Paper Folding to the Moon

Posted by Ethan on August 31, 2009
(47)
More »

Okay, as many of you had heard, I’ve got a new job as a full-time Professor. And not only am I pretty excited about it, I thought I’d share with you one of the more interesting things I taught on the first day. I got this idea from talking to Michael, the chair of the department (and this is not the first time he’s taught me something neat). Chances are, if you’re in a classroom, that one thing everyone has is a piece of paper.

If you folded this piece of paper in half, it would now be twice as thick as it was before:

So my question is this: how many times would you have to fold this paper onto itself to reach the Moon? I’ll give you a chance to guess, so pick the closest one from the options below.

Well, let’s see how we’d figure it out. I don’t know how thick one piece of paper is, but I know it’s pretty thin. I can, however, estimate how big those 500 page reams are. They’re about 2 inches high, so maybe that’s about 5 cm. That means one page is about 0.01 cm high. And what of the Moon?

Mean distance from the Earth is about 384,000 km, or about 3.84 x 1012 pages away. So you’d expect that you’ll need an awful lot of foldings to get there, right? Well, hang on for a second.

When I start with an unfolded page (zero foldings), it’s one page thick. When I fold a page once, it will be 2 pages thick. But — and this is key — when I fold it twice on itself, it’s not three, but 4 pages thick.

If I fold it a third time, I’ll see that it’s 8 pages thick. Can you see a pattern here? Paper folding is exponential, so that if I fold it a fourth time, it’ll be 16 pages thick (so that option is clearly wrong), a fifth time will give me 32 pages thick, and so on. By time I get to 9 foldings, my folded paper is bigger than my original ream of 500 sheets. By time I get to 20 foldings, my folded paper is more than 10 kilometers high, which surpasses Mt. Everest. 41 foldings will get me slightly more than halfway to the Moon, so that means that 42 foldings is all it takes! (Of course, good luck folding a real piece of paper more than 7 or 8 times…)

Pretty incredible, isn’t it? But that’s the power of an exponential, that it lets you turn small things into huge things by simply compounding what you have over and over again. And incredibly, it only takes 42 foldings of a paper to get from the Earth to the Moon, and only about 94 foldings of a paper to make something the size of the entire visible Universe! And how surprised are you that the answer is so small a number?

(47)
More »

Comments

  1. #1 Brandon
    August 31, 2009

    lol 42.

    Aw, I saw this on facebook and thought it was gonna be an open question type of post, so I figured it out before coming over here. Too bad you can’t actually fold a piece of paper more than 7 times.

  2. #2 Helioprogenus
    August 31, 2009

    We all knew this was the answer to life, the universe, and everything. I actually guessed, figuring Douglas Adams would have been right. In all seriousness though, it really puts a perspective on scale. We’re used to seeing such big numbers, but at some point, those numbers are not intuitive anymore. So our national debt is in the trillions, well, that’s a lot of money. I mean stacking a million dollars is quite a feat, now dealing with a trillion is easy mathematically, but intuitively, it’s about as difficult as dealing with exponential systems, universal scales, even physical distances. We throw around 380,000 km, but that’s still quite a number. Try walking that you triathletes.

  3. #3 frog
    August 31, 2009

    Cute. Did any of your students check for reasonableness?

    If the C bonds are about 150e-12, it would take 2.5e18 carbon bonds to go to the moon. Since that number is five orders of magnitude less than Avagadro’s number and a piece of paper is 5g or so, not only could you reorder a single piece of paper to the moon — you should be able to fold yourself beyond Pluto.

  4. #4 Brian
    August 31, 2009

    I remembered the wrong distance for the Moon’s orbit, and I guessed a bad value for the thickness of paper, and I still came up with 40 as my answer. A nice reminder that good order-of-magnitude estimates can be made even without precise data.

  5. #5 Katherine
    August 31, 2009

    But, you can only fold a piece of paper 6 times before it is too strong to fold! I don’t think even today’s technology would be able to fold it 42 times.

