TITLE: Is there a size of rectangle that retains its ratio when it's folded in half?
QUESTION [87 upvotes]: A hypothetical (and maybe practical) question has been nagging at me.
If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches and 3 inches (2:3), which wouldn't retain its ratio. For example, here is a piece of paper that doesn't retain its ratio when folded:

Is retaining the ratio technically possible? If so, what is the side length and ratio that fulfills this requirement? Any help would be appreciated.
Update:
I added "once" because I got an answer saying that any recectangle would work, as any rectangle folded twice has the original ratio. Nice answer, but not quite what I was looking for. As for the other answers, I got 3x as much information as I needed! Thanks!

REPLY [3 votes]: When you fold any rectangular sheet of paper through the middle, clearly the original area is the double of resulting (folded) area.
At the same time, if you have a little rectangle with sides $a$ and $b$ (so area $ab$) (let us say $a>b$), and you want a similar rectangle with area $2ab$, then clearly that larger rectangle must have sides $\sqrt{2}\cdot a$ and $\sqrt{2}\cdot b$, the factor of magnification being $\sqrt{2}$.
Combining these two simple observations, and looking at your illustrations, we see that the long side $a$ of the "half" rectangle is equal to the short side $\sqrt{2}\cdot b$ of the "unfolded" (full) rectangle, so $a = \sqrt{2}\cdot b \Rightarrow \frac{a}{b} = \sqrt{2}$.