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Why can't paper be folded more than 7 times? Paper formats and sizes - PB "Going uphill"
Perhaps it is, if you are strong!
Have you ever tried folding a regular piece of paper? Probably yes. One, two, three times is not a problem. Then it gets harder. It is unlikely that anyone will be able to fold a standard sheet of A4 paper more than 7 times without improvised means. All this is explained by the presence of a physical phenomenon - it is impossible to fold a sheet of paper repeatedly due to the rapid growth of the exponential function.
As Wikipedia says, the number of layers of paper is equal to two to the power of n, where n is the number of folds of paper. 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 ordinary paper you can derive an equation.
Equation for plain paper:
,
Where W- width of the square sheet, t- sheet thickness and n
When using a long strip of paper, an exact length is required L:
,
Where L- minimum possible length of material, t- sheet thickness and n- the number of bends performed is doubled. L And t must be expressed in the same units.
If you take not ordinary paper with a density of 90 g/dm3 (or a little more/less), but tracing paper or even gold foil, then you can fold such material a little more times - from 8 to 12.
The 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 tools (11 times using a roller and a loader). According to fans of the TV show, tracing paper from a 520×380 mm offset printing plate package folds eight times effortlessly when folded fairly casually, and nine times with effort. In this case, 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 fold a sheet of paper not with your hands, but use a hydraulic press as your assistant? Let's see what happens then. Just keep in mind that the video is in English, with a very strong accent (Arabic Finnish).
Typically, these sizes are obtained from cut or folded paper, as well as specially made paper (for example, cards, invitations).
| Format | Width x Length (mm) | Typical Use |
| 1/3 A3 | 105 x 297 | |
| 1/3 C3 | 114 x 229 (115 x 230) | Envelope under 1/3 A3 |
| 1/3 A4 | 99 x 210 (100 x 210) | Postcard for Euro envelope |
| 1/3 C4 | Euro DL = 110 x 220 (110 x 229) | Envelope "Euro" (under 1/3 A4) |
| 1/4 A4 | 74 x 210 | |
| 1/8 A4 | 13 x 17 | |
| 1/3 A5 | 70 x 148 |
Sizes of formats according to ISO 7810
The standard defines the dimensions of identification business cards.
| Format | Width x Length (mm) |
| ID-1 (CIS, Russia) | 90 x 50 mm (less commonly 90 x 55 or 60 mm) |
| ID-1 (Europe) | 85.60 x 53.98 |
| ID-2 (A7) | 105 x 74 |
| ID-3 (B7) | 125 x 88 |
ISO 623
The standard determines the dimensions of folders for storing A4 sheets and other printed products not exceeding the dimensions of A4 format when unfolded or folded. Maximum dimensions for folded folders are given.
| Format | Width x Length (mm) |
| Regular folders without removal | 220 x 315 |
| Folders with short stem(less than 25mm) | 240 x 320 (with or without clip) |
| Folders with wide extension(more than 25mm) | 250 x 320 (without clamp), 290 x 320 (with clamp) |
ISO 838
The standard defines holes in sheets for hemming. Two holes with a diameter of 6±0.5mm. The centers of the holes are at a distance of 80±0.5mm from each other and at a distance of 12±1mm to the edge of the sheet. The holes are located symmetrically relative to the axis of the sheet.
Russian standard publication formats according to GOST 5773-90
| Paper Sheet Size (mm) | Leaf share | Symbol | Trim format (mm) | |
| maximum | minimum | |||
| Book publications | ||||
| 600x900 | 1/8 | 60x90/8 | 220x290 | 205x275 |
| 840x1080 | 1/16 | 84x108/16 | 205x260 | 192x255 |
| 700x1000 | 1/16 | 70x100/16 | 170x240 | 158x230 |
| 700x900 | 1/16 | 70x90/16 | 170x215 | 155x210 |
| 600x900 | 1/16 | 60x90/16 | 145x215 | 132x205 |
| 600x840 | 1/16 | 60х84/16 | 145x200 | 130x195 |
| 840x1080 | 1/32 | 84x108/32 | 130x200 | 123x192 |
| 700x1000 | 1/32 | 70x100/32 | 120x162 | 112x158 |
| 750x900 | 1/32 | 75x90/32 | 107x177 | 100x170 |
| 700x900 | 1/32 | 70x90/32 | 107x165 | 100x155 |
| 600x840 | 1/32 | 60x84/32 | 100x140 | 95x130 |
| Magazine publications | ||||
| 700x1080 | 1/8 | 70x108/8 | 265x340 | 257x333 |
| 600x900 | 1/8 | 60x90/8 | 220x290 | 205x275 |
| 600x840 | 1/8 | 60x84/8 | 205x290 | 200x285 |
| 840x1080 | 1/16 | 84x108/16 | 205x260 | 192x255 |
| 700x1080 | 1/16 | 70x108/16 | 170x260 | 158x255 |
| 700x1000 | 1/16 | 70x100/16 | 170x240 | 158x230 |
| 600x900 | 1/16 | 60x90/16 | 145x215 | 132x205 |
| 840x1080 | 1/32 | 84x108/32 | 130x200 | 123x192 |
| 700x1080 | 1/32 | 70x108/32 | 130x165 | 125x165 |
American paper sizes
