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While procrastinating around the web I stumbled on a page that contained the image below, from cracked.com.

I can't help but believe that this is false… Even though the article header says:

22 Statistics That Will Change The Way You See the World

My question: is what the image below implies a mathematical impossibility? (…Just for procrastination's sake…)

mathematical-impossibility

If you could fold a piece of A4 paper just 42 times it would be thick enough to reach the moon

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21  
No, because of the finite thickness of an A4 paper, you cannot fold it 42 times. As a junior in high school Britney Gallivan has worked out a theorem of the upper bound of number of folds of a piece of paper given its thickness and width. –  achille hui Mar 27 at 14:13
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Instead of folding, imagine that you cut the piece of paper in half -- the short way -- and stacked it. Then cut the stack into two -- the short way -- and stacked it. And so on. If you did that 42 times then you'd end up with an extremely large quantity of extremely fine confetti that would stack up quite high. Of course, since it would be finer than fine dust, it would blow away before it got to the moon. In fact, each piece of confetti would have only a few atoms in it. You certainly couldn't do this a 50th time; you'd have to split atoms. –  Eric Lippert Mar 27 at 18:10
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@EricLippert In fact, you couldn't even get down to individual items. Once you reach the point where you have only a few monosaccharide units per piece, it's debatable whether they can still be called cellulose fibers, let alone paper. –  Superbest Mar 27 at 19:50
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@EricLippert instead of cutting the paper you could just go to the office store and pick up a few packs of A46 paper. –  CompuChip Mar 28 at 8:03
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The strange thing about the claim is that "A4" does not imply anything about how thick the paper is .... –  Henning Makholm Mar 28 at 19:36

11 Answers 11

Even if the sheet of paper were infinitely foldable, the answer is that no, you can't reach the moon by folding a sheet of A4 paper any number of times, for a reason that bears calling out (and in fact explains why a sheet of paper that size can only be folded a certain number of times — that is, why it's impossible to fold it 42 times in the first place): consider the last fold and imagine looking at the sheet in a cross-section perpendicular to this fold. The 'faces' of the folded paper that are at the top and at the bottom after the last fold must be connected along the fold edge, since they were part of a single 'face' before the fold — but this means that the distance along the paper between the top and bottom must be at least as long as the distance 'through' the paper on a straight line between them. In other words, you need to start with a sheet of paper that's at least 385,000km along at least one direction (using Sabyasachi's numbers) to be able to reach that far, regardless of what sequence of folds you use.

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8  
+1 but and why do you assume I don't know people who can supply that kind of paper? ;) –  Sabyasachi Mar 27 at 15:35
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@Sabyasachi that kind of paper would not be called "A4" anymore. –  Paŭlo Ebermann Mar 27 at 16:40
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@PaŭloEbermann don't you challenge my beliefs. –  Sabyasachi Mar 27 at 16:44
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For the record, your argument is inconsistent because you start by assuming an infinitely foldable sheet but then contradict that by proving that there cannot be an infinitely foldable sheet. As others have pointed out, the statement "if you could fold a sheet 42 times, it could reach the moon" is implicitly true. Thanks for the explanation though of why there is a folding limit in the first place. –  imjosh Mar 27 at 19:09
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@imjosh "because you start by assuming an infinitely foldable sheet but then contradict that by proving that there cannot be an infinitely foldable sheet." This is just proof by contradiction, and it's certainly a valid line of reasoning, no? –  AmadeusDrZaius Mar 27 at 20:34

The statement is true in two different senses. As Sabyasachi shows, the intended sense that $2^{42}$ times the thickness of a sheet of paper is greater than the distance to the moon is correct. In the spirit of achille hui's comment, the sentence is an implication with a false antecendent, so it is true in that sense as well. It is also true to say "If you could fold a piece of A4 paper 42 times then the moon is made of green cheese."

