TITLE: Dividing a square into 5 equal squares
QUESTION [11 upvotes]: Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.

REPLY [15 votes]: Since $1+2i$ has length $\sqrt5$, you can lift a square fundamental domain of $\mathbb C/\mathbb Z[i]$ to $\mathbb C/(1+2i)\mathbb Z[i]$. Overlay a square fundamental domain for the larger torus to get a way to divide a square into 5 smaller squares. 
It's pretty easy to decompose any rectangle into a square geometrically, but the general decomposition is not as nice.