| Format | Width x Length (mm) | Width x Length (inches) |
| Statement | 139.7 x 215.9 | 5.5 x 8.5 |
| Executive | 184.1 x 266.7 | 7.25 x 10.55 |
| Letter (Size A) | 215.9 x 279.4 | 8.5 x 11 |
| Folio | 215.9 x 330.2 | 8.5 x 13 |
| Legal | 215.9 x 355.6 | 8.5 x 14 |
| Arch 1 | 228.6 x 304.8 | 9 x 12 |
| 10 x 14 | 254 x 355.6 | 10 x 14 |
| Ledger (Size B) | 279.4 x 431.8 | 11 x 17 |
| Arch 2 | 304.8 x 457.2 | 12 x 18 |
| Tabloid | 431.8 x 279.4 | 17 x 11 |
| Size C | 431.8 x 558.8 | 17 x 22 |
| Arch 3 | 457.2 x 609.6 | 18 x 24 |
| Size D | 558.8 x 863.6 | 22 x 34 |
| Arch 4 | 609.6 x 914.4 | 24 x 36 |
| Arch 5 | 762 x 1066.8 | 30 x 42 |
| Size E (Arch 6) | 563.6 x 1117.6 | 34 x 44 |
English paper formats
Note
These sizes do not apply to the formats of albums, atlases, toy books, booklets, facsimile, bibliophile, music editions, calendars, publications produced for export, publications printed abroad, as well as miniature, unique and experimental publications.
The format of publications is conventionally designated by the size of a sheet of paper for printing in centimeters and fractions of a sheet.
The shape of the publication in millimeters is determined: for an edition with a cover - by its dimensions after trimming on three sides, for an edition under a binding cover - by the dimensions of a block trimmed on three sides, with the first number indicating the width, and the second - the height of the publication.
Maximum formats are preferred for application. It is allowed to reduce the format of the publication to the minimum height and (or) width when printing the publication on machines of outdated designs, imported equipment, as well as taking into account the technological features of production.
Maximum deviations of publication formats from those established for a given circulation should not be more than 1 mm in the width and height of the block.
We have never been able to find the original 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’s more surprising is that it belongs to the girl who mathematically substantiated this “riddle of a paper sheet.”
Of course, we are talking about real paper, which has a finite, and not zero, thickness. If you fold it carefully and completely, excluding tears (this is very important), then the “failure” to fold in half is usually detected after the sixth time. Less often - the seventh. Try this with a piece of paper from your notebook.
And, oddly enough, the limitation depends little on the size of the sheet and its thickness. That is, just taking a thin sheet of paper larger and folding it in half, since let’s say 30 or at least 15, doesn’t work, no matter how hard you try.
In popular collections such as “Did you know that...” or “The amazing thing is nearby”, this fact - that you can’t fold a piece of paper more than 8 times - can still be found in many places, online and off. But is this a fact?
Let's reason. Each fold doubles the thickness of the bale. If the thickness of the paper is taken to be 0.1 millimeters (we are not considering the size of the sheet now), then folding it in half “only” 51 times will give the thickness of the folded pack 226 million kilometers. Which is already obvious absurdity.
It seems that this is where 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 indefinitely and does not tear, but if it breaks - this is no longer folding). But still…
In 2001, one American schoolgirl decided to take a closer look at the problem of double folding, and this turned out to be a whole scientific study, and even a world record.
Britney Gallivan (note that she is now a student) initially reacted like Lewis Carroll's Alice: "It's no use trying." But the Queen said to Alice: “I dare say that you haven’t had much practice.”
So Gallivan started practicing. Having suffered quite a bit with various objects, she finally folded a 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 something in half 12 times!” Like, make sure that this is something completely impossible.
An example of folding a sheet in half four times. The dotted line is the previous position of 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 they do not occupy the same position as it might seem at a quick glance (illustration from the site pomonahistorical.org).
The girl did not calm down at this. In December 2001, she created a mathematical theory (or mathematical justification) for the double folding process, and in January 2002, she performed 12 folds in half with paper, using a number of rules and several folding directions.
Britney noted that mathematicians had already addressed this problem before, but no one had yet provided a correct and practice-tested solution to the problem.
