What stops a piece of paper from being folded more than seven times?
Category: Physics Published: December 7, 2012
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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 an initial thickness of t results in a total thickness of 2nt. At the same time, every two folds cuts the width in half, so that a series of n folds reduces the initial width w to (1/2)n/2w. This is assuming that you fold side-to-side, then top-to-bottom, then side-to-side, then top-to-bottom, and so forth. This also assumes that "initial width" means the initial width of the paper in the smaller dimension and the first fold is top-to-bottom. If we say that a paper cannot be folded when its total thickness equals its reduced width, then that means that: 2nt = (1/2)n/2w. Solving for n, we find that the maximum number of times a paper can be folded is:
n = 0.96 ln (w/t).
Using this equation, a standard piece of printer paper with a width of w = 8.5 inches and a thickness of t = 0.004 inches leads to n = 7. A standard piece of printer paper can indeed only be folded seven times. However, if we had an 8.5 x 11 in. sheet of paper that was half as thick as normal, using this equation, we could fold it eight times.
If you take a roll of toilet paper and roll it out into one long line you can fold it even more. However, with toilet paper you could only effectively fold one direction. Therefore, you would have to use (1/2)nw for the width, leading to the slightly different equation:
n = 0.72 ln (w/t).
If you take many unrolled pieces of toilet paper with a thickness of 0.004 inches and tape them together to make a single piece that is 170,000 inches long, using this equation we find that you could fold it thirteen times. This has actually been done in real life at MIT by students from St. Mark's School and the students indeed folded the toilet paper thirteen times, as reported by NPR.
Let's assume that we are sticking to regular folding, where you switch off the directions you are folding. The black curve in the image below shows the plot of the equation for the width of the paper as a functon of number of times folded, which is w' = (1/2)n/2w. The colored curves in the image below show the plots of the equation for the thickness of the paper as a function of number of times folded, which is t' = 2nt, where the different colored curves correspond to different initial paper thicknesses. The place where a colored line crosses the black line is where you have reached the point where the paper cannot be folded anymore. However, in reality, you cannot fold a paper 4.63 times, or whatever the result may be. The number of complete folds must be a whole number. This means that the actual answer will be the biggest whole number that is below the point where a colored line crosses the black line.
According to the image above, which I have verified experimentally, you cannot fold a thick, regular-width index card more than four times, you cannot fold a piece of standard cardstock more than six times, you cannot fold a piece of standard printer paper more than seven times, and you cannot fold a large piece of thin origami paper more than eight times (assuming that you are folding in the standard way: folding in half each time while alternating the folding directions).
Note that the paper width values have been plotted in the image above as a percentage of the original paper width, to make the numbers more immediately meaningful. Furthermore, in order to plot everything on the same scale, the paper thickness values have also been plotted as a percentage of the paper width. For instance, if a piece of paper was originally 20 cm wide, then after two folds it would have a width of 50% of its original width, which is 10 cm. After a total of four folds, it would have a width of 25% of its original width, which is 5 cm, and so forth. Also note that the curves in the image above technically should be just a series of unconnected data points, with a data point at each whole number for the number of times folded. This is because you can't do a fractional amount of folds. However, I have connected the data points into curves in order to make it visually easier to tell which data points go with which data set.