You find a lot of great stuff when browsing through arXiv, the repository of scientific research run by Cornell University, including very elaborate solutions to problems you didn’t necessarily know existed. But now that we think about it, we would like to know the ideal way to fold a notebook page in order to “bookmark” it, actually.Enter Chenguang Zhang, a postdoc in MIT’s math department and Earth Resources Laboratory, who crunched the numbers to find the perfect folding method. In a new paper posted to arXiv, Zhang says he was inspired by a square notebook with a top spiral that he’s been using.More Math!Can You Solve This Viral Triangle Brain Teaser?The Amazing Math Inside the Rubik’s CubeThe Math of the Perfect BubbleFolding as a way to bookmark pages, Zhang says, conjures the idea of origami. In recent decades, as the study of how materials can fold and deform becomes more important in micromaterials, spaceflight, and everything in between, researchers are studying origami and kirigami principles under the scrutiny of science. As Zhang explains, “Since the last century, it has seen significant developments that, interestingly, were far beyond its originally recreational and artistic nature.”So what’s the best way to fold a square page so that the maximum amount of area peeks out the side of your notebook? Let’s do an experiment that simplifies Zhang’s own experiment.Folding a rectangular notebook page.Caroline DelbertIf your notebook is a rectangle with a top spiral, like the classic quad-ruled steno books sprinkled all around my entire life, your fold is super easy. The maximum fold flap is just the entire area that peeks out the side of your notebook, and for a rectangle, that’s however much longer your page is than it is wide. My notebook is 6"×9", but loses some length to the spiral and perforation. The fold makes a tab that’s about 2.5"×6", so 15 square inches.But what about a square page? If you fold a square page the way I did above, it just makes a clean right triangle with no tab. Instead, we have to have a fold that’s less 45 degrees and more 35 degrees. (These are just reference approximations.)Rotating the fold counterclockwise so it’s about 35 degrees instead of 45 degrees.Caroline DelbertThis creates a tiny little fold that’s not really useful.What if we fold a bit further, to about 30 degrees?Rotating the fold more, from 35 degrees to 25 degrees.CarolineNow the tab is bigger and more of a right triangle.But can we do better?The optimum fold, at about 15 degrees.Caroline DelbertThis fold makes the largest tab of all, although it’s just a little bigger than our previous tab—the sweet spot is somewhere between the two.Here’s how the areas compare, very roughly:Areas of the tabs created by different folds.Caroline DelbertIn the paper, Zhang describes this kind of folding in a diagram:Case 2 shows the “bookmark fold.“Chenguang ZhangUsing trigonometry, Zhang derives a function for this fold:Chenguang ZhangThen he graphs the function:Chenguang ZhangThe sweet spot is when the b value is about 0.6, which maps with Zhang’s conclusion. “This means to have the most visible bookmark: first pick the top-left corner, then pick from the right edge a point 58.6 [percent] from the bottom, then fold,” he concludes. There you have it. There really is math for everything.Caroline DelbertCaroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She’s also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all.
