Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time.

Do you know of any other concepts like these?

I think if you look at this animation and think about it long enough, you'll understand:

• Why circles and right-angle triangles and angles are all related.
• Why sine is "opposite over hypotenuse" and so on.
• Why cosine is simply sine but offset by $\frac{\pi}{2}$ radians.

My favorite: tell someone that $$\sum_{n=1}^{\infty}\frac{1}{2^n}=1$$ and they probably won't believe you. However, show them the below:

and suddenly what had been obscure is now obvious.

This visualisation of the Fourier Transform was very enlightening for me:

The author, LucasVB, has a whole gallery of similar visualisations at his Wikipedia gallery and his tumblr blog.

Here is a classic: the sum of the first $n$ positive odd numbers $= n^2$.