**Mechanics inside 1 arm wiper**

alek_hiddel: Exponential growth. Each fold doubles the thickness of the paper. If I gave you 1 penny today, and promised to give you twice as much money tomorrow, and twice as much the day after, and so on and so on, on day 30 I’m giving you over $5 million dollars.

You can do the math on this yourself pretty easily though. Open the calculator on your computer, and type in 0.0039 (the thickness of a piece of paper). Press the Times button, and then the number 2, and press enter. Now press enter 102 more times. By the mid-50’s your calculator will actually reach a point where it has to start using exponents, and by 60 you’ll have surpassed the what the calculator is capable of.

Ceilibeag: And not to be pedantic, but wouldn’t the stack of folded paper be *thicker* than the diameter of the observable universe, but it’s *width* would be 1/2^103 times the size of the original width dimension, and it’s length unchanged? The point being the *folded paper* won’t be bigger than the observable universe; only one of it’s principal dimensions (thickness).

Heynony: It’s math. Theoretical in that I think 10 times is about the most that’s ever been achieved.

More interesting to me has always been how big a piece of paper would have to be to start with, assuming it possible to be in a position to fold it 103 times. That’s even simpler math but I’ve never gotten around to it.

afcagroo: When you fold something once, you make it 2x as thick. Fold it again, you make it 4x as thick. Fold it N times, you make it 2^N times as thick. 2^103 is a very, very, very large number. (Approximately 1 followed by 31 zeros.)

So even if you are folding something very thin, if you were able to fold it that many times, it would become 10^31 times thicker.

fart_shaped_box: As others have pointed, it’s not physically possible. If it were, the surface area of the paper would also shrink by a factor of 2 each time. So you’d end up with more like one very, very, very thin rod that’s longer than the observable universe.

The long and short of it is, exponential growth is *very* fast. The difference between a single byte and a kibibyte is just 10 powers of 2. Same with going up to a mebibyte, gibibyte, tebibyte.. That’s still “only” 40 powers of 2.

HungLI5: I believe this has to be folded in half 103 times. The folded as to how it’s folded makes a difference.

kodack10: It’s not so much an arbitrary number of folds, as it is an exponential increase in thickness. Imagine it took X folds of a single sheet to be taller than the empire state building. At X-1 (one away) it would only be half as tall. At X-2 it would only be 1/4 as tall.

Another way to think about it is each fold doubles the height, thus the 20th fold creates the same increase in size as all of the 19 folds that came before, and the 100th fold creates the same increase as all the 99 folds before it.

What is interesting is when it comes to the maximum size something can be, and the minimum size, our existence is quite a bit closer to the maximum, than the minimum.

Like if you changed your size by a factor of 10, you could do this farm more times getting smaller, than you could getting larger.

LuxuriousThrowAway: Because a width of 2^103 units is inconceivably large, even if the unit is the thickness of a piece of paper.

dalongbao: I always find myself wondering but unwilling to spend time calculating: if we took into account the “excess paper” in each fold, what would the surface area of a sheet of paper folded this many times be?

By excess I mean that bit of paper that’s on the outer “side” of the fold and therefore had a longer path than the inner sheet. Therefore, folded enough times, the outer and inner layers won’t meet at the edges because it took extra paper to go around the outer edge. The question is just how much and how to account for the amount used by each layer upon each folding.

To make it interesting, I’d assume that the edges do in fact meet and we have some sheet of paper which already has all the excess paper needed. An interesting follow up would be what would the shape of this sheet of paper be if unfolded? Would it even lie flat?

Canvasch: Basically, because it doubles every fold, and when you double something that many times, it gets reeeeeally big. Of course, this is all theoretical, it is impossible to fold paper that many times. Go try it, i doubt you could even get 10 folds.

usernumber36: if you fold a paper, it’s now two layers high. if you fld it again, it’s four layers high. Fold again and it’s 8 layers high..

eventually you get enough layers that it’s thicker than the universe is wide.

If course this can never happen, because we can never have paper big enough to make those folds

Loki-L: It is the power of exponential growth. It is basically the same effect that bankrupts you with compound interest.

Most people don’t realize just how quickly and how big things can get with exponential growth, because most examples of that we see in nature break down when things become untenable.

In real life you would never be able to fold a piece of paper more than a few times or get enough rice to fit on a chessboard or anything like that. Looking at it in pure math terms however just shows hob big things can really get.

If you fold a piece of paper you double its width. There is nothing about that statement that most people wouldn’t agree with.

A piece of paper folded in half has twice the width of a single sheet and if you fold it again it has four times that width.

If you try folding it again and again in the real world, the whole pattern breaks down because it quickly becomes to small and big to fold again.

However if you take the original math and continue it unrealistically as if you really had an infinitely big piece of paper that you could fold as often as you wanted you run into problems very quickly.

For every 10 times you fold the paper it becomes a 1000 time thicker. For every 20 times you fold it, it becomes a million times thicker.

A 0.1 m thick paper folded 10 times will be a meter thick. Folded another 10 times (20 in total) it will be a kilometer thick. Folded another 10 times (30 in total) it will be a 1000 kilometer thick.

When you fold it another 10 times (40 in total) it will be a million kilometer thick and several times thicker than the moon is away from us. Another 10 times (50 in total) and you are in the outer solar system past the orbit of Jupiter.

It grows really rapidly.

the observable universe has a diameter of 8.8×10^26 m according to Wikipedia and mathematically a piece of paper 0.1mm in thickness doubled a 103 times will have a thickness of over 10^27 m.

Practically that would never work of course, but it is a good way to show just how strong exponential growth can be and that you need to beware of it.

Exponential growth breaks down before it can can get to big in most practical examples, but if the practical example is compound interest and the breakdown point is the bank taking away your home or loanshark breaking your leg it is too late.

So keep in mind that anything that growth by a set percentage over a time can grow very big very fast you are in trans-lunar orbit or have broken legs before you know it if you don’t watch out.

infinitefeedback: I’ve got a two part question for clarification. 1: is this hypothetical piece of paper larger than the conceivable universe when unfolded only or even when folded is it that big? 2: just for curiosity’s sake, what would the presumed total mass of that piece of paper be assuming the paper is average legal paper thickness

merd2k: I had PhD in pper foldering so I k so when u folds paper is width 2**n which at 103 is 2**103 big long number which is big so big that it is very big

nammertl: it’s not literal. I’ve always wondered about this as well but from what people are saying it’s just a mathematical expression.

bitJericho: I calculated based on the thickness of one carbon atom and it’s not even close. 3.5 million lightyears.

jsbarrios: A piece of paper is 0.05 mm thick.

When folded in half the thickness doubles.

So written another way 0.05 to the 103 power = 9.860761e-135

Or 9.86 with 135 zeros after

**Paintball in Mini Tanks**