Beautiful Theorems and what constitutes as beautiful I often hear the phrase "mathematical beauty". That a proof or formula or theorem is beautiful. and I do agree I was awestruck when I first saw Euler's formula, connecting 3 seemingly unrelated branches of mathematics in a single formula $e^{i\pi}=-1$
But beauty is a rather subjective term. When I was taught Linear Algebra the instructor introduced Cayly-Hamilton theorem as beautiful, and I thought it was "nothing special".
I'm interested in theorems that are considered beautiful, and why they are so.
Just as an example to what I think is beautiful, last night a friend told me that the sum of the first $n$ odd numbers is equal to $n^2$. for example if $n=3$ then $1+3+5 =9=3^2$. if $n=5$ then $1+3+5+7+9 = 25 =5^2$ Simplistic. Surprising. Elegant. I liked it a lot.
I would be very much interested in learning more theorems / formulas like that.

If $M=M^2$ is a smooth compact $2$-dimensional Riemannian manifold with (smooth) boundary $\partial M$, $K$ denotes it's Gauß-curvature, $k_g$ the geodesic curvature of it's boundary und $\chi(M)$ the Euler-Characteristic, then the theorem of Gauß-Bonnnet states that
$$\int_M K dA + \int_{\partial M}k_g ds = 2\pi \chi(M)$$
(There are generalizations of this to higher dimensions. For me the beauty of this particular theorem originates from the fact that is one of the early insights of mathematicians into the deep relationships between topological invariants and analytical quantities)
