A4 (and related) proportions

The A4 rectangle is used all round the world, with the notable exception of America. It's a fascinating geometrical construction, with many hidden surprises for origami designers to take advantage of. The basic principle is that the long side of an A rectangle is the same length as the diagonal of the enclosed square. In the first figure on the right, you can see that the diagonal AC has been swung down to form the side AE.

Using Pythagorus' theorem, if AB = BC = 1, the hypotenuse Ac must be root 2, approximately 1.414

Extending the figure on the opposite side, we have the basis for a traditional method of doubling the size of a square. One fascinating aspect of an A rectangle is that if you cut it in half along the longest edge, you get two smaller rectangles that have the same proportions! This means, for example, if you need several small A rectangles for a modular design, you can simply cut A4 sheets into quarters.

There are a number of ways to produce an A proportioned paper with which to fold. Here are some you may find useful.

Clearly, all of these methods leave creases that we probably don't want. The simplest way around this is not to cut the extra strip off, but to fold it over and use as a template. Here I've used the method above from the shorter rectangle, but left the edge folded over. Slide the clean sheet into the gap, then fold the nearest edge over to line up with the lower edge. Finally, cut off the small strip, leaving an A rectangle.

artwork @ text © Nick Robinson 2004