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A hypothetical (and maybe practical) question has been nagging at me.

If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches and 3 inches (2:3), which wouldn't retain its ratio. For example, here is a piece of paper that doesn't retain its ratio when folded: enter image description here

Is retaining the ratio technically possible? If so, what is the side length and ratio that fulfills this requirement? Any help would be appreciated.

Update:

I added "once" because I got an answer saying that any recectangle would work, as any rectangle folded twice has the original ratio. Nice answer, but not quite what I was looking for. As for the other answers, I got 3x as much information as I needed! Thanks!

Pyraminx
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  • Do you want say that after one fold it has to satisfy the ration? Otherwise you can always have boxes with dimensions $2n:n,\ n\ge 1$ – Samrat Mukhopadhyay Feb 11 '15 at 05:53
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    This is exactly why the whole world use A series paper (except US, of course). – Derek 朕會功夫 Feb 12 '15 at 04:41
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    Well, I __have__ watched a video on YouTube about this and A series paper __does__ retain its ratio length to width when folded in half width to width and it retains the same ratio. BTW, you still made this question smokin' hot! – ReliableMathBoy Feb 14 '15 at 05:07
  • Why did you want paper with this property? Does this solve a particular problem? – Loki Clock Feb 14 '15 at 08:34
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    Yes. It solves a big problem: printing documents (or images) scaled up or down to different paper size. Good luck with US letter. BTW check out [NumberHub video](https://www.youtube.com/watch?v=mHeo62B0d0E) about paper sizes where it is explained quite good. – Hauleth Feb 15 '15 at 02:10
  • This question is not getting much more attention, even though it's that hot. – ReliableMathBoy Feb 15 '15 at 16:09
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    @LokiClock: Some of us are simply curious about the world in which we live. We do not require facing "a particular problem" in order to wonder about something. It's called... imagination! :) – Lightness Races in Orbit Feb 16 '15 at 11:34
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    @LightnessRacesinOrbit Others of us are interested in the people about us and their thought processes and style. The A series of paper sizes were chosen so as to have this property because it solves particular problems, mainly that if you design a print for one scale and have to drastically resize it, your rescaled print won't also have to be reformatted - nothing will get cut off, and it will fill the same portion of the page. Since the OP spoke in terms of halving paper, it's possible they could have rediscovered one of those problems, or found a new one. An artist might, for instance. – Loki Clock Feb 17 '15 at 12:23

8 Answers8

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The $1:\sqrt{2}$ ratio ensures exactly that. That is the idea behind the ISO 216 standard for paper sizes, which was adopted from the German DIN 476 standard.

Its most common usage is the A series which especially in Europe is a collection of very common paper sizes. The base size, A0, has an area of a square meter, and every next smaller paper size is constructed by folding it in half.

poke
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orangeskid
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    @Mehrdad: because you don't use it, or because you took the ability to fold A4 in half to create an A5 booklet totally for granted? ;-) – Steve Jessop Feb 11 '15 at 10:55
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    @SteveJessop: Neither -- I don't use it currently, but I've used it before, and I simply have never noticed this. It's not like when you fold a piece of A4 paper you suddenly see a glowing "A5" that tells you it's an A5 paper... I've never dealt with A5, so I never knew what its dimensions were and how they related to A4 and such. – user541686 Feb 11 '15 at 11:23
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    It also means you can print booklets (2/4 pages on 1 sheet) without ending up with loads of whitespace. – NickG Feb 11 '15 at 12:30
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    Being American, let me tell you: Not being able to scale a tabloid (11"x17") down to letter (8.5"X11") with no _added_ margin is a real pain. And before you ask why I don't just use multiple pages, I'd need to print 3 sheets (because the majority of printers don't print borderless, the center of the tabloid wouldn't be printed if I did 2 pages at 100%), trim margins, then place them side by side to see the result. – Cole Tobin Feb 11 '15 at 15:43
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    @Mehrdad Actually not only A4 has the property, but the whole A format sequence has it: the A0 format is defined as a sheet of area 1 square meter and sides in proportion $\sqrt 2 : 1$, then every next $A_{n+1}$ is a half of $A_n$, retaining $\sqrt 2 : 1$ sides' lengths ratio. – CiaPan Feb 11 '15 at 16:59
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    [ISO 216](http://en.wikipedia.org/wiki/ISO_216) is an international standard and not just used in Europe. – Frank Vel Feb 11 '15 at 17:47
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    "Actually not only A4 has the property, but the whole A format sequence has it" So do the B, C (and D) series. Because their side ratio is 1:sqrt(2), which is the key, as the other answers mention. As this is the accepted answer, it should mention that fact. – David Balažic Feb 11 '15 at 17:48
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    @fvel It originated in Europe though; it’s German engineering at its best ;) – poke Feb 11 '15 at 22:19
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    @poke Yes, but this answers fails to mention this. The answer make it seems like it's the standard paper only in Europe. It _originated_ in Europe (or Germany, more specifically) and _is the standard paper_ in most countries around the world. – Frank Vel Feb 11 '15 at 22:55
  • As a user of Stack Exchange for nearly 5 years now, I'll say rare is the answer that consists mostly of a link that I actually approve of. Part of me feels this should mention the 1:sqrt(2) ratio, but brevity is the soul of wit, after all. – corsiKa Feb 11 '15 at 23:02
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    This answer seems incomplete. A4 is a good example, but more detail on what it is and why seem necessary. – AlannaRose Feb 11 '15 at 23:36
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    Though the ratio is not _exactly_ $\sqrt{2}$ since they round the sizes to the nearest millimeter. – Mark Adler Feb 12 '15 at 02:27
  • This ratio is _that_ common that it came as a shock to me when I learned that not all ratios scale. – SQB Feb 13 '15 at 12:32
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    @MarkAdler: Even if the A-series would be measuring down to a femtometer, it would still not be an _exact_ $1:\sqrt{2}$ ratio. No realizable implementation of _any_ ratio is really "exact", after all: such luxury is often exclusive for mathematics :-) . – Daniel Andersson Feb 13 '15 at 13:02
  • [This article](http://www.royvanrijn.com/blog/2015/02/paper-sizes/) covers exactly your question. – franklin Feb 13 '15 at 16:59
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    @DanielAndersson: A spec could be written such that a request that a piece of paper be A0 size with any particular dimensional tolerance would be satisfied if its dimension were within that tolerance of $1:\sqrt{\sqrt{2}}$m by $1:\sqrt{\sqrt{8}}$m, even if it wasn't possible to construct a paper with perfect dimensions. – supercat Feb 14 '15 at 00:28
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A ratio of $1:\sqrt{2}$ will do the trick!

