Mathematical Methods in Origami: A Path From Math to Art

It feels a bit odd for me to be writing for SIAM News, as my education was in engineering and physics and my current career is a mixture of engineering and art. Nevertheless, math has been the key to making the intellectual connections and career transitions that led to my current situation, which combines mathematics, technology, and art: specifically, the art of origami.

In high school, Martin Gardner’s books and Scientific American columns inspired my passion for all things mathematical. A variety of mathematical activities soon followed: math team competitions in my home state of Georgia and beyond; a summer program at the University of Chicago, where I learned number theory from Arnold Ross; and, when the time came, an undergraduate education across the country at the California Institute of Technology (Caltech).

I arrived at Caltech with every intention of becoming a mathematician, but the spark wasn’t there for me in the core freshman mathematics class. Instead, I discovered digital electronics; it made computers make sense and allowed me to build tangible objects that did interesting things. I got hooked on engineering, became an electrical engineer, and put math aside — or so I thought.

After a brief stint at Stanford University to pick up a master’s degree in electrical engineering, I returned to Caltech to pursue lasers and optics for my Ph.D. Although I had been a fairly hardcore experimentalist as an undergrad, the first question I tackled as a new Ph.D. student was mathematical in nature: analyzing certain anomalies in the noise properties of diode lasers. Within a few months, I had a solution to the governing equations—followed by my first journal article—and thought, “Wow, this mathematical stuff is fun!” I didn’t fully abandon experimentation, but I did find immense satisfaction in building mathematical models — particularly when they helped me understand the functionality of real-world devices.

Upon earning my Ph.D. in applied physics, I spent a year in Germany conducting research on diode lasers before accepting a position back in Pasadena, Calif., at the Jet Propulsion Laboratory (JPL): NASA’s lead center for the unmanned space program. At JPL, those in the “science” part of the organization essentially study the workings of the Earth, planets, and universe. Undergirding that research is the “technology” side of the house, where engineers develop the underlying technologies that enable the scientists to do their science. Semiconductor lasers and optoelectronics comprise part of this technological base. When I first arrived, JPL was in the process of establishing a new laboratory, which looked like the ideal venue to pursue the types of lasers that interested me.

And so I pursued them, working on a phenomenon called filamentation (a nonlinear pattern of spatial oscillations similar to solitons) and the theory of complex diffraction gratings within diode lasers. After several years, I was approached by Spectra Diode Laboratories (SDL), a commercial laser diode company that needed a theoretician to support some of their research projects. I accepted the offer and spent more than nine years there, working on high-power diode lasers, grating-defined cavity lasers, unstable resonator lasers, and so forth. SDL was a small business when I joined but it grew over the years, as did my responsibilities. I advanced from Section Manager to Senior Section Manager and eventually to Chief Scientist, which seemed like my dream job; I was involved in a wide range of cutting-edge research and worked with many talented people.

It is the general nature of career growth that as one assumes increasingly higher levels of responsibility, there are fewer and fewer opportunities to get down and dirty with differential equations. And so when I became Vice President of Research and Development at SDL, I mainly managed contracts, funding, and special products for select customers while my group engaged in the exciting research and technical challenges that stemmed from the exponential growth of the late-1990s dot-com boom.

But during those years, I maintained one mathematical pursuit: the mathematics of origami. I had been creating new origami designs since childhood and began writing my first book with my own origami designs in graduate school. I continued to create and write about origami throughout my lasers and optoelectronics career. Many of my designs were quite complex; in fact, they included some of the most intricate origami figures of the time. Along the way, I developed several geometric techniques that I implemented in my designs — techniques that relate the intended shape to the crease pattern (CP) in the paper and ultimately give rise to the folded form (FF).

The goal of a mathematical physicist is often to follow some version of this procedure: determine how to mathematically describe the phenomenon in question, then use math-based tools to accomplish particular objectives (e.g., learn more about a physical phenomenon or achieve a specific design). In physics, this method works because the studied phenomena frequently obey rules whose simplicity allows known mathematics to provide a good approximation of what is actually happening. This also seemed to be the case in origami.

The following is a very simple rule for origami, particularly the relationship between CP and FF. If we mark two dots on an unfolded paper and subsequently fold the paper into a two- or three-dimensional shape, then the distance between those two points—measured by traveling along the FF—must be less than or equal to their original separation on the unfolded paper. Folding can reduce the distance between the dots but never increase it (only cutting or ripping can do that). A simple inequality thus governs the positions of any two points \(\mathbf{p}\) and \(\mathbf{q}\) in the CP and FF:

\[|\mathbf{p}_{\textrm{CP}}-\mathbf{q}_{\textrm{CP}}| \ge |\mathbf{p}_{\textrm{FF}}-\mathbf{q}_{\textrm{FF}}|.\]

This rule must hold for any pair of points in an origami figure: the head and tail of an animal, the tips of an arm and leg, or any two fingers. For all origami subjects—regardless of complexity—we can hence establish a system of these equations between all possible pairs of points that constrains their relative positions.

