Beautiful Theorems and what constitutes as beautiful

Question

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.

Answer 1

Theorem: There exist positive irrational $a,b$ such that $a^b\in\mathbb{Q}$.

$\square$ Consider $\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=2$. Then either $\sqrt{2}^{\sqrt{2}}\in\mathbb{Q}$ or $\ldots$ $\blacksquare$

Answer 2

This might sound silly, but the Quadratic Formula was the first formula I ever learned to prove, and I still have a soft spot for it. $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ And do you know what I thought was beautiful? I was so excited when I first learned it that I would solve linear equations as follows: $$ax+b=0$$ $$ax^2+bx+0=0$$ $$x=\frac{-b\pm{\sqrt{b^2}}}{2a}=-{b\over a}$$

Answer 3

$$\mathcal G(n)=\int_0^\infty e^{-x^n}dx\qquad=>\qquad n!=\mathcal G\bigg(\dfrac1n\bigg)$$ In particular, the Gaussian integral $$\int_{-\infty}^\infty e^{-x^2}dx=\sqrt\pi$$

Answer 4

$$\int_M d\omega = \int_{\partial M}\omega $$

Answer 5

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)

Answer 6

Often the beauty of a theorem is measured in terms of the brevity of its formulation. If one has a short easily understandable statement it is often considered a beautiful result particularly if the proof is not obvious or considerably longer than the statement of the theorem. The Cayley-Hamilton theorem is "beautiful" in this sense since the formulation is quite brief whereas the proof is not altogether obvious.

Answer 7

Does the Mandelbrot set being a fractal count as a beautiful theorem?