Metric paper which always has the same ratio of dimensions when folded in half that being 1/sqrt(2). How was the logic behind this derived?
In: Mathematics
The idea is the larger paper format is a multiple of smaller. A A3 paper is equal in size to two A4 side by side. A A4 is two A5.
The size is so that a A0 has an area of 1 square meter. That means a A4 has an area of 1/2^4= 0.0625 square meters
If you for example want to print a A5 booklet you can simply do it by printing it on A4 and folding it. If the booklet is 4 pages you need a single A4, for more pages you need some staples
We can set it up as a formula.
We want a piece of paper that has the same ratio of dimensions when it is folded in half. Lets say the short side has length 1 and we’ll call the long side we’re trying to find x. So the ratio of dimension is 1/x . If we fold it in half, the short side is now x/2 and the long side is 1. So the ratio of the folded piece of paper is (x/2) / 1, or just x/2. We want them equal to each other, so 1/x = x/2.
Multiply both sides by x, we get 1 = x^2 / 2.
Multiply both sides by 2, we get 2 = x^2
Take the squre root of both sides, we get sqrt(2) = x, and there’s your sqrt(2)
It’s not exactly difficult to think that the next size down should be one page folded in half, and the next size up should be too pages side by side.
The more baffling thing is that there are entire systems of paper sizes which don’t do this extremely obvious thing. Metric paper’s system has existed since 1786, the US has had long enough to switch to it.
The long side of a piece of paper is 1 unit. The short side is X units. Fold it in half, the long side is X units, and the short side is 0.5 units.
The ratio of the sides is long / short.
So we need a value of X where 1/X = X/0.5. We can rearrange this, and see X²=½ or X=1/sqrt (2). It’s the only number that works.
Everyone who have been doing a bit of mathematics have noticed this quirk. It can be a bit odd to work with fractions of strange numbers so we often end up normalizing a fraction when solving equations. With 1/sqrt(2) we can multiply with sqrt(2)/sqrt(2) as this is just 1 and you can multiply any number with 1. So you end up with sqrt(2)/sqrt(2)^2 which is just sqrt(2)/2. This is so common that you tend to do this automatically when solving equations.
But this is also the basis behind the metric papers, and were actually used before metrics were invented. We do not know exactly who came up with this first or even if it was just one idea. We have some letters from 1786 between two German scientists which mentions 1/sqrt(2) paper but they probably did not invent it. We also find mentions of it in a 1798 French tax law which shows that it was a common ratio.
Suppose I have a piece of paper that is 1m x 2m. How do I get the same 1:2 ratio but with the 1m as the long side? I take a ¼ of the paper.
Suppose I have a piece of paper that is 1m x 3m. How do I get the same 1:3 ratio but with the 1m as the long side? I take a ⅑ of the paper.
So if I have a piece of paper that is 1 by X in ratio, if I take 1 / X² of the paper, I will preserve the ratio.
If X is 1.414 or √2 then I will preserve the ratio when I take ½ of the paper.
US works the same with the 2x scale.
A= 8.5 × 11
B = 11×17
C = 17×22
D = 22×34
E = 34×44
F = 44×68
Those weirdo sizes like legal etc are just derivatives trying to screw up the system.
[https://www.reddit.com/user/Average_guy94/comments/1azy72w/this/](https://www.reddit.com/user/Average_guy94/comments/1azy72w/this/)
that’s weird I solved my own problem (bothered my relative about this)
The reason for this is that you sometimes need to make the contents of a page larger or smaller.
Two A4 pages reduced to half their original size will exactly fit to a new A4 paper for example.
If you want to copy a double page spread of of an A4 magazine to a single A4 page it will fit exactly without any space wasted.
You can design stuff on a small scale and blow it up to a larger scale and keep the same aspect ratio.
Like designing an A1 poster as and A4 page. Or you can draw on a larger scale with more detail and reduce the result later.
This all is very useful and the ratio and size of an A4 is also close to what people were using anyway for writing letters.
An A0 paper is one square meter with the dimensions of 1 by sqrt(2)
Each subsequent paper size A1, A2, A3, A4, A5 etc is arrived at by cutting the bigger paper in two along its longer axis.
This gives you a paper that is half the size and has the ratio of 1 by sqrt(2) / 2 which works out to sqrt(2) by 1 again.
The actual size of each paper size are rounded to the next millimeter.
You can derive that logic from algebra.
Let’s define a rectangle with its sides being x and y, x<y (so y is the long side). We could define the ratio of the length r = x/y
Here’s the important part: we want this ratio to stay the same when the longer side is folded aka r = (y/2)/x. Why? Because it makes it more practical to resize a A4 sheet into a A5 postcard without cutting parts of what you’re printing.
So by moving terms in both expressions, we want x = r * y AND y = 2 * r * x at the same time. Replace x in the 2nd expression with the 1st, you get y = 2 * r * r * y. Divide both sides by y, you get 1 = 2 * r^2
Now you just have to solve this expression, and you get r = 1/sqrt(2)
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