For Merkw, Urchun, nemo and other Mathmos....

What are some fascinating mathematical observations, theorems, conjectures and other things you've come across?

I've not gone too in-depth in too many sub-fileds but I find a lot of math facts fascinating.

Here is a short list:

1) The fact that there are many mathematical statements that are implicitly "undecidable."

2) The fact that group theory is so incredibly versatile at capturing symmetries in the universe.

3) The intimate connection between Gaussians and sinusoids.

4) The central limit theorem, linear regression and other reasons the Gaussian is so "natural"

I figure if one of us lists something another finds fascinating, we can take turns explaining (attempting to) them to each other, and or exploring the idea in depth when we find it fascinating but don't really understand it yet (most on my list are in the second category).

What do you say?