Exponential Origami

Raju Varghese (via Sploid) claims that if you fold a piece of paper 103 times, the thickness of your paper will be larger than the diameter of the observable universe: 93 billion light years. The explanation is simple enough – every time you fold a piece of paper you double its thickness and when you start doubling things they get very large very quickly – but I couldn’t leave this without checking the numbers for myself.

Of course, I couldn’t resist checking this for myself and pulled out a calculator. I soon found that the mental juggling needed to get from fractions of millimetres to kilometres was too much for my little brain and converting between millimetres and light years was going to be impossible.

So I wrote a script. The code is pretty simple, as you can see below, although I did have a four fold discrepancy when I first ran it (I came up with 107 folds needed, rather than 103). It turned out that my initial thickness of the paper was out by a factor of 10. Once I fixed this, everything matched.

#!/usr/bin/env python
""" Foldpaper
    Calculates the thickness of a piece of paper after n folds """

thickness = 0.1
folds = 0
meter = 1000
kilometer = 1000000
lightyear = 1000000 * 9000000000000
size_of_universe_in_mm = 93000000000 * lightyear

while thickness  1:
        print(folds, int(thickness/lightyear), 'light years')
    elif int(thickness / kilometer) > 1:
        print(folds, int(thickness/kilometer), 'kilometers')
    elif int(thickness / meter) > 1:
        print(folds, int(thickness/meter), 'meters')
    else:
        print(folds, thickness, 'mm')

And then, with a slight edit, I dumped the results into a table so that I could add a few comparative distances.

Folds Height Notes
15 3 metres Taller than the average human
22 419 metres Taller than The Shard in London
27 13 kilometres We’re now standing higher than Mount Everest
42 439804 kilometres Now we’ve just passed the Moon
51 225179981 kilometres And the Sun
56 7205759403 kilometres And finally we reach Pluto
69 6 light years With a single fold, we have shot past Alpha Centuri
83 107460 light years And now the thickness of our piece of paper is larger than the Milky Way
88 3438722 light years And in a few short folds, we pass Andromeda
103 112680053353 light years And with that final fold, we have exceeded the size of the Universe

Exponentiation is awesome.

5 thoughts on “Exponential Origami

  1. That is a nice overview! 🙂

    I can stop folding now! 😉

    Maybe there will be a binary paper folding competiton. Participants are given a large piece of paper, and the one who can fold it most times without any tools wins 🙂

    Like

  2. Wow, there is competition in such things. But why would there be a limit..? I guess if there are mathematical reasons, then I dont understand them.

    But if the size and thickness doesnt matter then one could fold a 1 cm thick 10×10 cm paper say 8 times, which seems rather impossible? Isnt it the thickness vs surface size ratio relevant?

    I need coffee 🙂

    Like

  3. Wherever there is a challenge, someone will accept it 😉

    There is no theoretical limit on how many times you can fold a piece if paper, as long as you can find a sheet large enough and apply enough force. But the practical challenges will make themselves known very quickly.

    If you have ten folds, for example, you have a stack of paper 1024 sheets high. Folding that many sheets is going to take a fair bit of effort.

    And, of course, once you’ve folded your sheet 103 times, you have a stack of paper larger than the universe and I’m not sure where you would put such a pile 😉

    Like

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