Why is A4 paper the most widely used printing paper?

A4 paper can be said to be the most used paper besides toilet paper, especially in the student days, printing review materials, writing a love letter, stacking a thousand paper cranes...

A4 paper is not only popular in the printing industry, but also in the fashion circle in the past few years. For a time, "A4 waist" has become the standard for judging people's body, and A4 paper is in the fashion circle. The scenery is infinite.

However, many people may still have questions about the appearance of A4 paper. Why do people love to use A4 paper?

Perhaps the popularity of A4 paper may be related to its "service spirit" for human customers: low-key, unassuming, but giving "customers" the ultimate use experience.

A3 is cut in half and becomes A4
A4 paper is the most commonly used paper size in life. The paper specifications of the A series are characterized by:

1A0, A2, …, A5, all sizes of paper have the same aspect ratio.
2. In the A series of paper, after cutting the paper with the front serial number, you can get two pieces of paper with the following serial number. For example, you can get 2 sheets of A1 after cutting A0, and you can get 2 sheets of A2 after cutting A1, and so on.

These two features make A-series paper very easy to use, and pictures drawn on A4 paper can be scaled up to A0 posters. As long as you have a certain A-series paper at hand, you can make A-series of any size. This is the special aspect ratio of the A-series paper, the special ratio √2:1 of the Lichtenberg ratio, which brings this characteristic.

Here are the actual numbers for you to see:

A0 is 84.1 cm × 118.9 cm 118.9/84.1=1.41;

A4 is 21 cm × 29.7 cm, 29.7/21=1.41;

A5 is 29.7/2=14.8cm×21cm, 21/14.8=1.41.

We can find that these ratios all approach √2.

In other words, we all enjoy the benefits of the Lichtenberg ratio, but few people know what the Lichtenberg ratio is. To a certain extent, this is the ultimate consideration, allowing people to enjoy its benefits. , but did not notice its existence.

Well, I'm not talking about the feminine line, although the two are very similar (by every distance).

Hiding in the A4, there are even more amazing proportions
The golden ratio of 1.618 can be called the king of ratios, and he is the most common ratio in terms of Google searches or the range of applications. However, in fact there are ratios similar to the golden ratio, hidden in A4 paper like the Lichtenberg ratio.

The name of this ratio happens to be the same as the rank of the saints in "Saint Seiya". It is the silver saint under the golden saint. No, the silver ratio is 2.414. As long as the short side of the A4 paper is used as the side length, draw a square and cut it out, and the aspect ratio of the remaining rectangle is the silver ratio. We can verify that the aspect ratio of A4 paper is √2:1. After drawing a square with the short side and cutting it, the aspect ratio of the remaining rectangle will be 1:[√2:1]=2.414.

We can also express the silver ratio as √2+1, which has many similarities with the golden ratio of (1+√5)/2. For example, they are all related to a certain sequence of numbers.

The golden ratio and the Fibonacci sequence were previously introduced:

1, 1, 2, 3, 5, 8, 13, 21...

Divide the latter set of numbers by the former set to obtain a ratio that approximates the golden ratio.

The silver ratio uses the Pell sequence:

1, 2, 5, 12, 29...

Divide the two groups of numbers before and after, you can get 2, 2.5, 2.4, 2.41, ... getting closer and closer to the silver ratio of 2.414. We can explain graphically why the Fibonacci and Pell sequences produce the golden or silver ratio.

First of all, the nth group of numbers in the Fibonacci sequence is the sum of the first two groups of numbers, which means Pn=Pn-1+Pn-2, for example 21=8+13.

The nth group of numbers in the Pell sequence is the double of the previous group of numbers plus the previous group of numbers, which means Pn=2Pn-1+Pn-2. For example 29=12×2+5.

Dividing the recursive representation of the Fibonacci and Pell sequences by Pn-1 yields:

Pn/Pn-1=1+Pn-2/Pn-1.

Pn/Pn-1=2+Pn-2/Pn-1.

The first formula says that a large golden ratio rectangle is a square plus a small golden ratio rectangle.

The second formula is that a large silver-ratio rectangle can be composed of two squares and a small silver-ratio rectangle, as shown in the figure below.

The spirit of service in the silver ratio
The Lichtenberg ratio made paper easier to use, and the golden and silver ratios were used more widely, one of which was for artistic contributions.

For example, in the green rhombus below, the aspect ratio of the diagonal is exactly the ratio of silver to silver. Therefore, we can use the geometric rule "two large plus one small" just now to place three such rhombus, two of them horizontally, and three They are placed side by side, creating a blue diamond that is proportionally enlarged, and the enlargement ratio is exactly the ratio of silver.

This particular rhombus and square are the basic elements of the complex Islamic-style collage shown below.

Compared with the square or honeycomb hexagonal inlay of ordinary floor tiles, this inlay is more gorgeous and beautiful. For such a complex inlay, in addition to the basic rhombus that requires the ratio of silver, the inlay process also needs to use the ratio of silver. The many regular octagons that are finally formed still have a ratio of silver.

It can be seen that the silver ratio also provides the best service, so that users can't perceive it, but because it benefits a lot. At least the local tyrant who used this inlay technique in the mansion must not know that there is a silver ratio behind it, otherwise he will definitely call the designer over and scold him:

"Why not use the golden ratio instead of the silver ratio?!"

Yes, there is the golden ratio inlay, which is called the Penrose puzzle. Mathematics is the science with the most service spirit and the most thoughtful service.