and for difficult problems to be made simple with changes in perspective." Here you can find the video he made on the Taylor Series Expansion [LINK] . I am also learning it right now and it is actually very helpful. Try to check his other videos to intuitively understand the quaternions [LINK] , the Basel problem [this video] , eigenvectors and eigenvalues [LINK] , etc. Note: This is by no means a paid advertisement, I am simply a huge fan and he has helped a lot of people understand and love mathematics, including me.

Strong rec ViktorStein · 2019-07 · Mathematics Stack Exchange · on “Where does $\pi^2$ appear spontaneously within Physical Phenomenon and Mathematics Equations?”

A great example in my opinion is the Basel problem [LINK] $$ \sum_{k = 1}^{\infty} \frac{1}{k^2} = \frac{\pi^2}{6}. $$ A great intuitive and geometric explanation can be found in this video [this video] , which, along with the whole channel, I cannot recommend enough. There's also a whole Stack Exchange post [LINK] dedicated to it.

Strong rec DonielF · 2019-11 · Physics Stack Exchange · on “Where does $\pi$ come from in the Heisenberg equation?”

@G.Smith [this video] - 3Blue1Brown has an amazing video that deals with this problem, and in other videos he deals with other physics and number theory problems where pi randomly shows up. Highly recommend his channel in general.

Strong rec RSpeciel · 2020-08 · Mathematics Stack Exchange · on “Riemann Zeta function with a complex exponent”

@optimistic_mathematician If this is for a school math project, I would highly suggest looking at $\zeta(2)$ instead of this. Despite its apparent simplicity, it is actually a very interesting problem (hence a very well documented problem). See this introduction [LINK] , as well as one of my all time favorite youtube videos here [this video] . Hope this helps.

Strong rec Dan · 2022-12 · Mathematics Stack Exchange · on “Is this just a coincidence? Expectation of product of areas in unit circle equals $\pi^2/6$, the answer to the Basel problem.”

@FShrike I guess you're referring to this [this video] . Great video, but I don't think it's the same as what I'm thinking of here (at least not obviously).

Strong rec Christian E. Ramirez · 2023-07 · Mathematics Stack Exchange · on “How to define $\pi$ from geometry to the limit of a sequence?”

The Basel problem does actually have somewhat of a geometric interpretation. 3Blue1Brown has an excellent video [this video] on the subject, as well as a video [LINK] on a geometric proof of the Wallis product.

Recommends 6infinity8 · 2018-03 · Mathematics Stack Exchange · on “Are there any visual proofs for $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$?”

Grant Sanderson just published a video on his channel 3Blue1Brown [LINK] in which he presents a very accessible way of visualizing the problem and its solution. Link to the video: [this video]

Recommends brainjam · 2018-03 · Mathematics Stack Exchange · on “Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)”

Another video that gives a nice geometric derivation, and shows why Pi (and indeed Pi^2) appears in the sum. [this video]