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folding a piece of paper
Can You Really Only Fold a Paper Seven Times?
Everybody has met someone who randomly puts forth a claim like “a piece of paper can only be folded on itself seven times.” Because YouTube wasn’t a thing when I was young, as soon as I heard this claim, I ripped out a piece of paper and started folding it. I folded in half once, then twice, and so on until I got to 7 folds. After the seventh fold, my classmates and I could not fold the paper an eighth time, no matter what we tried. We stepped on it, put it under a table with six of us on it, ran it over with a bike, and smashed it with the classroom door to no avail. All the while, the friend who sent us on this wild rampage just sat back and grinned, looking distantly to the future, imagining the Nobel prize he would win.
I believed in this fact until my sophomore year in university, when my mathematics professor mentioned the bedsheet problem and told us about Britney Galivan, who in 2002 had folded a massive sheet of paper 12 times, cementing her name in the Guinness Book of World Records. That day, not only had I learned that my friend’s bet was false, but that there was another Britney other than Britney Spears. Again, this was before the internet age, and Britney Spears was the only Britney I had heard about.
Instead of approaching the problem by folding the paper in two, Britney Gallivan devised another tactic. This is the principle that mathematical thinking teaches us, anyway. If you cannot solve a problem one way, no matter how many times you try, isn’t it best to find another method?There is a mathematical explanation for how Gallivan folded a piece of paper more than seven times; she even explains it below. However, before we dive into how it was done, we should explore why it even gets harder to fold the paper the more we do it.
Why is it so hard to fold a piece of paper this many times?
Folding a piece of paper isn’t inherently hard. The challenge arises when we try to fold it many times. This phenomenon occurs because as we fold the paper, it gets stronger and more resilient to folding. This resilience doesn’t increase linearly either. We call this behavior exponential growth. We will use the example below to solidify this in one’s imagination.
Every piece of paper has a thickness. For simplicity, I will assume that a piece of notebook paper will always be .01 cm in thickness. Now, if we fold this in half on itself, the result is a paper with a thickness of .02 cm. Now, we will fold it six more times and get the following thicknesses in order: .04 cm, .08 cm, .16 cm, .32 cm,.64 cm. Notice that the final thickness is nearly 128 times the starting thickness of the paper. You can also think of this as folding a 128-page notebook in half.Now, I can see you claiming that you can easily fold a 128-page notebook in half. However, you must understand that not only is our piece of paper 128 times its original thickness, but its area is also too small to apply enough force to fold it. This problem doesn’t arise with the notebook as only its thickness is fighting against us.
The question is whether someone with superhuman strength can fold the paper more than seven times. Yes, they can. Assuming that the paper doesn’t rip, people have proven that it can be folded 8, 9, or even 12 times.
I imagine now that you are asking yourself, “if any thickness of paper can be folded in half with enough strength, why hasn’t anyone folded one more than 12 times?” The answer, again, lies in exponential growth. If you remember from above, a piece of paper folded seven times had a thickness of .64 cm. If we continue folding it in half five more times, we get the following thicknesses: 1.28 cm, 2.56 cm, 5.12 cm, 10.24 cm, and 20.48 cm. As you can imagine, this is the width of a very thick book.
Although there is no physical proof, I imagine that if we want to fold a piece of paper at 20.48 cm thick one more time, we must start with a piece the size of a football field. While I imagine this is possible, nobody would be willing to produce a football field-sized piece of paper for you.
As a side note, folding a piece of paper is a very fun way to teach exponential growth. Folding paper is an extremely fast way to travel through outer space. If you think about it, at 12 folds, the paper is 20.48 cm. Following the pattern:
At fold 13, the paper is 40.96 cm thick.
At fold 14, it is 81.92 cm thick.
At fold 15, it is 163.84 cm thick.
At fold 16, it is 327.68 cm thick.
