The Beauty of Euler’s Formula
The Euler’s formula1 is usually considered as the most beautiful equation in mathematics, mostly expressed as:
\[\mathrm{e}^{\pi i}+1=0\notag\]The visual simplicity is one reason why it is viewed as the most beautiful. What’s more important is that it makes people who ever learned mathematics, more or less, feel amazing, because it perfectly combines together two most commonly used irrational numbers $\pi$ and $e$, imaginary unit $i$, the most fundamental integer “1”, and “0” which means nothing.
So, when today I happened to see that the Euler’s formula is written in this form:
\[\mathrm{e}^{\pi i}=-1\notag\]I felt it sort of odd, and lack aesthetic compared to the above form. Interesting. For the same mathematical notion, different degrees of regularity can be felt from different perspectives. Finding a good viewpoint at times helps to prove and understand mathematics.
That being said, there’s no a unified standard for beauty. Ramanujan’s formula to calculate $\pi$2:
\[\dfrac1\pi=\dfrac{2\sqrt2}{99^2}\sum_{k=0}^\infty\dfrac{(4k)!}{k!^4}\dfrac{26390k+1103}{396^{4k}}\notag\]is another kind, which, to my mind, is a beauty conveyed by carefully arranged complexity.
References