Last Friday night, students assembled in the upper CMC to see professor Gail Nelson give a demonstration on modular origami. While we were waiting, I overheard one student explaining how she’d just learned that an irrational number raised to an irrational number can be rational. Yep, that’s the Carleton math department.

Professor Nelson learned how to make modular origami at an annual joint conference between the American Mathematical Society and the Mathematics Association of America. In addition to all of the other events, the meetings offer a variety of math classes that are just for fun. One time, one of these classes was “Math and Origami,” and Professor Nelson explains, “I said, ooh, that sounds neat.” She signed up for it.

Unlike an origami crane, which is folded all out of one piece of paper, modular origami is made out of many repeating units of paper. To form the larger work, you must make several dozen identical paper shapes and then join them together. The results can be astounding. Nelson demonstrated how the power of modular origami could be harnessed to make models of buckyballs – formally called “fullerenes,” these are those really cool molecules that happen when carbon atoms bond together in a soccer ball shape. They’re a hot topic in nanotechnology right now.

The subunits for an origami buckyball are made out of Post-it notes. The sticky side of the note is folded under, then accordioned up. The trickiest part was getting the accordions to hook together, but Nelson went around to give assistance. (By the way, if you want to make an origami buckyball yourself, you can find the instructions to do it online. Here's one tutorial from the Royal Institute of Great Britain.)

Because she is a math professor, Gail Nelson couldn’t resist giving a little math lesson along with the folding. Buckyballs come in two families, 30n^{2} and 90n^{2}. That means that a 30-sided buckyball can exist (30 x 1^{2}), as well as 120-sided (30 x 2^{2}), 270-sided (30 x 3^{2}), and so on. She knows somebody who made an 810-sided buckyball. Remarkably, it doesn’t collapse under its own weight. “They’re remarkably stable,” she says. Then she used a little geometry along with Euler’s formula to prove that every buckyball, no matter how big, must have exactly 12 pentagons in it. The 30-sided buckyball, which is made out of 12 pentagons and nothing else, is the smallest buckyball that can exist.