Intuition behind $\zeta(2) = \frac{\pi^2}{6}$ [closed]

In literature, there are many proofs of the well-known result $$\zeta(2) = \frac{\pi^2}{6}.$$

However, as far as I know, they do not offer an intuitive explanation of why this result should be true.

So my question is:

What is the key intuition -- that is, the picture -- behind the result? Are there any visual of anyway intuitive proofs of the statement?

closed as off-topic by Will Jagy, Yemon Choi, Henry Cohn, Lucia, Felipe VolochOct 22 '14 at 0:53

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• "This question does not appear to be about research level mathematics within the scope defined in the help center." – Will Jagy, Henry Cohn, Felipe Voloch
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• Euler's first proof is very intuitive to me: find a "polynomial" whose roots are the non-zero integers (so your "polynomial" is sin/x) then use the relation between the sum of the inverse of the square of the roots and the first coefficients. Voilà. – Olivier Oct 21 '14 at 23:18
• @Olivier I see what you mean, but honestly, from my limited experience, that is not what I call intuitive. – user60665 Oct 21 '14 at 23:57
• Not everything in maths can be justified by a reason that feels "intuitive", and "intuition" is highly subjective – Yemon Choi Oct 22 '14 at 0:08
• @Dal, you don't need intuition, you need Euler. – Włodzimierz Holsztyński Oct 22 '14 at 0:24
• John von Neumann said something like this: young man, you don't understand (things) in mathematics, you get used to (them). – Włodzimierz Holsztyński Oct 22 '14 at 1:45

I like Euler's proof because it shows how it's related to the taylor series of $\sin(x)$ and so the mystery of $\pi$ and 6 becomes less mysterious.
On the other hand someone with lots of experience in Fourier analysis will see it as the Fourier transform of something and again the appearance of $\pi$, particular $\pi^2$ becomes apparent from Parseval's identity.
How Geometry implies $\sum \frac{1}{k^2} = \pi^2/6$