Folding the best bookmark with calculus

When Chenguang Zhang was sitting through a less-than-interesting seminar, he started toying with his square-shaped pocket notebook. He wondered if one of its pages could fold into a self-made bookmark, and against his intuition, he found that it could.

In a new study, Zhang describes the calculus behind this mundane phenomenon with the hopes that it can make a useful undergraduate classroom demonstration.

To qualify as a bookmark, the folded page must be visible and stay on the same side of the bind when the notebook is closed. Zhang determined that the best way of folding depends on the page’s aspect ratio, making it very different for square versus letter pages.

“You can define all folds by two parameters. Every time you change those values slightly, you change your way of folding slightly,” he said.

At some point, however, a small change can have a dramatic effect.

“Eventually, calculus leads you to the best way of folding, and tells you that the best way drastically changes with the page’s aspect ratio,” said Zhang.

The bookmark makes a great example of bifurcation, in which a sudden and qualitative change in the behavior of a system results from just a small change in its parameters.

“For me, the message to students is two-fold. First, your best solution to even a simple problem can change suddenly when the problem is varied slightly. The good news is that you can fully understand this behavior via calculus,” said Zhang. “The second message is that something nontrivial can be hiding right under your fingertips, literally. Therefore, keep your curiosity. I will keep my eyes open and my math ready for the next surprise.”

Source: “On the bifurcation behavior of a folded notebook page,” by Chenguang Zhang, American Journal of Physics (2023). The article can be accessed at https://doi.org/10.1119/5.0097340 .