In origami, aesthetics meet application

Origami and mathematics seem like an odd pairing, but in the world Robert J. Lang inhabits, the two pursuits align not just to create stunning works of art, but also technological advances that can deliver telescopes into space and life-saving drugs to targeted points inside the human body.

Lang, a world-renowned origami artist and laser physicist, spoke to a packed Solomon Center audience at Brown University Thursday, Nov. 12. The Institute for Computational and Experimental Research in Mathematics (ICERM) and Rhode Island School of Design (RISD) Edna Lawrence Nature Lab co-hosted his lecture, From Flapping Birds to Space Telescopes: The Mathematics of Origami.

The talk embodied the perfect juncture where art and design meet the science, technology, engineering, and math (STEM) fields, reinforcing the RISD-championed concept of STEAM — STEM plus Art — and leaving audience members awed by the possibilities.

“Look up in the sky. You may or may not see origami. But, in the future, you will see origami looking back down at you.”

Lang traced the history of the time-honored art and its method of folding a single piece of paper with two basic folds — valleys, which dip, and mountains, which ascend — into objects. The design of any origami object is encoded in the crease pattern, Lang explained, governed by rules that are fundamentally mathematic:

• 2 colorability: A crease pattern is two colors; any touching regions must be different colors.
• The number of folds always differs by 2: Mountain – valley = + or – 2
• Alternate angles add up to a straight line
• Paper cannot intersect itself

“Every origami figure is nothing more than valley and mountain folds,” said Lang, although the intricacies of the figures he presented — 1,500 pleats to create the lifelike scales of a rattlesnake — defied such simplicity.

The art of folding has been around for hundreds of years, Lang noted, but Akira Yoshizawa (1911- 2005), credited with inspiring a global renaissance of origami, created a language to convey the process.

“Once you understood the language, how to fold origami shapes, once you could communicate, you could build upon it,” said Lang. “This ability to grow and share kicked off exponential growth.”

Origami went from flapping birds and cootie catchers to mind-blowing figures with such precise details as cloven hoofs and insect pincers. What had been static for hundreds of years soon saw thousands of new designs. And although his artist friends don’t like the inference, Lang said, when math came to origami, the field transformed. Everyone — artists and mathematicians alike — began using mathematical concepts to engineer designs.

Using pleats to create texture, based on mathematical repeating patterns, had the greatest impact on origami, according to Lang, who wrote a computer program, TreeMaker, that can solve equations simultaneously and spit out a crease pattern.

The slide of an origami cactus Lang created — a green plant in a red pot — drew gasps from the audience when he noted that the two-colored figure sprung from just one sheet of paper. He also showed a series of bugs, each one more intricate than the one before, a so-called bug war of increasing complexity.

At the same time, these design capabilities are unleashing previously unimagined potential in exploration and discovery.

A telescope lens the size of a football field needs to be launched into space. Through origami, the lens can be folded “small for the journey and large for the destination,” said Lang. Or, for more recreational pursuits, think of a kayak folded up for easy transport and then unfolded for use.

The same theory applies in medicine. Lang reeled off a list — an implant that unfolds from a delivery tube, an origami heart stent, and a box that unfolds and releases an anti-cancer drug.

“How small can we go?” Lang asked. “The laws of folding can apply to paper or to molecules … at any scale.”

There is self-folding using light absorption and hydrogel programmed folding, swelling or unfolding in response to temperature change. NASA, too, is developing methods of origami applications in space.

“Look up in the sky,” Lang urged the audience as he closed out his talk. “You may or may not see origami. But, in the future, you will see origami looking back down at you.”

Story and photo by Amy Dunkle