Is the connection between $e$ and $\pi$ "arbitrary" or "natural"? 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.)
 A: 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.
A: It's just a natural consequence of the connection between complex exponentials and rotations. That is, $e^{i\theta}$ is a rotation through angle $\theta$ in radians. Since $-1$ is a half-turn and the angle of a half-turn is half the circumference divided by the radius, we have
$$
-1 =e^{iC/(2r)}  = e^{iC/d} ,
$$
where $d = 2r$ is the diameter. $\pi$ is defined as the ratio of a circle's circumference to its diameter, so $C/d = \pi$, and thus
$$
e^{i\pi} = -1
$$
and the identity follows.
A: This is going to be a very soft answer, and I will refer to the bible which will intrigue some and infuriate others, so take what you will.
Consider $ e ^ {i\theta} $. The formula suggests a circle since as $\theta$ varies it traces out a unit circle in a complex plane.
Also, if we set $\theta=\lambda t$ then $e^{i\lambda t}$ is reminiscent of a circular orbit for as t progresses its value orbits the origin according to a frequency of $\lambda \over {2\pi}$.
So in shape this connects to the circle of the earth, and to orbits at a solar or cosmic scale, and also if we imagine $e$ to relate to the electron, a naïve model of an atom, such as the Bohr model, which though is physically implausible because it is based on a 2D circle, it nevertheless gives useful answers.
This connection to an electron and the atom gives us an immediate connection to light. And in general there is a connection to any kind of wave such as sound or light.
So just by considering the basis for Euler's formula, we have a connection to important elements of creation. Euler's formula itself connects to a halfway position in an orbit, that is $e^{i\pi} = -1$ from which we more beautifully write $e^{i\pi}+1=0$, which uses 7 of the most fundamental symbols in mathematics, relating the transcendental numbers $\pi$ and $e$.
Now consider Gelfond's constant, $e^\pi = 23.140692632779269005729\dots $
Here there seems to be something weird going on because not only is $ e^\pi \approx 20+\pi$ to 2 decimal places. It agrees with $\pi = \textbf{3.14}\underline{15}\textbf{926}535897932384626\dots$ in five of the first seven digits which is a coincidence that seems specific to base 10.
$ e^{\pi }-\pi = 19.999099979189475767266442984669044496\dots $
Wikipedia explains:

Despite this being nearly the integer 20, no explanation has been given for this fact and it is believed to be a mathematical coincidence.

Also $\pi^e$ is closer to 22, more precisely, $\pi^e =  22.45915771836104547342715220\dots$
22 and 23 are also important number is human genetics.
Now how might one encode these concepts mentioned in a single word?
Numerologically, the word "In" consists of the 9th and 14th letters. 9+14=23 suggestive of Gelfond's number and includes the $i$ that Gelfond's number lacks. ("In" is the first word of the "AV" which is also 23 numerologically.)
But "In" is also suggestive of "ln", which is the natural log function, suggesting or confirming understanding 23 as a power of $e$ and hinting at Gelfond's constant. Even without the Gelfond connection we have , $ln(23) = 3.135494215929\dots \approx \pi$.
Now many people look at the digits of $\pi$ and wonder if they contain hidden meanings. The digits 1614 occur at position 1611 to 1614 (counting from the 3) of $\pi$. 1614 was the year Napier published his theory of logarithms, which are named after λόγος (lógos, “word, reason”) and ἀριθμός (arithmós, “number”)ref.
