The art of origami has existed in Japan since at least the 17th century, but there are hints of paper folding from long before. Initially, models were simple and—because paper was expensive—used largely for ceremonial purposes, such as the male and female paper butterflies known as Ocho and Mecho that festoon sake bottles at Shinto weddings. As paper prices fell, origami’s uses spread to gift wrap, playthings, and even geometry lessons for kids.
Then, in the mid-20th century, origami master Akira Yoshizawa helped elevate paper folding to a fine art. He breathed life and personality into each creature he designed, from a stern-faced gorilla glowering out of sunken eyes to a baby elephant joyfully swinging its trunk. With the publication of his first origami book in 1954, Yoshizawa also made the art form more accessible, establishing an easily understandable language of dotted lines, dashes, and arrows that contributed to systems still used today.
In the late 1950s, Yoshizawa’s delicate forms inspired Tomoko Fuse, now one of the foremost origami artists in Japan. Her father gave her Yoshizawa’s second origami book when she was recovering from diphtheria as a child. Fuse methodically crafted every model, and she’s been entranced with origami ever since. “It’s like magic,” she says. “Just one flat paper becomes something wonderful.”
(Graphic: See the innovations that origami has inspired in the world around us)
Among her many achievements, Fuse is famous for her advances in modular origami, which uses interlocking units to create models with greater flexibility and potential complexity. But she thinks of her work as less about creation than about discovering something that’s already there, “like a treasure hunter,” she says. She describes her process as if she’s watching from afar, following wherever the paper leads her. “Suddenly, beautiful patterns come out.”
Indeed, origami taps into patterns that echo throughout the universe, seen in natural forms such as leaves emerging from a bud or insects tucking their wings. For these exquisite folds to become scientifically useful, however, researchers must not only discover the patterns but also understand how they work. And that requires maths.
Putting numbers to origami’s intriguing patterns has long driven the work of Thomas Hull, a mathematician at Western New England University in Springfield, Massachusetts. When I walk into his school’s maths department, I know immediately which office is his. The door at the end of the hall is ajar, revealing boldly-coloured paper folded in all manner of geometric shapes. The models fill every nook of the small room—hanging from the ceiling, adorning the bookshelves, and surrounding the desktop computer. Hull himself is a riot of colour and pattern; black and white spirals dance across his shoes, which are tied with purple laces. He’s long been fascinated by patterns and still remembers unfolding a paper crane at age 10 and marvelling at the ordered creases in the flat sheet.
There are rules at play that allow this to work, he recalls thinking. Hull and others have spent decades working to understand the mathematics governing the world of origami.
As we chat, Hull pulls out an array of models that are folded in intriguing shapes or move in unexpected ways. One is an impossible-looking sheet folded with ridges of concentric squares, which cause the paper to twist in an elegant swoop known as a hyperbolic paraboloid. Another is a sheet folded in a series of mountains and valleys called the Miura-ori pattern, which collapses or opens with a single tug. Dreamed up by astrophysicist Koryo Miura in the 1970s, the pattern was used to compact the solar panels of Japan’s Space Flyer Unit, which launched in 1995.