  6. #6 Sweetwater Tom
    August 31, 2009

    From Mythbusters on Discovery.com:

    Episode 72: Underwater Car
    Meanwhile Grant, Tory and Kari roll out the Seven Paper Fold myth. Is it possible to fold a piece of paper in half more than seven times? Taking this myth to the outer limits, our crew sets up at a location that has plenty of space — NASA. Here, in the biggest build they have ever attempted, their mission is to put together a piece of paper that’s the size of a football field.
    Premiere: Jan. 24, 2007

  7. #7 Rich
    August 31, 2009

    re: the Mythbusters episode a few years ago, they did actually get more than seven folds (if I recall, it was something like 12 folds?) but they had a gigantic piece of paper, and they used a forklift to assist in their work….

    But I do like this problem because it has several layers of math, and I can look at it with my middle schoolers.

    I’m reminded of a very old “I Love Lucy” episode where she discovered that if she didn’t like a particular brand of baked beans, she could return it to the store and get DOUBLE her money back. She kept up with the scheme (of course with plenty of hijinx storing all of the cans and cases at home) until she actually ate some beans and realized that she really loved them after all. OK, so maybe that’s probably not as cool as folding paper, but it was a pretty good visual lesson for me 30+ years ago.

  8. #8 OneInterestedTeacher
    August 31, 2009

    I teach 9th grade earth science and will definitely use it to start our unit on the moon! Thanks for the awesome idea.

  9. #9 Tony P
    August 31, 2009

    That pattern btw, is powers of 2. Binary – 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32767, 65536, 131072…. 18 bits thus far.

    Being around computers for as long as I have I can also count in Base 8 and Base 16.

  10. #10 Katherine
    August 31, 2009

    So if a person can get 6 folds, and a forklift can get double that, what sort of technology would you need to get to 42? ;) Though I suppose giant paper isn’t as thin as regular paper so maybe you wouldn’t need so many folds?

  11. #11 jodie
    August 31, 2009

    I don’t know if today’s technology could fold a sheet of paper 42 times or not, but I am pretty sure that on a NASA program you could simply pile all the documentation you have to create into a stack and climb the stack to the moon.

  12. #12 archimedes
    August 31, 2009

    I hope you also point out to the students that the paper stack would become invisible to the human eye around the time it reached the edge of Earth’s atmosphere. By the time it reached the moon, the folded sheet would measure only a few hundred atoms on a side… give or take!

  13. #13 Robert
    September 1, 2009

    Each folding cuts the area of the paper in half.

    After 42 folds, assuming the paper started at 0.203 x 0.254 meters (8 x 10 inches) with zero folds, the area would be 1.17×10^-14 square meters.

  14. #14 archimedes
    September 1, 2009

    Which would be a square about 10^-7 m, or a hundred nanometres, on a side. And I’m told atoms are roughly 0.1 to 0.7 nm in diameter, hence the paper is a few hundred atoms on a side, give or take.

    Does this seem right?

  15. #15 lzell
    September 1, 2009

    Does that mean this article about folding paper 12 times incorrect? http://www.pomonahistorical.org/12times.htm

  16. #16 David Derrick
    September 1, 2009

    I’m told it doesn’t matter how big the piece of paper is initially. You can’t fold it more than 6 times. Is that because it’s paper? Surely that wouldn’t apply to every material?

  17. #17 Mu
    September 1, 2009

    Ok, please think for a second regarding the required paper size. When folding a square piece in the described way, you will have to cover the outer layer with a strip at least as long as the final thickness. I first thought you need to start with a 384,000 km square sheet (that’s 290 times the surface of the earth, and twice the surface area of Jupiter). Than I realized, the length of the sheet gets cut in half every time. So, to end up with a 384,000 gm stack, you need to start with 2^41 x 384,000 km length. That’s roughly 90,000 ly in length.

  18. #18 jace
    September 1, 2009

    The “possible number of foldings” depends on the ratio between the initial length of the unfolded paper and the thickness of the paper; once the stack becomes a cube (when the folded thickness equals the length of a side) the concept of a “fold” breaks down.

