A sheet of paper can be folded in half no more than a certain number of times.

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, if you fold a piece 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 becomes interesting. And many begin to try to fold sheets of paper: a notebook sheet, a standard A4 sheet, newspaper strips, napkins. Luckily everyone has paper at hand. AND Why can't paper be folded more than 7 times??

What happens when you fold paper 7 times?

Already when adding up for the fifth time, you begin 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 a paper multi-layered “rectangle”, which we can’t fold further in half.

There are many questions. Does such a limitation exist? Is there a limit to paper folding in half? And most importantly Why can't paper be folded more than 7 times?
Except practical way answer to this question, it is possible to explain the "phenomenon" theoretically. Let's try to count how many layers there are in this piece of "unyielding paper. First there was a single sheet of paper, then 2 layers, then 4 and so on. With a five-fold addition, we get already 32 layers, 6-fold 64, 7-fold - 128!. That is, with the eighth addition, we must simultaneously bend 128 layers of paper! Here's the thing, the number of layers of paper is growing exponentially. It is unlikely that anyone will be able to fold such a multi-layered “pie” the first time.

Who can fold paper more than 7 times?

But there were people who tried to refute such a statement. They reasoned like this: the larger the size of the original paper, the easier it will be to fold it later. It really is. Indeed, with an increase in the size of the paper, the shoulder of the force increases, 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, the researchers take sheets of paper as large as possible in area (up to the size of a football field) and fold it. True, at the same time they have to use technical means (skating rink and loader). In this experiment, they managed to fold the paper in half 8 times by hand, 11 times with the help of machinery.

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) folds 8 times, with effort.

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

For a long time there has been such a widespread theory that not a single sheet of paper can be folded twice more than seven (according to some sources - eight) times. The source of this statement is already difficult to find. Meanwhile, the current folding record is 12 times. And what is more surprising, it belongs to the girl who mathematically substantiated this “mystery of the paper sheet”.

Of course, we are talking about real paper, having a finite, not zero, thickness. If you fold it carefully and to the end, excluding breaks (this is very important), then the “refusal” to fold in half is detected, usually after the sixth time. Less often - the seventh.

Try to do it yourself 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 take a larger thin sheet, and fold it in half, let's say 30 or at least 15 times - it doesn't work, no matter how you fight.

In popular collections, such as "Do you know what ..." or "Amazing is 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 beyond. But is it a fact?

Let's reason. Each addition doubles the thickness of the bale. If the thickness of the paper is taken equal to 0.1 millimeters (we do not consider the size of the sheet now), then folding it in half “only” 51 times will give the thickness of the folded pack of 226 million kilometers. Which is an obvious absurdity.

Britney Gallivan world record holder paper tape folded in half (in one direction) 11 times

It seems that here we begin to understand where the well-known restriction of 7 or 8 times comes from (once again - we have real paper, it does not stretch to infinity and does not tear, but it 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.

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 that this is from the category of completely impossible.

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

So Gallivan took up the practice. Having suffered quite a bit with various objects, she folded a sheet of gold foil in half 12 times, which put her teacher to shame.

An example of folding a sheet in half four times. The dotted line is the previous position of the triple addition. The letters show that the points on the surface of the sheet are displaced (that is, the sheets slide relative to each other), and as a result, do not take the same position as it might seem at a cursory glance.

This girl did not calm down. In December 2001, she created a mathematical theory (well, or mathematical justification) for the process of double folding, and in January 2002 she did a 12-fold folding in half with paper, using a series of rules and several folding directions (for lovers of mathematics, a little more - here) .

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

Gallivan was the first person to correctly understand and justify the reason for the limits on addition. She studied the effects that accumulate when a real sheet is folded and the “loss” of paper (and any other material) on the fold itself. She obtained equations for the folding limit, for any given leaf parameters. Here they are.

The first equation refers to folding the strip in only one direction. L is the minimum possible length of the material, t is the thickness of the sheet, and n is the number of doubled folds. 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 realistic result.

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

In her work, the student determined strict rules 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 considered as part of a folded stack.

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

And in 2007, the MythBusters team decided to fold a huge sheet, the size of half a football field. As a result, they were able to fold such a sheet 8 times without special means and 11 times with a roller and loader.

And something else interesting:

sources

We have never been able to find the source of this widespread belief: no sheet of paper can be folded twice more than seven (according to some sources - eight) times. Meanwhile, the current record for folding is 12 times. And what is more surprising, it belongs to the girl who mathematically substantiated this “mystery of the paper sheet”.

Of course, we are talking about real paper, having a finite, not zero, thickness. If you fold it carefully and to the end, excluding breaks (this is very important), then the “refusal” to fold in half is detected, usually after the sixth time. Less often - the seventh. Try doing 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 take a larger thin sheet, and fold it in half, let's say 30 or at least 15 times - it doesn’t work, no matter how you fight.

In popular collections, such as "Do you know what ..." or "Amazing is 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 beyond. But is it a fact?

Let's reason. Each addition doubles the thickness of the bale. If the thickness of the paper is taken equal to 0.1 millimeters (we do not consider the size of the sheet now), then folding it in half “only” 51 times will give the thickness of the folded pack of 226 million kilometers. Which is an obvious absurdity.

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

It seems that here we begin to understand where the well-known limitation of 7 or 8 times comes from (once again - our paper is real, it does not stretch to infinity and does not tear, but it 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.

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 that this is from the category of completely impossible.

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

So Gallivan took up the practice. Having suffered quite a bit with various objects, she folded a sheet of gold foil in half 12 times, which put her teacher to shame.

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

This girl did not calm down. In December 2001, she created a mathematical theory (well, or mathematical justification) for the double-folding process, and in January 2002, she did a 12-fold folding in half with paper, using a series of rules and several folding directions (for lovers of mathematics, a little more -).

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

Gallivan was the first person to correctly understand and justify the reason for the limits on addition. She studied the effects that accumulate when a real sheet is folded and the “loss” of paper (and any other material) on the fold itself. She obtained equations for the folding limit, for any given leaf parameters. Here they are.

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

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 "alternative" directions is more complicated, but here is a form that gives a very realistic result.