What Is Origami? Folding as Geometry, Sculpture and Engineering

Paper reached Japan from China by the sixth century and was expensive enough that folding it was initially ceremonial, used for wrapping offerings and for the formal paper butterflies attached to sake vessels at Shinto weddings, a practice still followed. Recreational folding appears in Japanese sources from the seventeenth century, with a 1797 book giving instructions for folding a thousand cranes, and a separate European tradition developed independently in Spain and Germany, where Friedrich Froebel incorporated folding into the kindergarten curriculum he founded in the 1830s, which is how it entered schools worldwide. The modern era begins with Akira Yoshizawa in Japan, who from the 1950s created tens of thousands of original models, developed wet folding to produce curved organic forms, and, crucially, devised the system of dashed and dotted lines and arrows that became the international standard notation, without which designs could not be communicated between folders who shared no language.

The field changed when folding was treated as a formal system rather than a craft, and several results are both elegant and useful:

• •Maekawa's theorem: at any interior vertex in a flat-foldable pattern, the number of mountain folds and valley folds differs by exactly two
• •Kawasaki's theorem: at such a vertex, alternating angles around it sum to 180 degrees, which gives a simple test for whether a crease pattern can fold flat
• •The Huzita-Hatori axioms, seven operations defining what a single fold can construct, which turn out to be more powerful than compass and straightedge, since folding can trisect an angle and double a cube, both proved impossible with classical tools
• •The fold-and-cut theorem, proved by Erik Demaine and colleagues, showing that any shape with straight sides can be produced by folding a sheet flat and making one single straight cut
• •Deciding whether an arbitrary crease pattern folds flat is computationally hard, which is a formal statement of why complex designs are difficult
• •Circle packing, the insight by Robert Lang and Toshiyuki Meguro that each appendage of a model requires a circle of paper of a given radius, which turns designing a many-legged insect into a packing problem a computer can solve

Before the 1980s a folder invented models by trial and error and the results had a characteristic simplicity. Circle packing and the related tree theory made it possible to work backwards: specify a stick figure with the length and position of each limb, compute the crease pattern that allocates paper to each, and fold it. Robert Lang, a physicist who worked on lasers before turning to folding full time, wrote the software that does this, and the effect on the art was immediate and visible, producing insects with correct numbers of legs and segments, and figures of a complexity that earlier folders had regarded as impossible from an uncut square. The tradition now spans several approaches: technical folding of this kind, the curved-crease sculpture practised by Erik Demaine and others, modular work assembling many identical units, tessellations that cover a plane with repeating folded texture, and wet folding for soft realistic animals.

The applications follow from a simple fact: folding lets a large flat structure be packed small and deployed reliably with few moving parts. The Miura fold, devised by the Japanese astrophysicist Koryo Miura, packs a flat sheet into a compact parallelogram tessellation that opens completely by pulling two opposite corners, and it has flown on a Japanese satellite's solar array and appears in maps that unfold in one motion. NASA has developed a folded shade for a space telescope that deploys from a launch fairing into a structure many times wider. Medical devices use the principle for stents that are inserted narrow and expand in place, and for surgical instruments that fold through a small incision. Airbags are folded so that they deploy in a predictable sequence rather than bursting outward randomly, and the pattern is computed. Origami-inspired structures appear in architecture for deployable shelters, in robotics for machines that fold themselves from flat sheets, and at the molecular scale in DNA origami, where strands of DNA are designed to fold into specified shapes for use as scaffolds and drug carriers.

The traditional constraints are one uncut square, no glue, no marks, and the strictness with which they are held varies. Purists regard cuts as a failure of design, while others accept them as a legitimate technique, and the composite forms that use several sheets or supporting armatures are accepted or rejected depending on the community. A related argument concerns whether a model designed by software is really the designer's work, which is the same dispute that has occurred in every field where computation entered a craft, and the usual resolution is that the algorithm produces a crease pattern and the folding, shaping and judgement remain entirely human. What is not in dispute is the change of scale: the number of known original models has gone from a few hundred to a number nobody counts, and a craft once considered a children's activity now has dedicated journals, mathematical conferences and a place in aerospace engineering.

Folding began as ceremonial paper use in Japan and became a global craft through Froebel's kindergartens and Yoshizawa's notation system, which let designs be shared internationally. It was transformed by mathematical results about which crease patterns fold flat, by axioms showing that folding is more powerful than compass and straightedge, and above all by circle packing, which lets a designer specify a figure's limbs and compute the pattern. The same principles now deploy solar arrays, telescope shades, stents and airbags.