The Mathematical Beauty of Paper Size

I invite you to take a rectangular piece of paper that is not A4 shaped. You can always just tear a bit off your A4 paper and then neaten it up to a rectangle. With your non-A4 rectangle, try folding it in half along the shortest line of symmetry. You will observe, in a spectacular anticlimax, that you now have a piece of paper half the size, and a different shape. Possibly, you started with a ‘squarey’ rectangle and now you have a ‘long-thin-rectangle’, or vice versa.

Now do it with an A4 sheet. You probably already know what happens. You get an A5 piece of paper. It is half the size (of course it is, you just folded in half). What’s more, it is the same shape. Technically a similar shape, of course, but the sides are in the same ratio. This is something of a shock, if you ponder it, because rectangles do not normally behave like this.

And yet there are still people preferring "letter" size paper. Madness.