Eli5: Why can you only fold a paper in half seven times, no matter the size ?
In: 4
You can’t fold it in half 7 times physically. In the end you are folding 64 pieces of paper in half at the same time and your body lacks the strength to do it.
1 in half
2 in half
4 in half
8 in half
16 in half
32 in half
64 in half
As you fold paper its thickness increases exponentially. In other words every time you fold it, it gets twice as thick. After the 6th fold its the equivalent of having 64 pieces of paper stacked on top of each other. Folding it a 7th time results in a stack 128 which is just too much to reasonable fold.
It really wants to unfold itself. Each layer has some springiness and when you have 128 layers, it adds up.
However, Mythbusters managed to fold it [11 times](https://blog.thepapermillstore.com/seven-fold-limit-myth-fact-fiction/) using a steamroller and a forklift.
Also, there may be a limit where it becomes too thick to fold without ripping. Apparently that’s more than 7 though.
You can fold it more than 7 times. That’s not a hard limit. The higher surface area to thickness ratio you have the more times you’ll be able to fold.
But the more you fold it, the more it turns into wrapping the paper around itself.
Lets say you have a square piece of paper that is 100 mm x 100 mm. Its also 1mm thick. Commonly used math would tell you the first fold would result in a 100mm x 50 mm piece that is now 2 mm thick. But that’s not correct. The fold itself is still connected together. It will contain a little over 2 mm -ish of material (top edge to bottom edge along a curve). So you have closer to 100 mm x 49 mm. (49 + 49 = 98, so 2 mm missing)
*Edit to add that the first fold doesn’t normally use that much material since you can crease the paper, which breaks the fibers so it doesn’t actually curve the entire outer diameter. Later folds don’t crease though, so its useful for explaining whats happening.
Second fold. Common math says that you will end up with 50 mm x 50 mm that is 4 mm thick. But the fold is now even worse than the first fold. Its layered. The inside layer is similar to the first and uses a similar amount of material. The outside layer has to wrap around the inside though. So its going to use up roughly twice the material as the first fold. You end up with roughly 48 mm x 49 mm that is 4 mm thick. And the edges of this fold don’t line up. Because its only the OUTSIDE layer of that fold that is 48 mm. The inside is 49. This creates a lot of stress in the paper where its still connected and trying compress and stretch to make the folds.
The more folds you do, the worse the situation gets. You have a greater difference in the amount needed to make the fold for the inside layer vs the outside layer. Very quickly the outer layer no longer has enough material to cover much of the inner layers and you eventually reach the point where the material needed just to connect the top layer to the bottom layer is more than the material available for that layer. You have 12.5 mm of material for the layer left, but the stack is now 32 mm tall.
I think the important thing to point out here is that it’s not like 7 is some universal constant of paper folding. The idea behind this thought experiment is that folding paper becomes really really hard after a relatively short amount of fold, 7 is just a nice number that is probably already hard to reach with any normal piece of paper. How many times you can fold a piece of paper depend on its size, on the type of paper, and the tools at your disposal.
I love how nobody has pointed out yet that this “problem” doesn’t apply just to paper. It actually applies to any sheet-type material (copper sheets, lead sheets, gold leaf sheets, aluminum foil). And it seems to consistently stay at or around 7 folds for human-level strength, but if you up the strength factor, you can get more than 7, although there is (and must be) an upper limit for what can be folded given infinite strength before the material separates from itself on the outside edge of the fold OR becomes incompressable on the inside edge of the fold.
As a matter of pedantry, you might also consider the difference between a fold and a roll. It’s entirely possible that it is definitionally impossible to achieve more than 7 folds because every “fold” after that is actually a half-roll. How do you contrast a fold and a roll?
The seven times rule is not real, but it is a fairly practical limit for smaller pieces of paper. If you have a piece of paper or foil that is much thinner than usual, you could fold it more times. The same is true if you have a really gigantic piece of paper. For instance Mythbusters folded a piece of paper the size of a football field 11 times.
The limit comes from the fact that when you fold the paper, the outside layer has to be longer to wrap around the inside layer, so is under more stress than the inside layer. The more folds you make, the thicker the folded paper is and the more stress/force it takes to make the outer layers wrap around the inner layers. Eventually the force required to fold more layers is so high that you won’t be strong enough to fold it, or the paper will rip before it is able to fold.
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