We have been using DIN A4 sheets of paper for decades, and surely many of you have asked yourself the same question from time to time: why a DIN A4 sheet of paper? mide 210 x 297 mm? Why not other more “round” dimensions?
In fact, one might think that it would be much better to end up using sheets of, for example, 200 x 300 mm to make everything simpler. Perhaps it would be to remember those dimensions, but then the DIN A4 sheet of paper would not be so perfect. And it is thanks to mathematics.
The magic of maintaining the “aspect ratio”
If you take any piece of paper that is not a conventional sheet of paper in some DIN format, you will find yourself in a curious situation. You can fold it without problems, but when you do so, those halves will no longer have the format of the original paper. They will be more rectangular or more square, but they will not retain the “aspect ratio” of the original paper.
That is precisely the secret of the DIN A4 format, and that is where the mathematics comes in. In 1786 a letter from the German academic Georg Christoph Lichtenberg to Johann Beckmann – who coined the word ‘technology’, there’s nothing – formulated the idea of using a paper format that could be preserved by folding (or expanding it proportionally).
It was not until the beginning of the 20th century that Germany managed to standardize the idea. That standard is now known as ISO 216, and it defines the international standard paper size almost everywhere in the world. How was that standard defined?
Well, with only one objective: that the aspect ratio was maintained, and this is where a simple mathematical operation allowed us to solve the problem. As mathematician Ben Sparks explained, one can draw a rectangle with aspect ratio x:1. If one divides the rectangle in half, the new rectangle will have an aspect ratio of 1:x/2.
If one applies the math and wants both aspect ratios to be the same, just solve the equation x/1 = 1/(x/2)which in the end results in x = √2.
So, that is the only solution to maintain the aspect ratio. Since there is no pair of integers that would allow us to obtain an aspect ratio √2, approximations are used. Some approximations that, yes, start from an almost perfect number.
Thus, A0 paper (DIN A0) uses that aspect ratio and has 1 m² area. Or almost, because its dimensions (1,189 x 841 mm) are quite close to that “round” area (999,949 mm²).
From there we fold the A0 several times (one, two, three, four, five, …) to successively obtain paper A1, A2, A3, A4 or A5, etc., which have dimensions that are half of the previous format, and that, of course, maintain the aspect ratio. Magic. Or mathematics, rather.
There is some curiosity associated with that aspect ratio. The first is that the weight of the paper can be easily calculated: if 80 gsm (grams per square meter) paper is used, an A0 sheet of paper will weigh exactly 80 grams. An A4 sheet of paper with that density will weigh 5 g (because we have folded—divided—the A0 four times).
These rotrings advance in thickness by multiplying the previous one by 1.4. They are not exact, but the relationship remains quite stable.
The second, that the thicknesses of technical markers tend to also be increased while maintaining that ratio of √2, or what is (almost) the same, 1.4. That way the next thickness of a marker will be appropriate for drawing on the next size of paper.
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