# Paper folding math **URL:** \ **Category:** Factual Questions\ **Created:** [December 14, 2015, 5:50am UTC](https://boards.straightdope.com/t/paper-folding-math/740397 "2015-12-14T05:50:10Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Lucas\_Jackson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lucas_jackson/32/303_2.png) [@Lucas\_Jackson](https://boards.straightdope.com/u/Lucas_Jackson)\ **Post date:** [December 14, 2015, 5:50am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/1 "2015-12-14T05:50:10Z")
I hesitate to even ask this due to the source (quora), but they claim that: “2. If you could fold a piece of paper in half on itself 100 times, the height of the resulting folded paper would not fit within the current known universe.” That seems incredible. Is it even remotely true? Link: [https://www.quora.com/?digest\_story=1287380](https://www.quora.com/?digest_story=1287380) ---
**Author:** ![engineer\_comp\_geek](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/engineer_comp_geek/32/504_2.png) [@engineer\_comp\_geek](https://boards.straightdope.com/u/engineer_comp_geek)\ **Post date:** [December 14, 2015, 6:07am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/2 "2015-12-14T06:07:09Z")
It’s kinda true, and yet impossible. Every time you fold the paper in half, you double its thickness. So the first fold makes it 2 times as thick, then 4, then 8, then 16, then 32, then 64, etc. After 8 folds, the paper is 256 times as thick, the thickness of a decent sized book. After 16 folds, the paper is 65,536 times as thick. It gets a little difficult to imagine exactly how tall a stack of roughly 65 thousand sheets of paper is. After 32 folds, the paper is over 4 billion sheets thick, and we’re nowhere near done yet. So yeah, depending on the thickness of your paper, somewhere around 100 folds gets you in the neighborhood of the width of the observable universe. All we are doing though is taking the thickness of the paper and multiplying it by 2 to the power of x, where x is your number of folds. In the real world, you’re going to have a very difficult time folding the paper more than about 7 or 8 times, depending on the thickness of the paper. If you try to imagine what your piece of paper would look like, it becomes a lot more obvious about exactly how silly this is. When you fold the paper in half, it becomes half the size when looking down at its surface area. Fold it in half again, and it’s 1/4th. As the thickness of your folded paper goes up, the surface area when viewed looking down on it goes down by the same factor. So your paper that you folded 16 times is 65,536 times as wide, but 1/65,536th the surface area. Your “paper” is going to very quickly look more like a square rope, getting longer but narrower with each fold. Obviously, that’s not going to happen. This is why you aren’t going to get 16 folds out of it, let alone a hundred. This is called “exponential growth”. You can google that term for more details. ---
**Author:** ![RadicalPi](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/radicalpi/32/9806_2.png) [@RadicalPi](https://boards.straightdope.com/u/RadicalPi)\ **Post date:** [December 14, 2015, 6:13am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/3 "2015-12-14T06:13:34Z")
Well, if you fold a sheet of paper once, you double its thickness. If you fold it again, you double its thickness again. Continuing on in this way, folding a sheet of paper 100 times is the same as doubling its thickness 100 times. So, the total thickness would be 2[SUP]100[/SUP] × the thickness of one sheet. Now, 2[SUP]100[/SUP] is 1,267,650,600,228,229,401,496,703,205,376, and the thickness of a 20-pound sheet of paper is about 0.097 mm. Thus, the total thickness is 122,962,108,222,138,251,945,180,210,921.472 mm, or about 13 billion light years. Now, the diameter of the observable universe is about 93 billion light years, so the claim isn’t true, unless your paper were thicker. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [December 14, 2015, 7:46am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/4 "2015-12-14T07:46:56Z")