Gallivan became the first person to correctly understand and justify the reason for the restrictions on addition. She studied the effects that accumulate when folding a real sheet and the “loss” of paper (and any other material) to the fold itself. She obtained equations for the folding limit for any initial sheet parameters. Here they are.
The first equation applies 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 double folds made. 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 doubling each time). Here W is the width of the square sheet. The exact equation for folding in "alternate" directions is more complex, but here is a form that gives a very close result.
For paper that is not square, the above equation still gives a very accurate limit. If the paper has, say, 2 to 1 proportions (in length and width), it is easy to figure out that you need to fold it once and “reduce” it to a square of double thickness, and then use the above formula, mentally keeping in mind one extra fold.
In her work, the schoolgirl defined strict rules for double addition. For example, a sheet that is folded n times must have 2n unique layers lying in a row on one line. Sections of sheets that do not meet this criterion cannot be counted as part of the folded bundle.
So Britney became the first person in the world to fold a sheet of paper in half 9, 10, 11 and 12 times. One might say, not without the help of mathematics.
We have never been able to find the original 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’s more surprising is that it belongs to the girl who mathematically substantiated this “riddle of a sheet of paper.”Of course, we are talking about real paper, which has a finite, and not zero, thickness. If you fold it carefully and completely, excluding tears (this is very important), then the “failure” to fold in half is usually detected after the sixth time. Less often - the seventh. Try this with a piece of paper from your notebook.
And, oddly enough, the limitation depends little on the size of the sheet and its thickness. That is, just taking a thin sheet of paper larger and folding it in half, let’s say 30 or at least 15, doesn’t work, no matter how hard you try.
In popular collections such as “Did you know that...” or “The amazing thing is nearby”, this fact - that you can’t fold a piece of paper more than 8 times - can still be found in many places, online and off. But is this a fact?
Let's reason. Each fold doubles the thickness of the bale. If the thickness of the paper is taken to be 0.1 millimeters (we are not considering the size of the sheet now), then folding it in half “only” 51 times will give the thickness of the folded pack 226 million kilometers. Which is already obvious absurdity.
It seems that this is where 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 indefinitely and does not tear, but if it breaks, this is no longer folding). But still…
In 2001, one American schoolgirl decided to take a closer look at the problem of double folding, and this turned out to be 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 something in half 12 times!” Like, make sure that this is something completely impossible.
Britney Gallivan (note that she is now a student) initially reacted like Lewis Carroll's Alice: "It's no use trying." But the Queen said to Alice: “I dare say that you haven’t had much practice.”
So Gallivan started practicing. Having suffered quite a bit with various objects, she finally folded a sheet of gold foil in half 12 times, which put her teacher to shame.
The girl did not calm down at this. In December 2001, she created a mathematical theory (well, or a mathematical justification) for the double folding process, and in January 2002, she did a 12-fold folding in half with paper, using a number of rules and several folding directions (for math lovers, a little more detail -).
Britney noted that mathematicians had already addressed this problem before, but no one had yet provided a correct and practice-tested solution to the problem.
Gallivan became the first person to correctly understand and justify the reason for the restrictions on addition. She studied the effects that accumulate when folding a real sheet and the “loss” of paper (and any other material) to the fold itself. She obtained equations for the folding limit for any initial sheet parameters. Here they are:
The first equation applies 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 double folds made. 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 doubling each time). Here W is the width of the square sheet. The exact equation for folding in "alternate" directions is more complex, but here is a form that gives a very close result.
For paper that is not square, the above equation still gives a very accurate limit. If the paper has, say, 2 to 1 proportions (in length and width), it is easy to figure out that you need to fold it once and “reduce” it to a square of double thickness, and then use the above formula, mentally keeping in mind one extra fold.
In her work, the schoolgirl defined strict rules for double addition. For example, a sheet that is folded n times must have 2n unique layers lying in a row on one line. Sections of sheets that do not meet this criterion cannot be counted as part of the folded bundle.
So Britney became the first person in the world to fold a sheet of paper in half 9, 10, 11 and 12 times. One might say, not without the help of mathematics.
On January 24, 2007, in the 72nd episode of the TV show “MythBusters,” a team of researchers tried 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.
The law was confirmed on an ordinary A4 sheet, then the researchers tested the law on a huge sheet of paper. They managed to fold a sheet the size of a football field (51.8x67.1 m) 8 times without special tools (11 times using a roller and a loader). According to fans of the TV show, tracing paper from a 520×380 mm offset printing plate package folds eight times effortlessly when folded fairly casually, and nine times with effort.
An ordinary paper napkin is folded 8 times, if you violate the condition and fold once not perpendicular to the previous one (on the roller after the fourth - the fifth).
Headgear also tested this theory.
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