    The length of a side decreases as 1/sqrt(2) (assuming a square paper for simplicity), and the thickness doubles with each fold (assuming incompressibility). If I did the math right, then, assuming a paper that starts with length L and thickness T,

    N = max folds = 2/3/log(2) * log (L/T) = 2.2146 log(L/T)

    An 8.5×11 piece of paper would have an L of about 250 mm, and a T of about .1 mm, for a ratio of 2500; that means a maximum of 7 folds (they reach equality at 7.5).

    The sheet of paper used in Mythbusters was “the size of a football field”. Lets call that 70m for L, and assume thickness is unchanged; the L/T ratio is 700000, and you should be able to get 12 folds (12.94; just a little bigger or thinner and you’d make 13).

    To get 42 folds with a .1mm paper is going to require a starting side of 9.23*10^14 m, or 35.6 light-days, or 6170 AU.

  19. #19 multipath
    September 1, 2009

    jace, now that’s some good insight on the problem. And not so good English on my part.

  20. #20 SecondCobra
    September 1, 2009

    Mu, where did you get the idea that the surface area of the Earth is 1324Km square? It is 510,072,000Km square. A simple check would have shown this.

  21. #21 DD
    September 2, 2009

    Earth is not square! If you walk a straight line its a circle. There isn’t enough cellulose on Earth to make a paper that will reach the moon in any practical manner, but check out the space elevator.

  22. #22 J Todd DeShong
    September 2, 2009

    It’s kinda like the question:
    “How long was Margaret Thatcher Prime Minister?”
    a. 11 years
    b. 300 years
    c. 2,000 years
    UHM, it’s a trick question. I would guess b. 300 years.
    It was a very long time.
    JTD

  23. #23 MadScientist
    September 2, 2009

    Congratulations on your new job! Do we have to call you Prof. now?

    My notebook conveniently has about 16 pages to 1mm, so that’s 2^4 or a mere 4 folds to 1mm. 1Km is another 20 folds. 384403 is another 18 or 19. Gee, that’s tricky; I’d better backtrack and stop this rounding at each step. 42.48 – damn that half fold! Huh. To think all I had to do was read The Hitchiker’s Guide to The Galaxy and I would have discovered the answer.

  24. #24 MadScientist
    September 2, 2009

    Aaaahahaha … I had a look at the results and you can tell how many people just guess.

  25. #25 Mu
    September 2, 2009

    SecondCobra, 384,000 km square = 147,456,000,000 km^2 area, divide by 510,072,000 km^2 = 289.1. Want to try again?
    Jace, thanks for the side length correction, I missed the side length gets only cut in half every second fold.
    In any case, your folding will be limited by relativity :) . You need 50 + years to get the ends of the first fold to come together to keep the corner movement below light speed.

  26. #26 Chris' Wills
    September 4, 2009

    Posted by: J Todd DeShong

    “How long was Margaret Thatcher Prime Minister?”
    a. 11 years
    b. 300 years
    c. 2,000 years
    UHM, it’s a trick question. I would guess b. 300 years.
    It was a very long time.

    Correct answer is; not long enough.

  27. #27 Alex
    September 7, 2009

    Stop trolling Chris. Thatcher is and was PM for too long. Thatcher (the woman) was PM until 1990. Thatcher (the concept) hasn’t abdicated yet. Notice how neo-liberal New Labour has been, yet?

  28. #28 SecondCobra
    September 9, 2009

    Mu – I stand corrected :( I read your comment as square Km. Will read more carefully next time.

  29. #29 Olie
    September 29, 2009

    Sorry if this has been answered above but 20 times would only be just over 1KM, not 10KM like you have suggested.

  30. #30 JohnBruno
    November 5, 2009

    “By time I get to 20 foldings, my folded paper is more than 10 kilometers high, which surpasses Mt. Everest.”

    Not sure about this.

    correct me if I’m wrong, but by my calculations…

    2^20*0.01cm = 10485.76cm = 104.86m

    Great post!

  31. #31 mos
    December 2, 2009

    if you think of the piece of paper is space time, you would only fold the paper once.

  32. #32 Therese
    December 26, 2009

    I just saw your blog today, and I am keen over its name – PAPER FOLDING TO THE MOON! Care to share?

