Paper sizes
Two standard
Yes, this post is about the paper. He absolutely has nothing to do with programming, but it contains a fragment of entertaining mathematics.
In the world there are two basic formats of paper. The most popular - ISO-216 , better known as A-type formats, as A4. This system is used almost everywhere in the world except the United States and Canada, and a few others. In the United States used its own standard US Letter .
US Letter
Format US Letter size paper involves 216 × 279 mm (8, 5 × 11 inches) and an aspect ratio of 1, 291666666666667.
(I'll explain why it is important in the mathematical ratio of the below!).
Rationale size rather vague. Exactly this, no one knows.
Most of the sources (and Wikipedia) say:
Length 11 inches of standard paper is about a quarter of the "average maximum amplitude hands of an experienced worker». Blockquote> In general, US Letter is the standard because ... well, because it is so. Do not ask questions, just accept it.
ISO-216
Format ordinary office paper in ISO-standard - A4. Its size of 210 × 297 mm, ie √2 (math!).
Aspect Ratio - what makes such a great format. Naturally, it is no coincidence. Smart people think and develop these dimensions. They have become the standard because of its superiority. "Magic" A4 property that it consists of two sheets of A5. In turn, the two sheets of A4 folded into the larger format A3. This, for example, simplifies the manufacture of booklets A5, folding in half page A4. Conversely, it is convenient to make copies of booklets A5 in turn, they correspond exactly to A4 format copier. This can not be done with paper US Letter, you still have white space around the edges of the paper!
Mathematics aspect ratio
How it works mathematically? The magic lies in the fact that 2 / √2 = √2. Imagine that we have a piece of paper with the long side and the short side A B. If we sognёm it on the long side and create a new paper size with the parties B and C, which is the aspect ratio?
A / B = √2 (original ratio) C = A / 2 (new short side: A half) What is the new balance of B / C? B / C = B / (A / 2) B / (A / 2) = 2 / (A / B) 2 / (A / B) = 2 / √2 2 / √2 = √2! Pre> Since the aspect ratio of √2, the result of folding the sheet in half again will √2. You can continue over and over again.
Take the largest size paper A0: 841 × 1189 mm. What will happen when minimized in two? Thus 841 becomes long side and 1189/2 = 594, 5 short. This is the paper size A1 (594 × 841 mm). It retains the magic proportion √2.
Size A0
Now, when can we explain the aspect ratio, yet still do not understand where it came from specific dimensions of 210 × 297. It is derived from paper sizes A0, in which the ratio of √2, and the area is 1 square meter.
This is all you need to know: √2 and square meter. B>
A * B = 1 m ^ 2 (our paper size) A / B = √2 (our aspect ratio) A / B = √2: A = B * √2 So, the area is equal to B * B * √2 = 1 m ^ 2 B ^ 2 = 1 / √2 And, in the end, the result count: B = 1 / √ (√2) ≅ 840, 89 mm A = B × √2 ≅ 1189, 20 mm pre > We began with two simple values: √2 and square meter and calculate the size of the paper A0: 841 × 1189 mm! To calculate other formats A simply reduce them twice:
A '= B / 2 B' = A pre>
Continuing to reduce the size of
If you continue to fold the sheets in half, then the A4 size, we obtain the following paper sizes (aspect ratios vary slightly due to rounding to the nearest whole number of millimeters).
A4 210 × 297 mm, the ratio of 1: 414 148 A5 × 210 mm, the ratio of 1: 419 105 A6 × 148 mm, the ratio of 1: 409 A7 74 × 105 mm, the ratio of 1: 418 × 52 A8 74 mm, the ratio of 1: 423 A9 37 × 52 mm, the ratio of 1: 405 A10 26 × 37 mm, the ratio of 1: 423 pre> And what if folded sheets US Letter?
216 x 279 mm, the ratio of 1: 291 139 x 216 mm, the ratio of 1: 554 & lt; - What? 108 x 139 mm, ratio 1: 287 & lt; - Blin ... 69 x 108 mm, ratio 1: 565 & lt; - Yes !? 54 x 69 mm, the ratio of 1: 278 34 x 54 mm, the ratio of 1: 588 27 x 34 mm, the ratio of 1: 259 & lt; - Oh God ... pre> Changing the aspect ratio leads to a lot of wasted place on the page, wasting ink and paper. And it just looks disgusting! If you want to deploy something from A5 to A4, it automatically converges, and if scaled format US Letter ... will need something to cut off and leave the excess paper.
So US Letter harmful to the environment.
Should be banned standard US Letter, in the name of mathematics!
Source: geektimes.ru/post/249542/
