If you create a rectangle whose sides are in the ratio , it has a very interesting property. If you cut it in half across the narrow dimension, the two pieces also have their sides in the ratio . This can of course be continued indefinitely.

In Europe and much of the rest of the world, paper sizes are standardized using this mathematical fact. Each size can be obtained by cutting the next bigger size in half. Because the sizes all have the same ratio of their sides (called the aspect ratio) A copying machine can magnify or shrink a page to another size, and it will fit exactly.