The world record for folding a piece of paper in half is twelve times. That required using a roll of toilet paper. With an ordinary piece of paper, the most times you can fold it in half is probably seven: > **[Mathematics of paper folding](https://en.wikipedia.org/wiki/Britney_Gallivan)** > > The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper folds to solve up-to cubic mathematical equations. > Computational origami is a recent branch of computer science that is concerned with studying algorithms that solve paper-folding problems. The field of computational origami has also grown significantly ... ---
**Author:** ![Lucas\_Jackson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lucas_jackson/32/303_2.png) [@Lucas\_Jackson](https://boards.straightdope.com/u/Lucas_Jackson)\ **Post date:** [December 14, 2015, 10:49am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/5 "2015-12-14T10:49:08Z")
Ok, so I Googled exponential growth. I see now why 100 folds mathematically puts you in the ballpark. Thanks. I tested the folding part myself. A normal sheet of notebook paper is really tough to fold 5 times! But the surface area of that sheet is starting to get small. Suppose instead of folding you cut it then stacked it, I’m wondering if there is even enough material in a sheet of paper to stretch that far. Didn’t Carl Sagan once claim that if you cut an apple pie in half, then halve that half, and so on after 64 cuts you would be down to an atom? Anyway, thanks for the responses. . ---
**Author:** ![PSXer](https://avatars.discourse-cdn.com/v4/letter/p/48db29/32.png) [@PSXer](https://boards.straightdope.com/u/PSXer)\ **Post date:** [December 14, 2015, 11:00am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/6 "2015-12-14T11:00:19Z")
they did this on Mythbusters iirc and folded a piece of paper larger than the observable universe ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [December 14, 2015, 11:05am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/7 "2015-12-14T11:05:32Z")
> [@White SIFL](#): > > Ok, so I Googled exponential growth. I see now why 100 folds mathematically puts you in the ballpark. Thanks. > > I tested the folding part myself. A normal sheet of notebook paper is really tough to fold 5 times! But the surface area of that sheet is starting to get small. > > Suppose instead of folding you cut it then stacked it, I’m wondering if there is even enough material in a sheet of paper to stretch that far. Didn’t Carl Sagan once claim that if you cut an apple pie in half, then halve that half, and so on after 64 cuts you would be down to an atom? > > Anyway, thanks for the responses. > . A sheet of A4 paper is 210 x 297 mm, or 0.06237m[sup]2[/sup]. If you cut it in half repeatedly and stacked the pieces, 64 times, then you would have 2[sup]64[/sup] pieces, each of which would be 3.38 x 10[sup]-21[/sup]m[sup]2[/sup]. A carbon atom has a diameter of 1.7 x 10[sup]-10[/sup]m, so if your piece of paper was approximated to a “square” one carbon atom wide it would have an area of 2.9 x 10[sup]-20[/sup]m[sup]2[/sup]. In other words, you’re within about one order of magnitude of individual atoms if you cut a piece of paper in half 64 times - essentially each “piece” would be a tower of individual atoms the height of the thickness of the paper. If we assume the paper is 0.1mm thick then that is about 600,000 atoms high. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [December 14, 2015, 11:07am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/8 "2015-12-14T11:07:47Z")
There are something like 10^26 atoms in a pound of most materials. I don’t know how much an apple pie weighs, but it’s less than 10 pounds. So there are something like 10^27 atoms in an apple pie. 10^27 is about 2 ^ 90, so my calculations say that you’d have to cut the pie in half about 90 times. ---
**Author:** ![bob\_2](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bob_2/32/3341_2.png) [@bob\_2](https://boards.straightdope.com/u/bob_2)\ **Post date:** [December 14, 2015, 11:34am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/9 "2015-12-14T11:34:19Z")
It’s the same as the story about a [Chinese] wiseguy who had a chessboard. He wanted one grain of rice on the first square, two on the second, three on… and so on. Long before it reached 64 squares, he would own all of the possible grains of rice in the world. Maths can be fun 🙂 ---
