structural:expansion, quality:correction
Paper size

In 1786, German physics professor Georg Christoph Lichtenberg found that any sheet of paper whose long edge is 2 {\displaystyle {\sqrt {2}}} times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standardised paper sizes at the beginning of the 20th century, they used Lichtenberg's ratio to create the "A" series of paper sizes. Today, the (approximate) aspect ratio of paper sizes under ISO 216 (A4, A0, etc.) is 1: 2 {\displaystyle {\sqrt {2}}}.

Proof: Let S = {\displaystyle S=} shorter length and L = {\displaystyle L=} longer length of the sides of a sheet of paper, with

Let R ′ = L ′ S ′ {\displaystyle R'={\frac {L'}{S'}}} be the analogous ratio of the halved sheet, then