Origami? Mathematics!

Much of origami is mathematics. This article looks at the relationship between geometry and origami, and how the science of numbers can surprise us with paper shapes you probably never knew could exist.

Written by Takahashi Koki Photos by Kawada Masahiro and Sakai Nobuhiko

Applying modern mathematical principles to origami

Azuma Hideaki, with some of his creations.

“Unfold an origami creation and look at the creases—you'll see that it is just lots of polygons joined together. When the origami piece is complete, it is a polyhedron, a figure with many flat surfaces, and when the paper is unfolded to show the creases, it is what we mathematicians call a two-dimensional manifold. If you think of an origami creation as a manifold, some very interesting possibilities open up. That's why I got into origami in the first place,” says origami designer Azuma Hideaki.

Azuma majored in geometry at the Mathematics Department of Tohoku University's Graduate School of Science, and during the seven years of study it took him to get his master's degree, he focused almost exclusively on the theory of manifolds. He says manifolds are very important in the study of modern mathematics as a whole, not just geometry.

For family reasons he returned to his hometown, Nara, and happened to see an origami book he had used as a child. That is when he drew the link between manifolds and origami.

His approach to origami is highly unusual: “In origami, you generally start with a square piece of paper. But why not use, say, rectangular paper? And instead of folding the paper with a lot of symmetric lines, the usual origami way, how about going for symmetry around a single point? Origami instructions often involve a series of right-angled triangles, but it's still origami if you make other kinds of triangles, of course.”

This spiral avoids the right-angled triangles so common in ordinary origami—each of the triangles has one angle of more than 90 degrees. Azuma says the spiral is based on Fourier transform mathematical principles, and this led him to call it “Convolution.”

After a lot of trial and error applying ideas like these, Azuma made the spiral pictured on the right. This launched him into his own world of origami.

“Each time, I try to make it come close to the manifold model that I'm envisioning in my head.”

“Once I've decided on some of the angles to make, the final model seems to develop on its own. While making it, I can change an angle or the way a fold is done, but somehow I don't get the same feeling from the arbitrary method.”

Azuma's origami has a strange beauty, a beauty that springs from his study of mathematics.

Website of Azuma's blog (English): http://fine.ap.teacup.com/foldings/

Left: An example of a creation made with rectangular pieces of paper. If Azuma had used paper of a regular thickness, the model would have ended up more or less flat, but the unusual thicknesses of the paper made it curve in on itself.
The model on the right is a derivative form that reverses direction partway, because of a change in the orientation of the folds.

Origami keeps the brain alert

Professor Kawashima Ryuta specializes in the science of the brain and does research at Tohoku University's Institute of Development, Aging and Cancer. He has demonstrated that doing origami increases the amount of blood flowing in the prefrontal area of the brain's cerebrum, helping the brain function better. That is why many seniors' clubs have taken up origami. One club, the Sendai Seniors' Network, holds “chit-chat and origami sessions” once a week. Members, who are in their 60s, 70s and 80s, have this slogan: “Origami fun, three times in a lifetime.” The meaning? Children pick up origami at a young age. When they are parents they teach it to their children. And when they are seniors they take it up again.

NIPPONIA