## Why can't you fold paper more than 7 times?

Mythbusters was one of my favorite series growing up, and it was definitely one of my early inspirations to study experimental physics. But over the course of the show, Adam Savage and Jamie Hyneman did plenty of experiments that were basically unnecessary had they done a two line physics calculation. Yes, it was fun to see the guys make a football field-sized sheet of paper, and use a hydraulic press to fold it, but...well...you could just do the math instead. Standard printer paper is \( t=0.1\text{mm}\) thick, and \( d=25\text{cm}\) long. It gets twice as thick and half as long each time you fold it, so after \( N \) folds it is \( 2^Nt \) thick and \( d/2^N \) long. Roughly speaking, you can fold paper in half until it is as thick as it is long, i.e., \( 2^Nt = d/2^N \), which implies \( N = \frac{1}{2}\log_2 d/t\approx 5.6 \). So maybe if you really force it: 6 times. Because this result depends only logarithmically on the length and thickness, it is fairly insensitive to their values. For example, if the sheet of paper is as large as a football field, \( d=100\text{m} \), 400 times bigger than standard printer paper, then \( N=9.9 \). So experiments are great, but sometimes a spherical cow is all you need.

Posted by Abraham