<s>[INST] Interesting and unexpected applications of $\pi$ $\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$
Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
and the generalization of $\zeta (2k)$, my perception of $\pi$ has changed. I used to think of it as rather obscure and purely geometric (applying to circles and such), but it seems that is not the case since it pops up in things like this which have no known geometric connection as far as I know. What are some other cases of $\pi$ popping up in unexpected places, and is there an underlying geometric explanation for its appearance?
In other words, what are some examples of $\pi$ popping up in places we wouldn't expect?
 [/INST] $\pi$ as a matrix eigenvalue
For a given positive integer $m$, let us form an $(m-1)\times(m-1)$ dimensional matrix $M$.  The terms on the diagonal of $M$ will be $-2m^2$, the terms on the off diagonals of $M$ will be $m^2$ and all other terms will be zero.  Thus,
$$M=m^2\left(
\begin{array}{cccccc}
 -2 & 1 & 0 & \cdots  & \cdots  & 0 \\
 1 & -2 & 1 & \cdots  & \cdots  & 0 \\
 0 & 1 & -2 & 1 & \cdots  & 0 \\
 \vdots  & \cdots  & \vdots  & \vdots  & \cdots  & \vdots  \\
 0 & \cdots  & \cdots  & 1 & -2 & 1 \\
 0 & \cdots  & \cdots  & 0 & 1 & -2 \\
\end{array}
\right).
$$
For $m=10$, the matrix is
$$M = 100
\left(
\begin{array}{ccccccccc}
 -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 \\
\end{array}
\right).
$$
If we numerically estimate the eigenvalues of this $M$ (for $m=10$), we get
$$
-390.211, -361.803, -317.557, -261.803, -200., -138.197, -82.4429,
-38.1966, -9.7887.
$$
Of note, they are all negative.  If we take the square root of the absolute value of the largest eigenvalue (the last in the above list) we get $3.12869$.
If we perform this exact same process for $m=1000$, we find that the square root of the absolute value of largest eigenvalue is $3.14159$.
It can be proved that the limit as $m\rightarrow\infty$ of this process yields exactly $\pi$.
</s>