Origami
Based on Wikipedia: Origami
A Single Sheet of Paper, Infinite Possibilities
Take a flat square of paper. No scissors, no glue, no markings. Just folds. From this austere constraint, you can create a crane so lifelike it seems about to take flight, a dragon with individually articulated scales, or a praying mantis with antennae thinner than toothpicks. This is origami—and it's far stranger and more mathematically profound than most people realize.
The word itself is Japanese, combining "ori" (to fold) and "kami" (paper). But origami's history isn't a neat linear story of Japanese tradition. It's a tale of parallel evolution across continents, cross-pollination between cultures, and—most surprisingly—a modern transformation driven by mathematics and computational theory that has turned a craft into something approaching science.
Paper Arrives in Japan
Paper reached Japan from China via the Korean Peninsula around the seventh century. The Japanese didn't just adopt the technology—they improved it. During the Heian period, roughly spanning the ninth through twelfth centuries, they developed a technique called nagashi-suki. The innovation was adding mucilage to the papermaking process, creating stronger, more flexible sheets than the conventional method produced.
This Japanese paper, called washi, became the foundation for origami. Unlike modern wood-pulp paper, washi uses fibers from the bark of the gampi tree, the mitsumata shrub, or paper mulberry. It's tougher, more forgiving of repeated folds, and holds creases with almost architectural precision.
But here's what's fascinating: the first origami wasn't for fun. It was deadly serious.
When Folding Paper Was a Matter of Protocol
As paper became common in Japan, it started appearing in religious ceremonies. At Shinto shrines, folded paper decorations called gohei, ōnusa, and shide took on ritual significance. The way you wrapped a gift in folded paper communicated respect, intention, and social standing.
The Imperial court during the Heian period established formal codes of etiquette for wrapping money and ceremonial goods. This wasn't art—it was protocol, as rigid and consequential as any court ceremony. Fold the paper wrong, and you'd committed a social blunder that could damage relationships or offend powerful people.
By the Muromachi period, spanning the 1300s through the 1400s, two powerful clans—the Ogasawara and the Ise—had codified these practices into a complete system of ceremonial origami that persists to this day. The Ise clan oversaw etiquette inside the palace of the Ashikaga Shogunate, and under one particularly influential figure, Ise Sadachika, these folding protocols spread throughout the daimyo and samurai classes.
This ceremonial origami looked nothing like the paper cranes you might picture. The shapes were geometric and abstract—stylized rather than representational. The noshi wrapper attached to wedding gifts, the male and female butterfly shapes displayed on sake vessels at celebrations—these weren't meant to look like real objects. They were symbols, their meaning encoded in the precise geometry of their folds.
Japanese even has separate words to distinguish these traditions. Recreational folding became "origami," while ceremonial folding retained the older term "origata."
When Did Folding Become Fun?
Nobody knows exactly when Japanese people started folding paper just for enjoyment. But we have clues.
A kozuka—the decorative handle of a Japanese sword—made by the metalworker Gotō Eijō between the late 1500s and early 1600s features an engraved image of a folded paper crane. This suggests that recreational origami existed at least by the Sengoku period or early Edo era. The crane wasn't a ceremonial symbol being depicted. It was a paper toy, rendered in metal as decoration.
By 1747, recreational origami had clearly become mainstream. A book called Ranma zushiki, which featured designs for ranma (decorative transoms in Japanese architecture), included various origami patterns as ornamental elements. The paper crane appeared among them, recognizable to modern eyes.
Then came the first true origami instruction manual.
Published in 1797, Hiden senbazuru orikata—which translates roughly to "Secret Methods of Folding One Thousand Cranes"—contained forty-nine origami designs created by a Buddhist monk named Gidō from Ise Province. These weren't simple models. They were sophisticated, technically demanding pieces that required skill to execute. The existence of such advanced designs implies a mature culture of paper folding had been developing for some time.
Gidō didn't stop after publication. He continued creating, leaving behind at least 158 masterpieces. In 1976, Kuwana City in Mie Prefecture—Gidō's hometown—designated those original forty-nine designs as Intangible Cultural Properties. The city now certifies qualified practitioners who can correctly produce these works and understand their history. Some of these folding methods have even been published on YouTube, connecting an eighteenth-century monk's creations to twenty-first-century audiences.
Meanwhile, in Europe: The Forgotten Art of Napkin Folding
Paper folding wasn't exclusive to Japan. Europe had its own sophisticated tradition—except it used linen napkins instead of paper.
During the seventeenth and eighteenth centuries, napkin folding flourished as a serious art form. At aristocratic dinner tables, elaborately folded napkins served as status symbols, demonstrating wealth and sophistication. The shapes could be extraordinarily complex.
Then porcelain arrived, and everything changed.
According to historian Joan Sallas, the introduction of fine porcelain to European tables displaced napkin folding as the primary way nobles showed off their taste and resources. Why struggle with fabric when you could display expensive Chinese or European porcelain? The tradition faded, mostly forgotten.
