In a letter written in 1786, the physicist Georg Christoph Lichtenberg pointed out a small marvel hiding inside a rectangle. A sheet whose sides stand in the ratio of the square root of two to one is the only shape that, when halved across its long side, yields two smaller sheets of exactly the same proportion. Fold it, and the offspring is a perfect scale model of the parent. Cut that in half, and the pattern holds again, on and on down the chain, each rectangle a faithful echo of the last. For more than a century the observation stayed an elegant curiosity.

Then, in 1922, the engineer Walter Porstmann built the German DIN 476 standard around it, and the idea acquired a job. From there it spread outward to become the international standard ISO 216, the paper sizing most of the world now takes for granted. This is the quiet logic behind everyday paper. An A4 sheet folded once is precisely an A5, with nothing wasted and nothing left over. A photocopier can enlarge A4 to A3 and crop away not a single line, because the proportion never changes as the size grows.

The family runs from A0, at 841 by 1189 millimetres, down through A6 at 105 by 148 millimetres, every step a clean halving of the one above. US Letter, by contrast, carries no such trick. Its origins are murky, most often traced back to the working dimensions of hand papermaking moulds, and it holds its place in the world by inertia alone rather than by any inner mathematics. The square root of two asks for nothing but itself, and in return it gives paper a shape that can be divided forever without ever losing its likeness.