The original rectangle will have a ratio of $x:y$, where $y$ is the longer side and $x$ the shorter side. Then the folded rectangle will have a ratio of $\frac{1}{2}y:x$ and we want

$$\begin{align} \frac{x}{y} & =\frac{\frac{1}{2}y}{x} \\ x^2 & = \tfrac{1}{2}y^2 \\ x & = \tfrac{1}{\sqrt{2}} \cdot y \\ y & = \sqrt{2}\cdot x \end{align}$$

SQB
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Laars Helenius
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Well, suppose you have a rectangle of sides with $A$ and $B$ of length, $A$ being the bigger side.

When we fold it along the longest side we end up with a rectangle with sides of length $B$ and $A/2$.

So what you want to know is if there are any values of A and B that satisfy the following condition:

$$\frac{A}{B} = \frac{B}{A/2}$$

From solving the equation we get that $A = B \cdot \sqrt2$. So if the paper's height is $\sqrt2$ times its width, then we can make a rectangle with the properties that you wanted. And this is the only ratio that will work.

6005
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Arthur Rizzo
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$H:W = W:(H/2)$ resolves to $H:W = \sqrt 2$

CiaPan
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Standard Bond paper sizes A4, A3, A2 and A1 have been designed so that areas double up and sides increase by scale $ \sqrt 2 $.

$$ \frac LB = \frac 12 \cdot \frac bl $$

Narasimham
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A degenerate rectangle with an aspect ratio of $0$, also known as a line segment, can be folded in half either along its length or across its middle, yielding either the same segment or a segment having half the original length. Either way, the aspect ratio remains $0$. Algebraically, the solutions to the equations $$\frac{a}{b}=\frac{a/2}{b}$$ and $$\frac{a}{b}=\frac{a}{b/2}$$ are both $a=0$. (In order to speak of the ratio being a real number, I'm ruling out $b=0$.)

As the other answers observe, the non-degenerate cases are $$\frac{a}{b}=\frac{b}{a/2},$$ which has the solution $a=b\sqrt2$, and $$\frac{a}{b}=\frac{b/2}{a},$$ which has the solution $a=b/\sqrt2$.

Chris Culter
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When you fold any rectangular sheet of paper through the middle, clearly the original area is the double of resulting (folded) area.

At the same time, if you have a little rectangle with sides $a$ and $b$ (so area $ab$) (let us say $a>b$), and you want a similar rectangle with area $2ab$, then clearly that larger rectangle must have sides $\sqrt{2}\cdot a$ and $\sqrt{2}\cdot b$, the factor of magnification being $\sqrt{2}$.

Combining these two simple observations, and looking at your illustrations, we see that the long side $a$ of the "half" rectangle is equal to the short side $\sqrt{2}\cdot b$ of the "unfolded" (full) rectangle, so $a = \sqrt{2}\cdot b \Rightarrow \frac{a}{b} = \sqrt{2}$.

Jeppe Stig Nielsen
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Well, if you have paper that has a ratio equivalent to the square root of two with the length divided by the width, if you fold it in half hamburger style (from width to width), you will get another ratio that's equivalent to the square root of two of the same property, likewise if you fold it more in half that way.

How It Works:

$$\sqrt{2}/1=\sqrt{2}$$$$\sqrt{2}/2=1/\sqrt{2}=0.7071...$$

That's half the square root of two, and when one is divided by it, we get the square root of two again: $1/0.7071...=\sqrt{2}$ and it just keeps going on and on from number to number being divided by the square root of two to get that result. Just divide the dividend by the quotient in those things.

ReliableMathBoy
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