Sizes and sizes of paper - PB "Up the hill". A sheet of paper can be folded in half a certain number of times US paper sizes

16.09.2020 Communication

Perhaps it is strong if you are!

Have you ever tried folding a regular sheet of paper? Probably yes. One, two, three times is not a problem. Then it's harder. A standard A4 sheet of paper is unlikely to be folded more than 7 times without tools at hand. All this is explained by the presence of a physical phenomenon - it is impossible to fold a sheet of paper many times because of the rapidity of the growth of the exponential function.

As Wikipedia says, the number of layers of paper is two to the nth power, where n is the number of times the paper is folded. For example: if the paper is folded in half five times, then the number of layers will be two to the power of five, that is, thirty-two. And for plain paper, you can derive an equation.

Equation for plain paper:

Where W- width of a square sheet, t- sheet thickness and n
When using a long strip of paper, an accurate length is required L:

Where L- the minimum possible length of the material, t- sheet thickness and n- the number of bends performed in half. L and t must be expressed in the same units.

If you take not plain paper with a density of 90 g / dm3 (or slightly more / less), and tracing paper or even gold foil, then such material can be folded a little more times - from 8 to 12.

Mythbusters once decided to test the law by taking a sheet of paper the size of a football field (51.8 x 67.1 m). Using such a non-standard sheet, they managed to fold it 8 times without special means (11 times using a roller and a loader). According to fans of the TV show, the tracing paper from the packaging of an offset printing plate in the 520 × 380 mm format, when folded carelessly enough, can be folded eight times without effort, and nine times with effort. Moreover, each of the folds must be perpendicular to the previous one. If you bend at a different angle, you can achieve a slightly greater number of bends (but not always).

Here are some more attempts:

Well, what if you do not fold a sheet of paper with your hands, but take a hydraulic press as your assistant? Let's see what happens then. Please note that the video is in English, with a very strong accent (Arabic Finnish).

We never managed to find the primary source of this widespread belief: not a single sheet of paper can be folded twice more than seven (according to some sources - eight) times. Meanwhile, the current folding record is 12 times. And what is more surprising, it belongs to the girl who mathematically substantiated this "riddle of the paper sheet."

Of course, we are talking about real paper, which has a finite, and not zero, thickness. If you fold it carefully and to the end, excluding gaps (this is very important), then the "refusal" to fold in half is found, usually after the sixth time. Less often - the seventh. Try this with a piece of notebook paper.

And, oddly enough, the limitation depends little on the size of the sheet and its thickness. That is, just to take a thin sheet of more, and fold it in half, if we say 30 or at least 15 - it does not work, no matter how hard you fight.

In popular collections such as "Do you know what ..." or "Amazing nearby", this fact - that it is impossible to fold paper more than 8 times - can still be found in many places, on the Web and off. But is it a fact?

Let's reason. Each fold doubles the bale thickness. If the thickness of the paper is taken equal to 0.1 millimeter (we are not considering the size of the sheet now), then adding it in half "only" 51 times will give the thickness of the folded pack of 226 million kilometers. Which is already an obvious absurdity.

It seems that this is where we begin to understand where the well-known to many 7 or 8 times limitation comes from (once again - our paper is real, it does not stretch indefinitely and does not tear, but will tear - this is no longer folding). But still…

In 2001, an American schoolgirl decided to come to grips with the problem of double folding, and this resulted in a whole scientific study, and even a world record.

Britney Gallivan (note, she's now a student) initially reacted like Lewis Carroll's Alice: "No use trying." But the Queen said to Alice: "I dare say that you did not have a lot of practice."

So Gallivan started to practice. Having suffered a lot with different subjects, she folded the sheet of gold foil in half 12 times, which put her teacher to shame.

Actually, it all started with a challenge thrown by the teacher to the students: "But try to fold at least something in half 12 times!" Like, make sure it's completely impossible.

An example of folding a sheet in half four times. The dotted line is the previous position of the threefold addition. The letters indicate that the points on the surface of the sheet are displaced (that is, the sheets are sliding relative to each other), and as a result, they occupy a different position than it might seem at a quick glance (illustration from pomonahistorical.org).

The girl did not calm down on this. In December 2001, she created a mathematical theory (well, or mathematical justification) of the double folding process, and in January 2002 she did 12 folds in half with paper, using a number of rules and several directions of folding.

