Euler's identity $e^{i\pi}+1=0$ has always fascinated me, and at the same time freaks me out a bit. Like, they are two very fundamental constants which seem to have absolutely nothing in common, but still there mysteriously is an immediate mathematical connection between them.
Now I don't understand the math behind it, as such a thing as an imaginary number in an exponent does not make a lot of sense to me. However I'd still like to get a feeling for how "arbitrary" or "natural" the connection between $e$ and $\pi$ might be.
It's hard to find an accurate wording for what I mean, but I'm thinking that for example the definition of how to deal with an imaginary exponent might be rather "forced" and more being just a decision by some human, and less a fundamental property of the universe. So, another phrasing of the question could be something like "Was the connection made by man or God?". (I'm aware this might be a pretty subjective topic, but to me it just seems too interesting to drop it without trying.)
The Taylor series of $e^x,\,\cos x,\,\sin x$ for real $x$ provide a natural identification of $e^{ix}=\cos x+i\sin x$, i.e. the unit complex number of argument $x$. So $e^{\pi i}=-1+0i$ is just the statement that $\pi$ is our name for half the number of radians in a revolution. Well, $\pi$ is also our name for the circumference-diameter ratio, i.e. half the circumference-radius ratio, so the claimed result is trivial.
This relation of $e$ to $\pi$ is very natural: it's really just saying that rotations in the plane are exponentiations in complex numbers. Which makes sense, because complex numbers admit the matrix representation $x+yi=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right)$, making $\cos x+i\sin x$ the $x$-anticlockwise rotation. This gives us $e^{ix}e^{iy}=e^{i(x+y)},\,(e^{ix})^n=e^{inx}$ etc. for free.
You might like to mull why the even weirder result $\int_{-\infty}^\infty e^{-x^2}dx=\sqrt{\pi}$ is also natural instead of arbitrary.