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Why is the aspect ratio of How You Really Use Mathematics To Define Paper SizeA4 paper irrational? The rich history and interesting mathematics behind the paper size!

George Kristof Lichtenberg Portrait and A Series Paper size Diagram have you ever thought about how the paper size is defined? Our most commonly used paper size is usually the A4 size. If you are engaged in or come into contact with technical disciplines, you will learn about other paper sizes, such as A0, A1, A2, etc. ISO 216ISO 2007 defines most of the paper sizes commonly used internationally, while other ISO standards such as ISO 269 and ISO 217 cover the rest.

But do you know where these standards come from?

How do we describe the size of paper before these standards and definitions?

Who chose these sizes for what purpose?

This is a series of interesting questions. It turns out that the invention of paper size standards contains a rich and mysterious history of mathematics.

Mathematics itself is subtle and profound. In this article, I will start with the history of mathematics and explore the mathematical problems behind paper size standards. Finally, I explained how beautiful human nature is cleverly embedded in this technology. Please sit down, relax and enjoy this pleasant journey!

Historical George Christopher Lichtenberg (Georg Christoph Lichtenberg) was a famous German physicist and satirist in the 18th century. He is an outstanding professor of physics at the University of Gottingen and one of the first scientists to introduce instrument experiments in the lecture.

He also maintained close contact with other great German figures of the time, such as John Wolfgang von Goethe (Johann Wolfgang von Goethe) and Emmanuel Kant (Immanuel Kant). It is well known that legendary mathematician Carl Friedrich Gauss (Karl Friedrich Gauss) also attended Lichtenberg's lectures. He is also famous for discovering tree-shaped discharge patterns, which later became known as Lichtenberg figures.

The portrait of Lichtenberg was particularly interested in the standardization of paper size among a series of scientific problems he was solving at the time. He wants to find a standard that can scale the contents of the paper perfectly. Perfect scaling here means that within a range of possible standardized magnification, the paper will neither be wasted nor cramped.

He used this as an exercise to ask one of his British algebra students. The student proposed a specific ratio that would meet the characteristics Lichtenberg wanted (which will be described in more detail later). When Lichtenberg began to try to actually apply this ratio to a piece of paper, he was pleasantly surprised to find that German newspapers had already adopted it.

In a letter to John Johann Beckmann in 1786, Lichtenberg said he was not sure whether the ratio came naturally for historical reasons or from accurate mathematical calculations. In any case, for the first time in this story, there is a record of mathematical knowledge behind paper size standards.

About the proportion of √ 2, √ 2 is a very interesting number in mathematics. When the Pythagorean theorem is applied to a right triangle with unit length and unit height, the length of its oblique edge is √ 2.

Therefore, the diagonal of the unit square is √ 2. √ 2 is an irrational number. If expressed in decimal, √ 2 is 1.4142135623730950488016887.

Lichtenberg (and his students) found that when a piece of paper is seen as a rectangle and its long side is the short side of √ 2, it meets the requirement that it will neither waste paper nor make the paper appear crowded when zoomed in.

Later, we will see how this is achieved mathematically. But first, it may be helpful to understand the concept from a geometric point of view.

The size of Series A paper displayed at the Science Museum in Barcelona comes from a rectangular piece of A0 paper. Its size is 841 mm × 1189 mm. If you fold it in half along the longer side, the two folded parts are each a piece of A1 paper. If you fold two pieces of A1 paper along their long edges, you will get four pieces of A2 paper. Each repetition of the process will result in 8 sheets of A3, 16 sheets of A4, 32 sheets of A5, 64 sheets of A6, 128 sheets of A7 and 256 sheets of A8 paper.

This is why modern printers can quickly scale the print to fit different paper sizes. For example, if we want to save paper, we can first reduce the page size of the book (electronic version) to the size of A6 paper, and then print on A4 paper. In this way, each piece of A4 paper will contain 4 pages of the book (each side), thus increasing the information density on the paper.

What's so special about √ 2 proves that the mathematical principle behind this ratio is very clear.

The author considers a piece of paper whose long side is "a" unit length and its short side is "b" unit length. If we fold this piece of paper along the long side, we will get two sheets of paper, each with a b unit on the long side and a / 2 unit on the short side.

Now, copy Lichtenberg's exercises to his algebra students, asking them to keep the ratio between the long and short edges in one Jang Dae-Il-dotted paper and two smaller sheets (folded). Then, this becomes a simple mathematical problem that can be solved in the following ways:

Derivation from the author when we use mathematical methods to solve this problem, we can clearly see that under the premise that the side length of the paper must be positive, the ratio can only be √ 2.

Historical reasons after Lichtenberg, France published a law on paper taxation in 1798, which proved to be the direct source of the current ISO standard.

Bostman (W. Porstmann) pointed out in a 1918 article that paper size standards also need to include the surface area involved. He also believes that the envelopes used in the above paper should be 10% larger than the paper itself. Under his influence, the German Industrial Standardization Committee (Deutsches Institute f ü r Normang-DIN) issued DIN 476, which involves four sizes of paper, but the aspect ratio of each paper is √ 2. 5.

A0 paper is defined as the surface area of A0 paper is 1 square meter (841 mm x 1189 mm) when the size is accurate to millimeters. A4 is recommended as the standard paper size for business and administrative activities; they also suggest that A6 paper be used in postcard making. For B-series paper, the width of B0 paper is 1 meter. C series paper is developed based on the envelope format.

Today, except for a few countries such as North America, Peru, Colombia and so on, almost all countries have adopted these standards.

Further mathematical meaning ratio √ 2 has some counterintuitive characteristics.

Vertically and horizontally, so far, we have seen that two sheets of A5 can be placed in one piece of A4 paper. But suppose we are interested in landscape printing instead of vertical printing. In order to place two pages horizontally on a piece of A4 paper, how much do we need to reduce the original A4 (vertical) content? Our gut tells us it's 50%.

However, since the aspect ratio is √ 2, we only need to reduce the content to 70% instead of 50%. This is because (1 / √ 2) = 0.7071... , which is about 70% (0.70).

Geometric averages have proved to be very useful in packaging paper of different sizes. For example, the size of C2 is the geometric average between A2 and B2. Similarly, the entire C-series format is the geometric average between the corresponding A-series and B-series numbers.

Last thought when I began to investigate this topic, I always had a ruler and tape measure around me. I found myself measuring the aspect ratio of any rectangular object, and the results were aesthetically satisfactory.

The objects I measure include: my writing desk, monitor, mouse pad, tablet, peppermint box, rectangular plate, physical book, etc. The aspect ratio of my most satisfactory rectangular shape is between 1.31 and 1.64 (√ 2 = 1.4142.).

Source: Ksenia Chernaya from Pexels first of all, I was surprised to find that so many objects around are rectangular. Before I started to study this problem, I paid less attention to their shapes; secondly, I was surprised that my aesthetic taste was consistent with the aspect ratio of √ 2.

I am not sure whether such aesthetic taste is learned in a particular cultural background or occurs spontaneously in nature. Now, all I can do is appreciate the fact that √ 2 has something to do with human nature, in addition to pure mathematics.

Sometimes, the deepest understanding comes from the simplest observation! I hope you will find this article interesting and useful.

Original text link:

Https://www.cantorsparadise.com/how-you-really-use-mathematics-to-define-paper-size-c2928ba551ec

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This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: Hemanth Translation: Nothing revision: zhenni

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