Geometric Interpretation of the Basel Problem?

Question

Does the Pi in the solution to the Basel problem have any geometric significance? Every time I see Pi, I have to think of a circle. I would love to see a nice intuitive picture connecting the Basel problem with geometrical figures. Anyone here been lucky enough to stumble upon such a thing?

Answer 1

Here's a reference that might interest you: Elementary Proof of Basel's Problem

His work is also discussed in this lecture: Why Pi?

Answer 2

I do not know if this is the kind of geometric interpretation you are after, but a $\zeta(2)$ proof by Beukers, Kolk, and Calabi has some geometry involved (relating to a triangle). The double integral: $$\int_{0}^{1}\int_{0}^{1}\frac{1}{1-x^2y^2}dydx,$$ evaluates a similar sum: $$\sum_{n=0}^{\infty} \frac{1}{(2n+1)^2}.$$ If you make the change of variables $x=\frac{\sin(u)}{\cos(v)},y=\frac{\sin(v)}{\cos(u)}$ and apply the change of variables formula, you end up having to compute the area of an isosceles triangle with vertices $(0,0),(\frac{\pi}{2},0),(0,\frac{\pi}{2})$. This area is $\frac{\pi^2}{8}$, the exact value of the sum above. From there you can derive $\zeta(2)=\frac{\pi^2}{6}$. However, a generalized version of the Beukers, Kolk, Calabi proof relates the volume of a $k$-dimensional polytope (defined by several inequalities) to the generalized sum: $$\sum_{n=0}^{\infty}\frac{(-1)^{nk}}{(2n+1)^k}, k\in\mathbb{N}.$$

Check out these papers on the computations and details: https://www.maa.org/sites/default/files/pdf/news/Elkies.pdf http://www.staff.science.uu.nl/~kolk0101/Publications/calabi.pdf