**Author:** ![Your\_Great\_Darsh\_Face](https://avatars.discourse-cdn.com/v4/letter/y/ac8455/32.png) [@Your\_Great\_Darsh\_Face](https://boards.straightdope.com/u/Your_Great_Darsh_Face)\ **Post date:** [December 14, 2015, 11:36am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/10 "2015-12-14T11:36:47Z")
> [@bob\_2](#): > > It’s the same as the story about a [Chinese] wiseguy who had a chessboard. He wanted one grain of rice on the first square, two on the second, three on… and so on. > > Long before it reached 64 squares, he would own all of the possible grains of rice in the world. > > Maths can be fun 🙂 _Four_ on the third, and so on. 1 + 2 + 3 + 4 + …+ 64 comes to a much more manageable 65 x 32 = 2080. ---
**Author:** ![GaryM](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/garym/32/241_2.png) [@GaryM](https://boards.straightdope.com/u/GaryM)\ **Post date:** [December 14, 2015, 2:06pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/11 "2015-12-14T14:06:32Z")
Here’s a link to a video of Mythbusters doing 11 folds. They used a DynaPac roller to help get the trapped air out. [![]( " - YouTube") ](https://www.youtube.com/watch?v=kRAEBbotuIE) ---
**Author:** ![sbunny8](https://avatars.discourse-cdn.com/v4/letter/s/c77e96/32.png) [@sbunny8](https://boards.straightdope.com/u/sbunny8)\ **Post date:** [December 14, 2015, 3:24pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/12 "2015-12-14T15:24:06Z")
> [@RadicalPi](#): > > Well, if you fold a sheet of paper once, you double its thickness. If you fold it again, you double its thickness again. Continuing on in this way, folding a sheet of paper 100 times is the same as doubling its thickness 100 times. So, the total thickness would be 2[SUP]100[/SUP] × the thickness of one sheet. Now, 2[SUP]100[/SUP] is 1,267,650,600,228,229,401,496,703,205,376, and the thickness of a 20-pound sheet of paper is about 0.097 mm. Thus, the total thickness is 122,962,108,222,138,251,945,180,210,921.472 mm, or about 13 billion light years. Now, the diameter of the observable universe is about 93 billion light years, so the claim isn’t true, unless your paper were thicker. Perhaps they would try to weasel on the definition of “known universe”. However, RadicalPi’s math looks right to me, so it appears that 103 folds would get you up to 104.3 billion light years. So the claim should have said “103 times”. ---
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [December 14, 2015, 9:09pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/13 "2015-12-14T21:09:37Z")
> [@bob\_2](#): > > He wanted one grain of rice on the first square, two on the second, three on… and so on. One on the first square, two on the second, _four_ on the third, eight on the fourth, doubling each time. The total would be 2[sup]64[/sup]-1 grains of rice, or about 1.8 x 10[sup]19[/sup] grains or roughly 6 x 10[sup]11[/sup] pounds. That would fill over 10,000,000 semi trucks, unless I messed up an exponent someplace. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [December 14, 2015, 9:24pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/14 "2015-12-14T21:24:46Z")
> [@bob\_2](#): > > It’s the same as the story about a [Chinese] wiseguy who had a chessboard. He wanted one grain of rice on the first square, two on the second, three on… and so on. > > Long before it reached 64 squares, he would own all of the possible grains of rice in the world. > > Maths can be fun 🙂 > [@CookingWithGas](#): > > One on the first square, two on the second, _four_ on the third, eight on the fourth, doubling each time. > > The total would be 2[sup]64[/sup]-1 grains of rice, or about 1.8 x 10[sup]19[/sup] grains or roughly 6 x 10[sup]11[/sup] pounds. That would fill over 10,000,000 semi trucks, unless I messed up an exponent someplace. Yeah, without the doubling, it’s not a lot of rice at all; **bob++**'s scenario requires only 2080 grains of rice - less than a cupful. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [December 14, 2015, 9:55pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/15 "2015-12-14T21:55:55Z")