While the constraint equations are necessary, they are not necessarily sufficient for the existence of a design. But it turns out that there are folding patterns within the world of origami models that we can apply to make them sufficient! In particular, I (and others) identified families of patterns—now called molecules—that were key to the sufficiency condition. If we have an arrangement of points that satisfy the distance conditions, we can then decorate the arrangement with these molecules and achieve a CP that is guaranteed to be foldable into the desired shape.

The theory that arose from this line of inquiry revolutionized the world of origami design. Figures that had previously seemed near-impossible—insects, spiders, and multi-legged, -horned, -winged, and -scaled creatures of all kinds—were now readily designable and foldable. Origami artists around the world, including myself, went on a designing and folding spree.

In my laser work, I didn’t just write equations to describe lasers; I wrote computer codes to assist others with their designs. I did the same thing with my origami mathematics and wrote a computer program called TreeMaker for the design of origami figures. Users would first construct a mathematical graph to represent the shape, after which the program solved all of the inequality conditions that were linked to the graph and filled in the resulting arrangement of points with the appropriate molecules to complete the CP.

I worked on TreeMaker over a period of years, utilizing it to test my theories and find and plug logical holes that resulted from unwarranted assumptions. By the mid-1990s, TreeMaker was powerful enough that I could use it to design figures that were more complex than basic intuition would have allowed. I showed the program to a computer science friend at the Xerox Palo Alto Research Center, who suggested that I submit it to a computational geometry conference. So I did, and it was accepted. That paper pulled me back into mathematics, particularly computational geometry. It got my name into the scientific literature, and presently people who were working on folding problems began to ask for my help.

Throughout a decade of writing about origami, I had always envisioned a book that was more than a collection of directions, e.g., “here’s how to fold an X.” I wanted to write a book about designing your own origami — and not necessarily using explicit mathematics. Many concepts in origami design can rely on simple geometric ideas, such as packing balls in a box rather than crafting algebraic inequalities. But during my years of laser work, I made very little headway towards this goal. I therefore concluded that writing such a book would need to be a full-time project with no other distractions.

Thus came a decision point: do I continue what I’ve been doing, or do I write this book? I had done pretty well as a laser physicist, but there are a lot of laser physicists in the world. Anything that I could accomplish in lasers could probably be accomplished by someone else (perhaps even better). But I felt like I was the only person who could write the book that I wanted to write — a book that I felt needed to be written.

So in 2001, I left my job in physics (the dot-com bust provided additional incentive) and set out to write my envisioned text. I also hung out my shingle as a professional origami artist and began to respond to opportunities that I’d previously needed to turn down when I had a full-time job.

It turned out that origami has quite a few applications. In addition to artistic projects—e.g., private art commissions and commercial art for advertising purposes—some companies and products need folding to accomplish their goals: medical devices for which origami’s no-cuts requirement preserves sterility, or the deployable aspect enables laparoscopic surgical implements; collapsible forms for backpacking and shelters; and space applications like the Eyeglass telescope, where large, flat shapes (lenses, reflectors, antennas, and shrouds) must be compactly folded for stowage and travel.

During the early 2000s, the mathematical applications of origami were also coming into the mainstream. Mathematicians had been occasionally looking at folding for decades, and the first conference about origami’s use in math and science took place in 1989. But by the 2000s, mathematicians like Thomas Hull and computational geometers like Erik Demaine were establishing the fundamental laws of folding and fundamental rules of its algorithms and complexity. New design tools entered the field, ranging from Tomohiro Tachi’s Origamizer—which did for shell-like structures what TreeMaker had done for tree-graph-like structures—to Jun Mitani’s ORI-REVO, which took on solids of revolution and curved shapes with axial symmetry. For many years, the mathematics of origami had been a side project for researchers (including myself) and was not deemed a worthy academic pursuit. But that mindset was changing.

In 2011 and 2012, the U.S. National Science Foundation (NSF) put out a call for proposals in their Emerging Frontiers in Research and Innovation program that specifically asked for research in folding: underlying theory, materials, devices, and applications. It turned out to be one of the most popular initiatives of all time, with more than 100 initial responses. Over the course of two years, the NSF funded a total of 14 awards, each of which supported several professors and their students at various universities for a total of four to five years.

I ended up working on a few of these award programs. But most importantly, this funding lit a fire within the world of mathematical, scientific, and engineering-related study of origami and folding. It brought professors directly into the realm of mathematical origami and inspired many graduate students to become professors themselves and pass the torch to subsequent generations. The NSF funding eventually ended, of course, but it prompted continued interest from other government agencies and commercial companies — all of which have witnessed folding’s potential to solve real-world practical problems. In tandem with these applications, there is a continued need for mathematical tools to solve such problems as they arise.