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14  
+1 for "green cheese" –  Sabyasachi Mar 27 at 14:19
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A point regarding the last statement "If $false$, then $anything$": Given that the question is real life-motivated and not about a mathematical theory or axiomatic framework, I think it's not good a idea to internalize the accidental flaws of the connective "$\Rightarrow$". Logicians and philosophers work hard for more than 100 years to improve the situation, e.g. via Relevance logic. The material implication $\Rightarrow$ is just sufficiently convenient for doing much of math, it can be translated into a binary operation in Boolean algebra etc. –  NikolajK Mar 28 at 9:25
    
NiftyKitty: Good point. The "would" (replaced by Ross with "is") specifically asks us to consider relevance. –  shannon Mar 28 at 11:58
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The image is false from the two points of view, even if the person who made it was "inspired" by a theoretically true (but practically unfeasible) statement. Here are the errors : The text says "if you could fold a piece of paper 42 times", but it does not say how it is folded. It need to be folded in such a way that the thickness doubles at every fold. The drawing is also incorrect as it shows an accordeon fold, so in this case the final thickness would be 42*thickness and not (2^42)*thickness. This is quite a difference, isn't it. –  radouxju Mar 28 at 13:53
    
You're conflating the subjunctive mood with propositional logic. It's true that in propositional logic, the statement "if you can fold a piece of A4 paper 42 times, then the moon is made of green cheese" is true. But the phrase "if you could" implies that the first part of the statement (folding a piece of A4 paper 42 times") necessitates the truth of the latter part of the statement. –  Kyle Strand Mar 28 at 19:31

Unless you tear the paper while you fold it, no two points of the paper can become farther from each other (in three dimensions) after folding than when the paper was flat.

Okay, perhaps there is some give in the paper, so let's generously say the folded paper forms a Lipschitz continuous embedding of the original flat paper into physical space, with Lipschitz constant $2$.

This still means that no two points on the folded (or scrunched or whatever) A4 paper can be farther apart than twice the diagonal of the flat paper, or about 72 centimeters. That's a far way from the distance to the moon.

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This comment is severely underrated. Many of the "well, theoretically, yes" answers are really pretty poor, considering that this is a very clear "well, theoretically, NO". I'm a little disappointed in a bunch of upvotes for naively multiplying the thickness of the paper by 2^42. –  msouth Mar 30 at 1:59
    
This is an even clearer way of making my point that doesn't rely on any specific properties of the fold in question - very well-put. –  Steven Stadnicki 15 hours ago

In my experience a standard sheet of paper, has thickness around $0.1$ mm.

Folding $42$ times, the thickness is,

$$2^{42}\times0.1\approx 439804 \,km$$

Wolfram Alpha tells us that the average distance is, $385000$ kilometers which makes the claim most certainly valid.

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7  
+1 Ah, the power of the exponent... –  imranfat Mar 27 at 14:17
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The shortest side (assuming you always fold along the widest side) would be $210/2^{21}\approx0.0001$ millimeters long, that is $0.1$ micrometers. ;-) –  egreg Mar 27 at 15:22
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Note that A4 paper has an area of $2^{-4}m^2$, so the folded (or let's better assume cut) paper tower would have an area of $2^{-46}\mathrm m^2=(2^{-23}\mathrm m)^2\approx (10^{-7}\mathrm m)^2=(0.1\mathrm\mu\mathrm m)^2$. –  Carsten Schultz Mar 27 at 15:26
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@CarstenSchultz why is everyone so seriously assuming that I plan to go to the moon by folding paper? –  Sabyasachi Mar 27 at 15:31
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Just to verify, I measured a package of 500 sheets of paper, and it was about 5 cm thick. So, 0.1 mm per sheet is a good estimate. –  Dan Mar 28 at 5:01

Just for the sake of discussion, lets consider how skinny the paper would get after folding it 42 times.

