# Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice looking visualizations. However, the underlying mathematics should not be too trivial, otherwise it will not seem challenging to the students. I already looked into Chaos-Math and Dimensions-Math, which provided me with useful material, and I was wondering if anyone knows of similar things? I’m sure there have to be things in differential geometry, or topology, that are equally inviting!

Thanks!

I like $e^{i\pi}=-1$ for making people stop and go “What? Really?”

Besides the simple explanation “It’s just $\cos(\theta) + i \sin(\theta)$” you can watch whichever definition of the exponential function you start with converge to the unit circle.

Definition 1: $\exp(z)=\sum_{i=0}^\infty \frac{z^i}{i!}$ Definition 2: $\exp(z)=\lim_{n\rightarrow \infty} (1 + \frac{z}{n})^n$ 