But not entirely. Some techniques survived, particularly in education. Friedrich Fröbel, the German educator who invented kindergarten, incorporated paper folding into his curriculum. The designs he published show clear stylistic similarities to the old napkin-folding repertoire. Fröbel's system standardized something that would later reshape origami worldwide: the bicolored square as the starting shape, and the prohibition against cutting.
Spain preserved another survivor. The "pajarita"—a stylized bird—traces back to at least the nineteenth century and remains a national symbol of paper folding.
East Meets West, and Origami Transforms
In the 1860s, Japan ended its long isolation and began rapidly modernizing. As part of this transformation, they imported Fröbel's kindergarten system—and with it, German ideas about paper folding.
This cross-pollination changed origami fundamentally.
Traditional Japanese origami had been relatively loose in its rules. Practitioners used various starting shapes, not just squares. They sometimes cut the paper. If they wanted color or patterns, they added these after folding. The German approach was stricter: start with a bicolored square, never cut, fold only.
Japan's first kindergarten opened in 1875, and origami became part of early childhood education. The 1877 regulations listed twenty-five activities, with origami among them. Children's magazines began regularly publishing origami articles. A 1908 book, Origami Zusetsu, made explicit the distinction between ceremonial and recreational origami.
The art form was becoming codified, standardized, and—crucially—documented in ways it never had been before.
The Yoshizawa Revolution
In the early twentieth century, a few practitioners began doing something new: creating and recording original designs rather than simply passing down traditional ones. Among them, one name towers above the rest.
Akira Yoshizawa transformed origami from a folk craft into a recognized art form. He developed wet-folding, a technique where dampened paper can be molded into gentle curves rather than sharp geometric edges. When the paper dries, it holds these organic shapes, enabling lifelike animal models that seem to breathe.
Even more influential was his contribution to documentation. Along with American folder Sam Randlett, Yoshizawa developed a diagramming system that could clearly communicate folding instructions across language barriers. The Yoshizawa-Randlett system uses standardized symbols for valley folds, mountain folds, and other techniques—a kind of musical notation for paper.
Before this system, learning origami meant watching someone fold in person or puzzling through vague written descriptions. Afterward, a folder in Brazil could precisely follow instructions created by a folder in Japan. The art could spread globally with unprecedented fidelity.
The Mathematical Turn
During the 1980s, something unexpected happened. Folders began studying folded forms mathematically, asking questions that had never been systematically explored. What determines whether a particular shape can be folded from a flat sheet? What are the fundamental rules governing crease patterns?
The answers turned out to be surprisingly deep.
A flat-foldable crease pattern—one where the paper can be completely flattened—must satisfy certain mathematical constraints. At any vertex where creases meet, the number of valley folds and mountain folds must differ by exactly two. The angles around each vertex must add up to 360 degrees in a specific alternating pattern.
These discoveries enabled computational origami. Instead of relying on intuition and trial-and-error, folders could use algorithms to design crease patterns for arbitrary shapes. Want to fold a specific insect with twenty legs? There's now a mathematical procedure to generate the crease pattern that will produce it.
Robert J. Lang, a former physicist, became perhaps the most prominent figure in this computational revolution. His software, TreeMaker, can generate crease patterns for folding any tree-like structure—which turns out to describe most animals and insects. Erik Demaine, a mathematician at the Massachusetts Institute of Technology (MIT), explored the theoretical limits of foldability, proving theorems about what shapes can and cannot be achieved.
The complexity of origami models exploded. Designs that would have been inconceivable in the 1970s—insects with compound eyes, dragons with hundreds of scales—became achievable. The art form that had started with paper cranes was now producing sculptures that could take dozens of hours to fold from single uncut squares.
The Materials of Folding
While mathematics enabled new designs, materials science determined what could actually be folded.
The only fundamental requirement for origami is that the material holds a crease. Beyond that, folders have explored remarkably diverse options.
Standard origami paper, often called kami, comes in prepackaged squares ranging from one inch to ten inches or more. It's thinner than copy paper, colored on one side and white on the other, though dual-colored and patterned versions exist. This paper works well for simple to moderately complex models.
For wet-folding, heavier paper is essential—at least 100 grams per square meter. The paper must hold moisture without disintegrating and maintain its new shape when dry. The results can be startlingly lifelike, with soft curves that geometric folds can never achieve.
The most extreme models require specialty materials. Foil-backed paper—thin metal foil laminated to thin paper—can hold extremely tight folds and maintain complex shapes. Even more specialized is tissue foil: kitchen aluminum foil sandwiched between layers of tissue paper, which must be made by hand since no commercial version exists. These materials enable the insect models with paper-thin legs that would be impossible in standard paper.
Traditional washi remains prized for its strength and forgiveness. Artisan papers like unryu, lokta, hanji, and kozo offer extremely long fibers that create strong, compressible sheets. These are often treated with methylcellulose or wheat paste before folding to give them body.
