Editing Square root of 2 (section)

==Applications==
===Paper size===
[[File:A size illustration2.svg|thumb|200px|The A series of paper sizes]]

In 1786, German physics professor [[Georg Christoph Lichtenberg]]<ref name=":0" /> found that any sheet of paper whose long edge is <math>\sqrt{2}</math> 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 size]]s at the beginning of the 20th century, they used Lichtenberg's ratio to create the [[ISO 216#A series|"A" series]] of paper sizes.<ref name=":0">{{citation |title=The Book: A Cover-to-Cover Exploration of the Most Powerful Object of Our Time|last=Houston|first=Keith|publisher=W. W. Norton & Company|year=2016|isbn=978-0393244809|pages=324}}</ref> Today, the (approximate) [[aspect ratio]] of paper sizes under [[ISO 216]] (A4, A0, etc.) is 1:<math>\sqrt{2}</math>.

Proof:

Let <math>S = </math> shorter length and <math>L = </math> longer length of the sides of a sheet of paper, with
:<math>R = \frac{L}{S} = \sqrt{2}</math> as required by ISO 216.
Let <math>R' = \frac{L'}{S'}</math> be the analogous ratio of the halved sheet, then
:<math>R' = \frac{S}{L/2} = \frac{2S}{L} = \frac{2}{(L/S)} = \frac{2}{\sqrt{2}} = \sqrt{2} = R. </math>

===Physical sciences===
There are some interesting properties involving the square root of 2 in the [[physical sciences]]:
* The square root of two is the [[interval (music)#Frequency ratios|frequency ratio]] of a [[tritone]] interval in twelve-tone [[equal temperament]] music.
* The square root of two forms the relationship of [[F-number|f-stops]] in photographic lenses, which in turn means that the ratio of ''areas'' between two successive [[aperture]]s is 2.
* The celestial latitude (declination) of the Sun during a planet's astronomical [[cross-quarter day]] points equals the tilt of the planet's axis divided by <math>\sqrt{2}</math>.
{{distances_between_double_cube_corners.svg}}
* In the brain there are lattice cells, discovered in 2005 by a group led by May-Britt and Edvard Moser.  "The grid cells were found in the cortical area located right next to the hippocampus [...] At one end of this cortical area the mesh size is small and at the other it is very large. However, the increase in mesh size is not left to chance, but increases by the squareroot of two from one area to the next."<ref name="Hjernen er stjernen" >{{citation |title=The Book: Hjernen er sternen|last=Nordengen|first=Kaja|publisher=2016 Kagge Forlag AS|year=2016|isbn=978-82-489-2018-2|page=81}}</ref>