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This origami structure, called “Green Cycles,” by Erik Demaine and his father Martin required a week of improvisation to assemble. Credit: Renwick Gallery

The shape of a Pringle, mathematically speaking, is called a hyperbolic paraboloid. Artists have been folding paper into this shape for years. The twist? Hyperbolic paraboloids shouldn’t exist in origami—it’s impossible to make such a 3D shape using only the creases pressed into paper by hand.

By that logic, some of Erik Demaine’s artwork shouldn’t exist either.

Demaine, the world’s top computational origami theorist, has created a series of sculptures by folding concentric squares into square pieces of paper, alternating mountain and valley, and folding the diagonals. With each sculpture, the paper pops into a saddle shape called a hyperbolic paraboloid and stays there. Its accordion-like folds are pretty to look at, but Demaine, a computer science professor at MIT, isn’t sure how it works.

A paper structure featuring multiple hyperbolic paraboloids. Credit: Erik Demaine

Once the paper is folded, the entire structure settles into a natural form. “Physics finds that balance,” Demaine says. But, the mechanisms of the Pringle-like shape are still poorly understood. Demaine posits there must be little creases in the paper invisible to the naked eye, as handmade folds alone can’t account for the end shape.

Trying to solve this mystery means marrying sculpture and mathematics.

“We’ve come up with a math problem that inspires new art—and an art problem that inspires new math,” says Demaine. The 31-year-old artist creates his origami sculptures with his father Martin.

The final product, “Green Cycles” (pictured at top), was created using two different colored sheets of French-made Mi-Teintes watercolor paper, bonded together. Using a ball burnisher, which is essentially a ballpoint pen without the ink, the Demaines pushed the two-layer sheet into rings of concentric circles carved into a wood template. The paper is scored along the circular creases and cut into a donut shape, before it springs into a three-dimensional form. The artist creates several of these models and loops them together into an interlocking paper sculpture. The younger Demaine says the hardest part is assembly, which takes up to a week, because they can’t predict if the resulting shapes will twist around one another to create a solid, aesthetically pleasing piece.

“We get them to interlock, let go and let them relax, sometimes overnight, if we think we have a candidate sculpture,” he says. If the structure droops or falls apart, the pair tries again.

“Natural Cycles” by Erik and Martin Demaine

Written instructions for paper folding first appeared in 1797 in Japan. Akisato Rito published a book, Sembazuru Orikata, with lessons for 1,000 paper cranes. Adachi Kazuyuki published a more comprehensive how-to collection in 1845. By the late 1800s, kindergarteners across Europe began folding colored squares in class.

The concept was simple: no scissors, no glue, no tape—just nimble fingers bending and twisting paper into novel shapes. Origami became a modern art form in the 1950s, when Akira Yoshizawa, a Japanese artist, combined the mechanics of the craft with the aesthetic of sculpture. He created more than 50,000 paper models, never selling one. Since then, artist Eric Joisel’s crinkled lifelike animal and human figures appeared on display at the Louvre and physicist-artist Robert Lang’s detailed compositions have been exhibited at the Museum of Modern Art.

But paper folding doesn’t just create something we can ooh and aah at. It also plays a role in answering long-standing questions in math, like the fold-and-cut problem.

The first known record of the problem appeared in 1721 in a Japanese book of brain teasers, one of which asked the reader to fold a rectangular piece of paper flat and make only one straight cut to produce a Japanese crest called sangaibisi, which translates to “three-folded rhombics.” The author offered a solution through a diagram, but the problem remained an open question for centuries—how many shapes are possible?—until Demaine solved it.

As it turns out, any shape is possible—swans, horses, five-pointed stars. All that’s needed is a geometric blueprint, a guide about folding here and bending there.

“Hugging Circles” by Erik and Martin Demaine

The use of such blueprints added complexity to origami. In the 1960s, folding diagrams involved 20 to 30 steps. Now, a model could require 200 to 300 steps from start to finish. That’s a lot of folding for a single piece of paper. But, the trick is using super thin paper with long fibers, which give it strength to withstand all the pulling and tugging.