At fold 17, it is 655. 36 cm thick. (6.55 meters)
At fold 18, it is 1310.72 cm thick. (13.1 meters)
At fold 19, it is 2621.44 cm thick. (26.2 meters)
At fold 20, it is 5242.88 cm thick. (52.4 meters)
At fold 21, it is 10485.76 cm thick. (104.8 meters)
At fold 22, it is 20971. 52x cm thick. (209.7 meters)
At fold 23, it is 41943.04 cm thick. (419.4 meters)
At fold 24, it is 83886.08 cm thick. (838.9 meters)
At fold 25, it is 167772.16 cm thick. (1677.7 meters)(1.7 km)
At fold 26, it is 335544.32 cm thick. (3355.4 meters)(3.4 km)
At fold 27, it is 671088.64 cm thick. (6710.9 m)(6.7 km)
At fold 28, it is 1342177.28 cm thick. (13421.7 m)(13.4 km)
At fold 29, it is 2684354.56 cm thick. (26843.5 m)(26.8 km)
At fold 30, it is 10737418.24 cm thick. (107374.1 m)(107.3 km)If you continue this brain teaser, you’ll find that by fold 42, you have reached the moon, and by fold 51, you have arrived at the Sun itself. Technically the paper would burn far before we reach the Sun but let’s assume we have a magical fireproof paper. I’d love to end this scenario here, but I must make one final remark. At fold 81, the thickness of the paper will have reached 127.786 lightyears in width. Knowing that one light year equals 9 trillion kilometers, I’ll leave you to imagine that distance.
Imagining even for a moment that God’s power is neverending and that he could fold a paper this many times has fried my brain.Now, onto the mathematics behind paper folding.
Firstly, paper folding doesn’t have a universal theorem. Just as the problem of who gets more wet in the rain, the man who runs or the man who walks, which I previously covered in an article, does not have a definitive answer, neither does the paper folding question. That is because to answer the question we must take into account endless parameters such as the thickness of the paper, the size of the paper, the material it is made from, the process by which it is made, whether or not the paper will tear, and so forth.
Knowing this, how did the myth that a piece of paper can only be folded seven times arise? The answer to this question is quite simple. The most widely used paper format is A4, which is what most people think about when they mention this myth. If we are bound to the confines of a piece of A4 paper, then yes, it can only be folded seven times before you need a ridiculous amount of force. Even with that amount of force, the issue of the paper tearing arises.That doesn’t mean that all pieces of paper can only be folded seven times, though. In fact, if you have a much larger piece of paper and a press machine, you can fold it many more times.
Above I mentioned that there is no universal theorem concerning paper folding. However, we can formulate an equation for this problem. The second Britney I ever learned about, Britney Gallivan, achieved this as a high school student.
In 2002, during Gallivan’s Honors Pre-Calculus, her teacher gave the students each a piece of paper and asked them if they could or could not fold it more than seven times. Taking the challenge to heart, Gallivan quickly began folding. Shortly after, she hit the road bump many of us have in our childhood, the seven folds limit. However, Gallivan also concludes that she would need a much larger piece of paper, but how large? Gallivan, taking into account that the size of the paper shrinks as she folds it, sets out to formulate an equation for this problem and finds the below solution.
In this formula, n is the number of folds, and t is the thickness of the paper. When you substitute real values for n and t, W is the length of the paper needed to achieve the folds. For example, to fold a .01 cm thick piece of paper 12 times, you would need a piece of paper 29.12 meters in length. As I mentioned before, however, this method is not very logical as the probability of the piece of paper tearing is very high.That is why Gallivan discovered a different method and formulated a more logical equation that considered the paper’s length and thickness. In this new method, she decided to fold the paper in the same direction every time.
With this new method came a much larger problem, however. While the paper would not tear with this new formula, she would need a piece 880 meters in length to achieve the folds. Yet there was no piece of paper that long. Using her practical knowledge, Gallivan came up with the genius solution to use a piece of “paper” that came in long rolls, toilet paper. Gallivan took six rolls of toilet paper and attached them so they would not tear. Going to a massive space, she unrolled the paper, and the rolls came to exactly 880 meters in length. Gallivan was exhausted after the first fold as she needed to walk 440 meters to make it. For the second fold, she walked 220 meters. Every fold she made, the distance needed to walk for the next one was halved. By fold 12, her job had become much easier as she only had to walk 40 centimeters.To wrap up, I congratulate and honor Britney Gallivan not because she broke a record and became a part of the Guinness Book of World Records but because when her teacher asked her a question, she obsessively chased after its solution.
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