 Dr. Christopher S. Baird # What stops a piece of paper from being folded more than seven times?

Category: Physics      Published: December 7, 2012 Public Domain Image, source: Christopher S. Baird.

Nothing stops a piece of paper from being folded more than seven times if the paper is thin enough. Depending on the thickness and width of the paper, after a certain number of folds, the paper stack becomes thicker than it is wide. After that point, there simply is nothing left to fold, so the limit is reached. Each fold in half makes a paper twice as thick, so that n folds of a paper that has a thickness of t results in a total thickness of 2nt. At the same time, every two folds cuts the width in half, so that n folds reduce the width w to (1/2)n/2w. If we say that a paper can't be folded when its total thickness equals its width, then 2nt = (1/2)w. Solving for n, we find that the maximum number of times a paper can be folded is:

n = 0.96 ln (w/t)

This equation assumes we fold side to side, then top to bottom, then side to side, etc. Using this equation, a standard piece of printer paper has w = 8.5 inches and t = 0.004 inches, so that n = 7. So a standard piece of paper can indeed only be folded seven times. However, if we had an 8.5 x 11 in. sheet of paper that was a quarter as thick as normal, using this equation, we could fold it nine times. If you take a roll of toilet paper and roll it out into one long line you can fold it even more. If you are folding in one direction however, you have to use (1/2)nw for the width, leading to the slightly different equation, n = 0.72 ln (w/t). If we take a super-jumbo-sized roll of toilet paper with a thickness of 0.004 inches and unrolled length of 170000 inches, using this equation we find we could fold it thirteen times. This has actually been done in real life at MIT and the students indeed folded the toilet paper thirteen times, as reported in the Boston Globe.