Folding paper

You can’t fold a paper more than seven or eight times. Don’t believe me? Then try it. Thinner paper? Longer paper? It doesn’t matter; you just can’t do it. I used to play this game with my friends which were always amazed and asked for an explanation. Is there any physical or mathematical constrain to do it? Nope: it is simply a matter of scale. If you have a paper of length L and you fold it, the length now is L/2. Do it again and it goes down to L/22. In general, if you fold it n times, the remaining length is

ln=L/2n

Obviously, this folding can be repeated until the remaining length of the paper is approximatelly the width of the paper. Thus, the maximum number of times you can fold a piece of paper of length L and width d is given by

d=L/2n => n = Log[L/d]/Log[2] 

Now get a piece of paper (an A4 sheet for example), which is 297mm long and more or less 0.1mm of width. Thus n = 11 which is much larger than the 7 or 8 times expected. The explanation for this miscalculation is that when folding paper, the remaining length of the paper is not given by simply dividing by 2 the previous lenght (see figure).

Paper folding after two iterations. Note that the remaining
length l2 is not equal to L/22 due to the paper lenght
“used” in the arcs

One should account for the paper in the arcs conecting different layers and the series of remaining lengths was calculatedby high school student Britney Gallivan in December of 2001 (see a detailed explanation here):

ln = 6L/[pi (2n+4)(2n-1)]

In this case we get

n = 1/2 Log[6L/pi d]/Log[2] 

which gives n = 6 for a A4 sheet of paper, as expected. In fact, longer paper can be folded more times: the same Britney Gallivan has recently folded 12 times a piece of paper, which accordingly to the equations above, was of (at least) length 879 meters.

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