Hey, you never know when it'll come in handy.

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.

Folding 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.

If 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.)

This creates a tiny little fold that's not really useful.

What if we fold a bit further, to about 30 degrees?

Now the tab is bigger and more of a right triangle.

But can we do better?

This 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:

In the paper, Zhang describes this kind of folding in a diagram:

Using trigonometry, Zhang derives a function for this fold:

Then he graphs the function:

The 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 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. 

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