Computer programs have only added to the fun. TreeMaker, a free software program created by artist Robert Lang, takes user-generated line drawing and churns out patterns that can be printed out and folded to create the shapes. Origamizer allows users to design a 3D model and alter its crease patterns on the screen, exploring different shapes and forms.

With the help of computer software, origami has expanded beyond the art world. Scientists and engineers have found practical applications for paper folding. Car manufacturers, for instance, use origami mathematics to compute a crease pattern for folding airbags into flattened shapes. Demaine says origami structures could influence nanomanufacture, spurring the creation of flat intel chips that can spring into 3D shapes. He also met with members of the National Institutes of Health last year to discuss how the craft could help design synthetic virus-fighting proteins.

Linking mathematics and art does carry some occupational hazards, though.

“A few paper cuts a year,” Demaine says.

Three works by the father-son team are on display in “4o Under 40: Craft Futures,” an exhibition at the Smithsonian’s Renwick Gallery through February 3, 2013.

Zachary Copfer with Einstein. Courtesy of Zachary Copfer.

“I’m a microbiologist masquerading as an artist. Or am I an artist masquerading as a microbiologist?” says Zachary Copfer on his personal Web site, Science to the Power of Art. “I can’t seem to remember anymore.”

His confusion over how to describe himself is understandable. Copfer is an artist in a lab coat.

Copfer graduated from Northern Kentucky University with a bachelor’s degree in biological sciences and secondary education in 2006. He then worked as a microbiologist for Proctor & Gamble and Teva Pharmaceuticals for five years. However, he quickly learned that the commercial lab setting wasn’t the best fit for him. ”I began to lose sight of all that I had found romantic about science,” says Copfer, on his site.

Copfer instead channeled his creative energies into art, pursuing a masters in fine art in photography at the University of Cincinnati. “Photography developed into my new method of inquiry. Everything that I had missed about science I rediscovered in photography,” he adds. He completed his coursework in June.

My Favorite Artist Series: Albert Einstein. By Zachary Copfer.

Already, Copfer’s experimentations have led to the creation of a medium he calls “bacteriography.” Essentially, the microbiologist-turned-artist borrows techniques from traditional darkroom photography to develop recognizable images in growing colonies of—yes, you got it—bacteria.

My Favorite Scientist Series: Pablo Picasso. By Zachary Copfer.

Copfer has created a series of “bacteria portraits” of famous artists and scientists, including Leonardo da Vinci, Pablo Picasso, Charles Darwin and Albert Einstein. For each one, he covered a large petri dish, measuring 9.5 by 9.5 inches, in Serratia marcescens, a bacteria responsible for some hospital-acquired infections. “I use it because it is red and it pops and it gives you that great color,” Copfer told Cincinnati Public Radio.

Then, the artist placed a photograph in the dish. For instance, in one, he laid the famous photograph of Einstein sticking his tongue out, captured by UPI photographer Arthur Sasse on the scientist’s 72nd birthday. Instead of exposing the setup to ultraviolet light, as you would when developing a photograph in a darkroom, Copfer exposed it to radiation. The image cast a shadow on the bacteria. In that shadow, the bacteria grew, but in areas where the radiation passed through, they did not. Once those colonies of bacteria grew to his liking, and the piece was finished, so to speak, Copfer irradiated the portrait, killing the bacteria. Finally, he sealed the portrait with a layer of acrylic, so that it could be safely displayed.

My Favorite Artist Series: Charles Darwin. By Zachary Copfer.

The resulting portraits are bold, pop art-like reproductions of the original photographs. Comprised of red dots—each a tiny colony of bacteria—the images call to mind Roy Lichtenstein‘s comic-strip style of portraiture.

Favorite Scientist Series: Leonardo da Vinci. By Zachary Copfer.

In the titles of his works, Copfer refers to artists da Vinci and Picasso as “scientists” and scientists Darwin and Einstein as “artists.” He believes that for many others, like himself, the titles are interchangeable.

“For me, the two seemingly disparate fields of study serve the same purpose, a way to explore my connection to everything else around me,” he says, on his site.