A4 Paper

A4 paper is an amazing standardisation of countries * to have a uniquely (and perfectly) propertied ratio.

If you want to have a piece of paper such that when you fold it in half you get something similar (in the maths sense of the word, i.e. something with the same ratio of sides) then we can work out that ratio:

By trying to make the two ratios equal, we get that k =2/k and because we want a positive value we get the unique answer that k = √2. This means that subsequent foldings of a piece of paper will all be in the same ratio.

This is useful, since that blowing up, or reducing an image by a factor of two using a photocopier is an easy process. Leaflets and books can get made easily and folded up newspapers can be stacked.

But this only fixes the ratio of the sides, it doesn't make any constraints on the absolute scale. A0 paper was chosen so that it had an area of 1 meter squared. This means that each subsequent sheet (A1, A2 etc) had half the area of the previous one, forming a nice geometric series of binary fractions.

B paper also exists, but is much rarer. B0 is half way between A0 and A1 in a timesy kind of way (geometric mean), in the same way that 10 is half way between 1 and 100 in a timesy kind of way. Equally each B lies between the A scale in the same pattern. It is the same ratio, but a different absolute scale.

* It would be America that was the exception of course. American paper is standardis(z)ed to be some arbitrary number of inches. Imperial rubbish which ignores mathematics.