A sheet of A4 paper is 30 cm long. If you fold it in half 42 times and alternate directions, you'll get down to a length of 30 cm / 2^21 = 1430 angstroms. ("cut in half" might be more accurate.) Your paper would mathematically reach the moon, but since paper is made of long cellulose fibers (thousands of units), it wouldn't really be paper any more. The dimensions of the paper would be under the length of a single cellulose fiber.

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1  
Indeed. Why reach the moon by folding it and not by stretching it? –  mirkastath Mar 28 at 12:42
    
According to www1.lsbu.ac.uk/water/hycel.html#mol it looks like 1430 angstroms is approximately the minimum length of a single fiber. –  Kyle Strand Mar 28 at 19:55

I have been reading all these theoretical answers, but no one made a comment about taking a piece of paper and actually doing the folding.

I bet that you can not fold a standard sheet of paper (75 g/cm2) with your bare hands more than 6-7 times. And you will end up with a total height of about 1cm.

If you ask a group, most of them will think we can fold the paper 20, 30, 40 or even more times.

This exercise is a good one to show the disparity between the physical world and the abstraction of it inside our mind.

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aha, don't think about, just do it! –  vp_arth Mar 27 at 20:15
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idk...@vp_arth can't you just fold it with your mind? Not sure about you but my mind powers are pretty dang strong. :p –  jollypianoman Mar 27 at 22:40
    
Interesting point. This exercise is a good one to show the disparity between the physical world and the abstraction of it inside our mind. When I was at school, a teacher asked the folding paper question to a group, and most people thought the right number should be around 40-50 times..... Pretty high folding paper mind powers for the real world..... :) –  Enrique Haro Mar 27 at 23:10

It depends on whether the sheet can be compressed.

The other answers all assume that the sheet has a constant thickness. If this is the case, then consider "folding" the paper in such a way that we don't care if the edge of the crease rips; i.e., we ignore the paradox demonstrated in Steven Stadnicki's answer. So our "folded" stack is really equivalent to cutting the paper into tiny rectangles and stacking them, as long as with each cut we separate every rectangle into two new rectangles (i.e. we double the number of rectangles each time). (This is a pretty loose definition of "folding," of course, but we're trying to reach the moon with a piece of paper, so that's hardly surprising.) If we use this definition of "folding", and we're able to perform the cuts at the atomic level and ensure that all the rectangles are perfectly stacked on top of one another, and the rectangles still have the same width as the original piece of paper (which is, at this point, a ridiculous assumption; see blah's answer), then yes, we'll reach the moon (as per Sabyasachi's answer).

If, however, the pressure created by making the folds (and cutting the paper into tiny rectangles and whatnot) compresses the paper so that it becomes less than ~0.1mm thick, then our exponent will no longer be valid. Say that during the cutting process, the cellulose fibers unravel somewhat, leaving only two layers of fiber. Since the fibers are 2-20 nm in diameter, let's say that the two-fiber-layer sheets are about 10nm thick. $2^{42} \times 10nm = 43,980 m$, which, according to Wolfram Alpha, is about five times the height of Mount Everest. Impressive, but only about 1.14% of the distance to the moon.

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I think the question has been misunderstood by those who offer a straight "no".

As already pointed out, it has been proven that you can only fold paper 7 times, if you always fold it in half every other direction.

You can fold it 11 times if you always fold it in the same direction, see here:

http://en.wikipedia.org/wiki/Britney_Gallivan

But if you fold it like an accordion, as demonstrated in the sketch, well, who says you cannot reach the moon? :-)

I don't think we have the machinery to perform such a fine folding task--see also @blah's comment. Anyone interested?