> [@sbunny8](#): > > Perhaps they would try to weasel on the definition of “known universe”. However, RadicalPi’s math looks right to me, so it appears that 103 folds would get you up to 104.3 billion light years. So the claim should have said “103 times”. Well we all know that 100 = 103 for sufficiently large values of 100. 😃 ---
**Author:** ![teflon\_grut](https://avatars.discourse-cdn.com/v4/letter/t/e95f7d/32.png) [@teflon\_grut](https://boards.straightdope.com/u/teflon_grut)\ **Post date:** [December 14, 2015, 10:04pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/16 "2015-12-14T22:04:08Z")
> [@RadicalPi](#): > > Well, if you fold a sheet of paper once, you double its thickness. If you fold it again, you double its thickness again. Continuing on in this way, folding a sheet of paper 100 times is the same as doubling its thickness 100 times. So, the total thickness would be 2[SUP]100[/SUP] × the thickness of one sheet. Now, 2[SUP]100[/SUP] is 1,267,650,600,228,229,401,496,703,205,376, and the thickness of a 20-pound sheet of paper is about 0.097 mm. Thus, the total thickness is 122,962,108,222,138,251,945,180,210,921.472 mm, or about 13 billion light years. Now, the diameter of the observable universe is about 93 billion light years, so the claim isn’t true, unless your paper were thicker. Okay… so 103 folds. ---
**Author:** ![penultima\_thule](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/penultima_thule/32/3833_2.png) [@penultima\_thule](https://boards.straightdope.com/u/penultima_thule)\ **Post date:** [December 14, 2015, 11:00pm UTC](https://boards.straightdope.com/t/paper-folding-math/740397/17 "2015-12-14T23:00:16Z")
> [@Wendell\_Wagner](#): > > There are something like 10^26 atoms in a pound of most materials. By Avogadro’s Constant, there are 6.023 x 10^23 molecules in a mole, Working out the molecular weight of apple pie or an apple is complicated. There are 6 x 10^23 molecules of water in 18g and hence 5.1x10^24 in a pound of water There are 6 x 10^23 molecules of iron (Fe) in 55.84g iron and hence 1.6 x10^24 in a pound of iron (if pure Fe). The larger the atomic mass the less there will be in a pound, but there are “only” 10^25 atoms of hydrogen in a pound of hydrogen . ---
**Author:** ![kaylasdad99](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kaylasdad99/32/3398_2.png) [@kaylasdad99](https://boards.straightdope.com/u/kaylasdad99)\ **Post date:** [December 15, 2015, 1:54am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/18 "2015-12-15T01:54:18Z")
> [@sbunny8](#): > > Perhaps they would try to weasel on the definition of “known universe”. However, RadicalPi’s math looks right to me, so it appears that 103 folds would get you up to 104.3 billion light years. So the claim should have said “103 times”. Maybe it’s that heavy construction paper like they gave us in kindergarten. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [December 15, 2015, 2:14am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/19 "2015-12-15T02:14:25Z")
So, penultima thule, how many atoms are there in an apple pie and how many times would you have to cut it in half to get down to one atom? ---
**Author:** ![penultima\_thule](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/penultima_thule/32/3833_2.png) [@penultima\_thule](https://boards.straightdope.com/u/penultima_thule)\ **Post date:** [December 15, 2015, 2:38am UTC](https://boards.straightdope.com/t/paper-folding-math/740397/20 "2015-12-15T02:38:40Z")
> [@Wendell\_Wagner](#): > > So, penultima thule, how many atoms are there in an apple pie and how many times would you have to cut it in half to get down to one atom? You give me the recipe and I’ll see I can give an estimate. As to the number of cuts needed then the answer provided by **Colophon** would be a reasonable starting point. > [@Colophon](#): > > … A carbon atom has a diameter of 1.7 x 10-10m, so if your piece of paper was approximated to a “square” one carbon atom wide it would have an area of 2.9 x 10-20m2. > In other words, you’re within about one order of magnitude of individual atoms if you cut a piece of paper in half 64 times… [Next page](https://boards.straightdope.com/t/paper-folding-math/740397.md?page=2)