Interesting math-facts that are visually attractive

Solution 1:

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: $\exp(z)=\lim_{n\rightarrow \infty} (1 + \frac{z}{n})^n$

Solution 2:

Here is a beautiful video about sphere eversion.

Here is a beautiful video about Möbius transformations.

Here is a gallery of surfaces in differential/algebraic geometry. There are dozens of beautiful images here -- and there is so much to say about all of them. Examples:

Here are pictures of the Weierstrass function and of $f(z) = \text{exp}(1/z)$. The Weierstrass function is continuous but nowhere differentiable. The second function provides a visual example of Picard's Theorem in action. Both of these are pretty mind-blowing, I think.

And lastly, here is a picture of the phenomenon of holonomy, which is a topic I'm considering researching. Notice that the north-pointing vector at $A$ is parallel transported in a loop, yet returns to point $A$ rotated.