Britney noticed that mathematicians had previously addressed this problem, but no one had yet provided a correct and proven solution to the problem.

Gallivan became the first person to correctly understand and substantiate the reason for addition restrictions. She studied the effects accumulated when folding a real sheet and the "loss" of paper (and any other material) on the fold itself. She received equations for the folding limit, for any initial sheet parameters. Here they are.

The first equation refers to folding the strip in one direction only. L is the minimum possible length of the material, t is the thickness of the sheet, and n is the number of folds made in half. Of course, L and t must be expressed in the same units.

In the second equation, we are talking about folding in different, variable, directions (but still - twice each time). Here W is the width of the square sheet. The exact equation for folding in "alternative" directions is more complicated, but here is a form that gives a very close to reality result.

For paper that is not a square, the above equation still gives a very precise limit. If the paper, say, has a ratio of 2 to 1 (in length and width), it is easy to figure out that you need to fold it once and "bring" it to a square of double thickness, and then use the above formula, mentally keeping one extra folding in mind.

In her work, the schoolgirl defined strict rules for double addition. For example, for a sheet that is folded n times, 2n unique layers must lie in a row on the same line. Sheet sections that do not meet this criterion cannot be counted as part of a folded pack.

So Britney became the first person in the world to fold a sheet of paper in half 9, 10, 11 and 12 times. We can say, not without the help of mathematics.

The phrase “a sheet of paper cannot be folded more than seven times” can be understood in two ways. Firstly, in the sense that it is forbidden or there is some kind of belief like, if you fold a sheet of paper 7 times, misfortune will happen. There is no information about this anywhere.

Then this phrase will sound like this: "It is impossible to fold any sheet of paper more than 7 times." It's getting interesting. And many begin to try to fold sheets of paper: a notebook sheet, a standard A4 sheet, newspaper strips, napkins. Fortunately, paper is at hand for everyone. AND why paper cannot be folded more than 7 times?

What happens if you fold the paper 7 times?

Already with the addition of the fifth time you start to experience problems, the sixth is also obtained with effort. We fold it for the seventh time, and with difficulty, we get a thick piece of paper multilayer "rectangle", which then cannot be folded in half.

Many questions arise. Is there such a limitation? Is there a limit to folding paper in half? And most importantly, why can't the paper be folded more than 7 times?
Besides the practical way of answering this question, the “phenomenon” can be explained theoretically. Let's try to count how many layers there are in this piece of "stubborn paper." First there was a single sheet of paper, then 2 layers, then 4, and so on. With a 5-fold addition, we get 32 layers, 6-fold 64, 7-fold - 128 !. That is, with the eighth fold, we must simultaneously bend 128 layers of paper! Here's the thing, the number of layers of paper is growing exponentially. Hardly anyone will succeed in putting together such a multi-layered "pie" the first time.

Who can fold the paper more than 7 times?

But there were people who tried to refute this statement. They reasoned like this: the larger the size of the original paper, the easier it will be to fold later. This is indeed the case. Indeed, with the increase in the size of the paper, the shoulder of force grows, with which we apply the effort to fold the paper in half. This is the well-known rule of the lever: the longer the lever, the greater the moment of force, that is, our strength increases by the same amount. Therefore, researchers take sheets of paper as large as possible (up to the size of a football field) and fold it. However, they have to use technical means (roller and loader). In this experiment, they were able to fold the paper in half 8 times by hand, 11 times using the technique.

Another way to dispel this "myth" is to take as thin a sheet of paper as possible. And in this experiment, the researchers managed to surpass the limit of seven. Thin tracing paper (from offset paper) is folded 8 times, with effort.

So the conclusions. The belief that paper cannot be folded more than 7 times in half did not arise from scratch. It really gets harder and harder to fold the paper. In any case, there is a limit to folding paper, some say that it is equal to 7, others 8 or more, but the essence is the same: paper cannot be folded in half an infinite number of times.

And, oddly enough, the limitation depends little on the size of the sheet and its thickness. That is, just to take a thin sheet of a larger one, and fold it in half, since let's say 30 or at least 15 - it does not work, no matter how hard you fight.

In popular collections such as "Do you know what ..." or "Amazing nearby", this fact - that it is impossible to fold the paper more than 8 times - can still be found in very many places, on the Web and off. But is it a fact?

