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!

Answer

I like eiπ=1 for making people stop and go “What? Really?”

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

Definition 1: exp(z)=i=0zii!

sum definition

Definition 2: exp(z)=lim

lim definition

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