  33. #33 jordynwallace
    April 1, 2010

    WHO DO ORbit MENT.

  34. #34 crd2
    August 25, 2010

    So can anyone figure out how many folds it would take to get to the edge of our observable universe? and how wide the paper would be once it got there?

  35. #35 mike
    September 23, 2010

    you guys have to decide if you stay in inches and join Liberia and Burma as the only countries still using this vintage system or you choose to join the world that are using the metric system. Using both in the same article is insane.

  36. #36 Joey
    September 23, 2010

    You are all gotdamn wunderkinds. First off: Thatcher Ruled, Rules and Always Will Rule. Now, Paper is meant for fingerpainting and paper-airplanes and sometimes anal-hygiene, not incredibly thin paper-chains to the moon. Not to mention that greenpeace would never allow it. Now, the real question is, how many times would you have to fold your iPhone/Android-phone onto itself before you reached the hight/width of Steve jobs pile of cash?

  37. #37 Aditya
    November 11, 2010

    42 is the Ultimate Answer to the Ultimate Question of Life, The Universe, and Everything. :)

  38. #38 Lincoln
    January 10, 2011

    Depending on the thickness, 50 folds goes to the sun.

    But a 12 yr old girl took a whole roll of toilet paper and brought it to the mall where she set a record of 12 folds that was 40 cm thick. She did it after her teacher told her folding gold foil was a cop out.

  39. #39 Elendilm
    January 15, 2012

    Article above is absolutely wrong. 20 foldings of 0.01 cm (0.1mm) will give you 104.8576 metres ( 0.1 km).

    0.1mm * 2 ^ 20 = 104.8576 metres i.e 0.1 km approx.
    For reaching mount everest with a thickness of 0.1mm, a paper should be folded 27 times

    0.1mm * 2 ^ 27 = 13.4 km which surpasses mount everest by approx 5 km.

  40. #40 peterd
    Armidale, NSW, Australia
    August 19, 2012

    Loved the post and the discussion surrounding it. I think that John is correct and Everest would be surpassed with 26 folds not 20. I have used this in the past with my Year 9 and they loved it. Goes well with the power of ten video.

  41. #41 Anthony M
    Ohio
    August 31, 2012

    I’m glad some people know how to add. Props to John Bruno and Elendilm for adding correctly. I think author was in a hurry and when added to 10485.76, forgot they were adding centimeters and not converted to kilometers yet. :)

  42. #42 wierd
    penrith cumbria
    October 15, 2012

    u r not showing how to do it

  43. #43 Jephze
    Philippines
    December 24, 2012

    This is really cool man…42 folds only to reach the moon…exponential matters really amazed me a lot…but what planes?? paper airplanes??? what kind of plane can i fold to reach to the moon? i found this also really cool…shared by my classmate…:) :) try to visit guy’s
    http://extremepaperairplanes.com/how-to-fold-the-puma-paper-airplane/

  44. #44 Foldy Foldy
    Moon
    December 28, 2012

    Just amused at the thought of tne 42d fold… You’d have 2 halves each half the distance to the moon.. Need some serious swinging space to manoever that around.. Hope it’s not raining (or snowy or windy) anywhere for that distance.. Or any buildings, trees, people… Might be easier to start from the moon (handy if you’re stuck there and need to slide back to earth?) .. Then again, the fold in the paper itself need be the length of the folded paper itself so you really would need to start with a pretty long sheet…

  45. #45 Daniel
    January 5, 2013

    that’s ridiculously wrong oO you’d need a giant paper, because a notebook page can’t possibly have enough material to reach the moon

  46. #46 Graham Bagshaw
    Virginia
    February 7, 2013

    Although folding may be impossible more than a few times just do cutting in half. You can demonstarte using scissors or just rip for the first few iterations. Although putting them on top of each other and repeating still very tricky but easier than folding multiple times

  47. #47 Graham Bagshaw
    virginia
    February 7, 2013

    Talking about students making common sense checks I did have one student ask me “Has anyone done this yet?”