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OK, so you fold it like an accordion and you end up with a cylinder of folded paper that reaches to the moon. The original length of your paper is now oriented in two directions: some parts are parallel to the base of the cylinder and some are parallel to the long sides. However, your original piece of paper was only 297mm long so the total amount of that length that is parallel to the long sides of the cylinder cannot be more than 297mm, so your cylinder is at most 297mm tall. That leaves you approximately 384399.999703km short of the moon. Better buy some more paper! –  David Richerby Mar 28 at 12:13
    
@DavidRicherby Thank you for solving this ridiculous puzzle! –  mirkastath Mar 28 at 12:39

Mathematically, it is possible. however realistically you will loss paper every time you fold, because you have to count the width of the paper being folded.

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It is mathematically possible to fold something to a length longer than it is? Even given an idealized piece of paper. Think about it. After folding an idealized piece of paper can it be longer than it started out? –  msouth Mar 30 at 2:01
    
@msouth well, i'm no mathematician, but when you fold it, it decreases x and y and increases height. so it looks thicker, but if you keep going, then if the paper could maintain its structural integrity which it couldn't, it'd start looking like a piece of elastic or rather, a thin strand. And if it were theoretically possible to stretch an "elastic band" very far, I wonder how long till one splits an atom? –  barlop Mar 30 at 8:15
    
@barlop what I was getting at is that, say you have a solid object with a length, width, and a height. Find the longest straight line inside that volume, call it the "diameter" of the volume. Then, start folding that volume. Once you have folded it, the longest straight line contained in that volume will be less than or equal to the diameter. Folding something like paper doesn't increase its diameter. You might increase the measurement in one direction (e.g. you can increase the "thickness" measurement by folding it in half), but you do so at the expense of some other dimension. –  msouth Mar 31 at 11:32
    
@msouth yes that's what i'm saying, and as you say "you do so at the expense of some other dimension." hence I said, it ends up like a thin strand(of what was paper and is 'now' I don't know what), and you end up with a thin strand, because it is SO (as you say) at the expense of the other dimensions. –  barlop Mar 31 at 18:50
    
@barlop No, we're still not talking about the same thing. Changing it to a thin strand is either via cutting into teeny tiny strips or stretching, neither of which is what is commonly understood by the idea of "folding". It's not just that it's any transformation you want at the expense of the other dimensions--you can't increase the diameter by folding. If you fold, you will decrease the diameter. The longest "paper stick" you can fit inside the boundaries of the folded object is always shorter after a fold. –  msouth Apr 1 at 19:24

There are $2$ more problems most people did not adress here.

  1. If you folded a paper $42$ times, its area would decrease accordingly, and after $42$ folds, the area in contact with earth would be $1.4\cdot 10^{-14}$ metres. This means you would have a column of paper roughly $0.1$ of a micrometer in width (if it was square). If you always folded paper over the same axis (keeping the width constant in one direction), this would mean the paper is still less than $10^{-13}m$ thick in the other dimension. This is smaller than one atom.

  2. A column of paper reaching to the moon would have to spin along with the earth. Its centre of gravity would be $219,902,325$ kilometres above the surface of the earth, way above the geostacionary satelite height. This means the paper would actually be pulled upwards by the centrifugal force of earth's spinning.

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If, however, you alternate folding directions, then the rectangles you produce are juuuuuuuust wide enough to be still "papery," probably, let alone bigger than a single atom. See blah's and my answers. –  Kyle Strand Mar 31 at 17:10

If folding it 42 times is possible then indeed it is!

public static void main(String[] args){
    long thickness = 1l;
    for(int i=1; i<=42; i++)
        thickness = thickness * 2l;
    System.out.println("thickness = : "+thickness+" x [thickness of paper]");
}
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where fold radius corrections? also distance between paper layers can't be zero... :) –  vp_arth Mar 27 at 20:18
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You know Math.pow(2, 42) exists, right? –  Rahul Mar 28 at 0:35
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1L << 42 is faster than calling a function ;-) –  Dan Mar 28 at 5:02
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@Dan bits power :D –  Silviu Burcea Mar 28 at 7:43
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Nice example of how Java programmers are trained by their language to take the long way around :) –  CompuChip Mar 28 at 8:02

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