# The Cleverness of Paper Size

Struggling on a problem can make the piece of paper you write on more interesting than the actual contents. The beauty lies in the question, why do we have the paper format we are used to?

If we were to design a paper format, we could think of a rectangle with an aspect ratio of \(a\) to \(b\).

When we now fold the paper in the middle, we would like to have a piece of paper with the same aspect ratio, to be able to cut the paper without waste. We therfore require that the aspect ratio of the new long side is \(b\) to \(\frac{a}{2}\). It follows that

\[\begin{array}{rrl}&\frac{a}{b} &= \frac{b}{\frac{a}{2}}\\\Leftrightarrow & \frac{a^2}{b^2} &= 2\\\Leftrightarrow & \frac{a}{b} &= \sqrt{2}\end{array}\]

This is interesting: To be able to fold a sheet of paper one time to get the same aspect ratio as before, the ratio of \(a : b\) must be the square root of two. In the A-series, which is pretty wide spread across the world, the largest sheet of paper A0 is now defined to have an area of \(1m^2\) with an aspect ratio of \(1:\sqrt{2}\). The series then goes like two A1 are A0, two A2 are two A1 and so on. It follows that A4, which is typically used has only \(\frac{1}{16}\) of the area of A0. To calculate the size of an A4 sheet, we state that the area \(A=\sqrt{2}\cdot a^2 = \frac{1}{16}m^2\), from which follows that the short side \(a\) of an A4 sheet is \(a=2^{-\frac{9}{4}}m = 210mm\). Multipling by \(\sqrt{2}\) leads to \(b=a\sqrt{2} = 2^{-\frac{9}{4}}m\cdot 2^{\frac{1}{2}} = 2^{-\frac{7}{4}}m=297mm\).

## Generalization of paper folding

Getting the same aspect ratio after halving \(\sqrt{2}\) works as we abuse the identity \(\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\) - which says half of square root of two is the same as the rotated aspect ratio. This can be generalized to \(\frac{1}{\sqrt{n}} = \frac{\sqrt{n}}{n}\), since \(n^{-\frac{1}{2}} = n^{\frac{1}{2}}\cdot n^{-1}\). This means that a sheet in aspect ratio \(1:\sqrt{3}\) keeps the ratio when divided by three:

But what this also means is that a sheet of paper with an aspect ratio of \(1:2\) must be quartered in order to transfer the aspect ratio to the smaller size:

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