And then there's money origami. Currency from various countries has become a popular medium, spawning its own subgenre with names like Dollar Origami and Orikane. The standardized size and distinctive markings of banknotes create unique design possibilities—and the results make memorable tips.
Tools and Techniques
Purists insist that origami should be folded in mid-air, using only hands, with no flat surface and no tools. There's something almost meditative about this approach—the paper exists only in relation to your fingers, rotating and transforming through pure manipulation.
But for complex models, tools help. A bone folder—a smooth, flat implement traditionally made from animal bone—creates crisp, sharp creases that fingers alone struggle to match. Paper clips serve as extra fingers, holding flaps in place while you work on distant parts of the model. Tweezers become essential when folds grow too small for fingertips.
When working from crease patterns—the flat diagrams showing where all folds will eventually go—a ruler and ballpoint embosser help score lines accurately. For wet-folding, spray bottles dampen the paper, and fixative sprays help finished models hold their shape.
Beyond Still Life: Origami That Moves
Not all origami sits quietly on a shelf. Some of it flies, inflates, or moves in response to touch.
Action origami encompasses these kinetic designs. The most famous is the traditional Japanese flapping bird: pull its tail, and its wings beat. Robert Lang has created origami instrumentalists whose hands move as if playing music when you manipulate their heads.
Some designs require inflation to complete—you fold the structure flat, then blow into an opening to expand it into three dimensions. Others capture kinetic energy from hands pushing or pulling at specific points, translating that motion into movement elsewhere on the model.
Then there's modular origami, which challenges the single-sheet constraint in a different way. Instead of one sheet folded many times, modular origami uses many identical sheets folded simply, then assembled into complex structures. The individual units, called modules, are often geometrically elegant on their own. But when combined—sometimes hundreds together—they create elaborate polyhedra, spheres, and decorative balls called kusudama.
A related tradition called golden venture folding, or 3D origami, uses large numbers of simple triangular units to build elaborate sculptures. The technique emerged from Chinese refugees detained in America, named after the ship that brought them. It's now practiced worldwide, producing models that can contain thousands of individually folded pieces.
From Craft to Engineering
The mathematics that enabled more complex art also opened doors to practical applications.
Medical stents—the tiny tubes inserted into blood vessels to hold them open—now sometimes use origami-inspired folding patterns. The stent can be compressed small enough to travel through vessels on a catheter, then expand to full size when deployed. The folding pattern determines both how compactly it can be stored and how uniformly it expands.
Space engineering has embraced origami with particular enthusiasm. Satellite solar panels must be large enough to generate adequate power but compact enough to fit inside rocket fairings during launch. Origami folding patterns solve this problem elegantly. The Miura fold, developed by Japanese astrophysicist Koryo Miura, allows a large flat surface to collapse into a compact bundle along one smooth motion and expand just as easily.
Packaging engineers study origami to create boxes, containers, and protective structures that fold flat for shipping and storage. Architects explore origami principles for deployable structures, emergency shelters, and facades that can change shape in response to sun or wind.
The deeper insight is that origami isn't really about paper at all. It's about the geometry of folding—a field that applies to any flat material, from metal sheets to solar sails to cell membranes.
The Social Dimension
The internet transformed how origami spreads. Before social networks, learning new designs meant buying books, attending conventions, or finding local clubs. Now, folders share crease patterns on forums, post video tutorials on YouTube, and display finished models on Instagram.
This connectivity accelerated innovation dramatically. A technique developed in Tokyo can reach folders in São Paulo within hours. Computational tools can be shared freely. The community of serious folders, once isolated by geography, now operates as a global network.
Conventions bring this community together physically. Origami USA, the Japan Origami Academic Society, and similar organizations host annual gatherings where folders share techniques, display models, and push each other to new achievements. The atmosphere combines the rigor of mathematical conferences with the creativity of art exhibitions.
The Continuing Appeal
What draws people to origami? The answers vary.
For some, it's the meditative quality of folding—the focus required to execute precise creases, the satisfaction of watching a flat sheet transform step by step into something recognizable. For others, it's the mathematical elegance, the deep questions about what shapes can emerge from systematic folding.
There's also the accessibility. Unlike sculpture or painting, origami requires almost no materials—a single sheet of paper, readily available anywhere, contains infinite possibilities. A commuter on a train can fold. A prisoner with limited resources can fold. A child learning geometry can fold.
Perhaps most fundamentally, origami demonstrates a principle that feels almost magical: that constraints enable creativity rather than limiting it. By accepting the rule that you cannot cut, cannot glue, cannot mark the paper, you are forced into solutions that would never occur otherwise. The limitations become liberation.
A flat square of paper, through nothing but folds, becomes a crane, a dragon, a molecule, a satellite array. The transformation seems impossible until you see it happen. And then you want to try it yourself.