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: 



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> What is the key intuition -- that is, the picture -- behind the result? Are there any *visual* of anyway *intuitive* proofs of the statement?
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Intuition behind $\zeta(2) = \frac{\pi^2}{6}$
The word intuitive by definition means that something "feels correct", and is very myopic. The fact that there are so many different proofs is in itself a gift because you can pick your favorite and intuit all you want. 


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. 