World record holder Britney Gallivan and paper tape folded in half (in one direction) 11 times (photo from mathworld.wolfram.com).

In 2001, an American schoolgirl decided to come to grips with the problem of double folding, and this resulted in a whole scientific study, and even a world record.

Actually, it all started with a challenge thrown by the teacher to the students: "But try to fold at least something in half 12 times!" Like, make sure it's completely impossible.

Britney Gallivan (note, she is now a student) initially reacted like Lewis Carroll's Alice: "It's useless to try." But the Queen told Alice: "I dare say that you did not have a lot of practice."

So Gallivan started to practice. Having suffered a lot with different subjects, she folded the sheet of gold foil in half 12 times, which put her teacher to shame.

Britney noticed that mathematicians had previously addressed this problem, but no one had yet provided a correct and proven solution to the problem.

Gallivan became the first person to correctly understand and substantiate the reason for addition restrictions. She studied the effects that accumulate when folding a real sheet and the "loss" of paper (and any other material) on the fold itself. She received equations for the folding limit, for any initial sheet parameters. Here they are.

Gallivan and her record (photo from pomonahistorical.org).

In the second equation, we are talking about folding in different, variable, directions (but still - twice each time). Here W is the width of the square sheet. The exact equation for folding in the "alternative" directions is more complicated, but here is a form that gives a very close to reality result.

In popular collections, such as “Do you know what ...” or “Amazing nearby,” this fact - that it is impossible to fold paper more than 8 times - can still be found in very many places, on the Web and off. But is it a fact?

In 2001, an American schoolgirl decided to come to grips with the problem of double folding, and this resulted in a whole scientific study, and even a world record.

Actually, it all started with a challenge thrown by the teacher to the students: "But try to fold at least something in half 12 times!" Like, make sure it's completely impossible.

Britney Gallivan (note, she is now a student) initially reacted like Lewis Carroll's Alice: "It's useless to try." But the Queen told Alice: "I dare say that you did not have a lot of practice."

So Gallivan started to practice. Having suffered a lot with different subjects, she folded the sheet of gold foil in half 12 times, which put her teacher to shame.

Britney noticed that mathematicians had previously addressed this problem, but no one had yet provided a correct and proven solution to the problem.

In the second equation, we are talking about folding in different, variable, directions (but still - twice each time). Here W is the width of the square sheet. The exact equation for folding in the "alternative" directions is more complicated, but here is a form that gives a very close to reality result.

So Britney became the first person in the world to fold a sheet of paper in half 9, 10, 11 and 12 times. We can say, not without the help of mathematics.

On January 24, 2007, in the 72nd episode of the TV show MythBusters, a team of researchers attempted to refute the law. They formulated it more precisely:

Even a very large dry sheet of paper cannot be folded twice more than seven times, making each fold perpendicular to the previous one.

On an ordinary A4 sheet, the law was confirmed, then the researchers checked the law on a huge sheet of paper. They managed to fold a sheet the size of a football field (51.8 × 67.1 m) 8 times without special means (11 times using a roller and a loader). According to fans of the TV show, the tracing paper from the packaging of an offset printing plate in the 520 × 380 mm format, when folded carelessly enough, can be folded eight times without effort, and nine times with effort.

Regular paper napkin folds 8 times, if the condition is violated and once folded not perpendicular to the previous one (on the video after the fourth - the fifth).

The Puzzles also tested this theory.

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V. B. Gubin

Mathematics studies the principles and results of activity in general, as if developing blanks for describing real activity and its results, and this is one of the sources of its universality.

Your attention is invited to a research program that consistently revives Neo-Pythagorean philosophy in theoretical physics and is based on the belief in the non-randomness of physical laws, in the existence of a single primary principle that determines the structure (visible and invisible) of the World and is written in an abstract mathematical language, in the language of Numbers (wholes, real and possibly their generalizations).

Richard Feynman

Imagine electric and magnetic fields. What have you done for this? Do you know how to do this? And how do I myself imagine electric and magnetic fields? What do I actually see when doing this? What is required of the scientific imagination? Is it any different from trying to imagine a room full of invisible angels? No, it doesn't seem like such an attempt.

Sizes and sizes of paper - PB "Up the hill". A sheet of paper can be folded in half a certain number of times US paper sizes

What happens if you fold the paper 7 times?

Who can fold the paper more than 7 times?

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