Origami: The Art of Paper Folding

A blank sheet of paper.

It’s the birthplace of so much art, so many ideas. Sketching, painting, writing - these are just some of the things that come to mind when we think about paper as a medium. But there are other forms too that one may not necessarily associate with the paper itself: Sheet music, movie scripts, and so much more. All these art forms are in some way shape or form indebted to the inviting nature of a blank sheet of paper - the way it inspires color, the way it inspires creativity. It is quite possibly a medium for expression, much like language. And yet, you could argue, even with these well-known forms of art, we haven’t witnessed paper’s true eloquence. 

You’d be forgiven to not think too much of this unassuming material. It is so thin and seemingly fragile after all. Yet, the properties of paper might surprise you. For example, after just 7 or 8 folds, paper has structural integrity comparable to that of steel. If you can amass enough strength and fold it, say, 20 more times, even if you were to start off with a thinner than the average piece of paper, you will have a block of paper that is well over a quarter mile tall - taller than the empire state building! From almost negligible thickness to taller than the empire state building in less than 25 steps. Just. Like. That. You could say that simply folding a piece of paper over and over again is boring. And, well, if you keep folding in it in half all the time, that might as well be true. But what if you folded the two edges of the paper down, then folded everything inward and folded half of the edges outward. You would have a plane, and this is just the beginning.

These are a much more graceful demonstration of paper’s ability, and they are specifically examples of what we know as origami, or ‘the art of paper folding.’ 

The oldest documentation of origami is from the 17th century, although it almost certainly originated before that. The word Origami is Japanese, with "Ori" meaning folding, and "kami" (which changes to "gami") meaning paper. Despite that, some believe origami may have originated in China considering that’s where paper was first created. 

Recreational folding may have started with cloth rather than paper. They were considered merely decorative at the time, and the art was reserved for only the wealthy. However, cloth rarely held their folds as well as paper did, and as paper became cheaper and more available, it slowly became the primary medium of this art form. As with most forms of art, origami, too, had humble beginnings. The earliest models, which is what they are called, were only somewhat reminiscent of the objects they were inspired by. Arguably the most famous origami model is the ‘crane’ bird. Kids in Japan learn how to make paper cranes at a young age as Origami is integrated early into the education system. Of course, Origami has come a long way since those days. While we are all probably somewhat familiar with origami, some of the models are much more detailed than one may think. So much more in fact that they take even the most experienced folders not days, not weeks, but years to complete. 

The Cactus, opus 680, for example, took Robert J. Lang, considered a master of the field, nearly 7 years from start to finish. And I should say that the entire object, pot included, is from a 1 meter-squared uncut piece of paper. And that is really not all that unusual for origami. Even Akira Yoshizawa, considered to be the greatest folder of all time and a modern legend of the art form, took many years to create some of his more famous models. Which begs the question, how complicated can these things really get? 

Well, as with the strength of half-folding paper, their complexity is also rather deceptive. You start with only a few rules. Rules that center around the idea of flat foldability - or in other words, an origami model that can fold flat, without adding a crease that already wasn’t there. If you were to unfold an origami model, what you would have is a crease pattern.
This pattern is the “underlying blueprint,” as Lang likes to call it, of an origami piece - it’s canvas if you will. There are 4 laws that flat foldable crease patterns obey. 

First, there’s 2 colorability - or the ability to color a crease pattern using only 2 colors without having the same color meeting.

Next, the difference between the number of mountain and valley folds will always be 2.

If you number the angles around a point where multiple crease lines intersect, both the even-numbered angles and the odd-numbered angles will add up to 180 degrees each. 
And finally, the impermeable nature of layers - meaning folds cannot be made through layers. If a crease pattern is flat foldable, it will follow these 4 laws and vice versa.

These laws are fundamental to this framework. If you remove these laws, will you not have origami? It is like asking if you were to remove certain axioms from Euclid’s elements, would you not have geometry? In both cases, the answer is a resounding yes. Of course, you would. But the framework you will be left with will lead to a different world of Origami. 

The shapes and models that will emerge out of that framework will be different from one where these are obeyed. Nevertheless, these simple 4 laws are by no means inadequate, for out of them emerges a rich framework of paper folding that can accommodate the design for nearly anything. A framework so rich, in fact, that it can be used to construct geometry that was classically impossible - by this I mean solving problems like trisecting the cube, which is a problem Euclid struggled with only to conclude that it was impossible. He was right in the sense that it was impossible to do so using the tools that were at his disposal - a straightedge and a compass. However, using the power of origami, and a sheet of paper - a modest tool in comparison - one can trisect an arbitrary angle. 

In addition, one can also double the cube, solve cubic equations, understand  combinatorics, and so much more. One may even demonstrate the Pythagorean theorem with just a square sheet of paper. That poses an interesting question. Should other education systems also incorporate Origami into the learning experience?  

This structured study of origami as something more than just a form of art has led origami to open a new avenue for engineering. Medical science is benefiting from having to make smaller incisions because the instruments they need to use can fold into much smaller, compact shapes outside of the body. But once they are inside, they can unfold into larger, more functional units and get to work. Astronomy and solar panel engineering are also benefiting by folding telescopes and panels alike into much more manageable sizes that rockets can carry. Origami design has actually been implemented in the design of large structures like stadiums as well.

Just the simple property of being able to reduce dimensions of a 3D object and allow it to be folded into a 1D flat sheet has shown incredible utility in the world of engineering and material science.

It is no wonder, then, that the modern fascination with origami is not so much from a cultural standpoint, but rather from a physics and mathematical standpoint. Most of the people developing origami today are indeed mathematicians. 

That’s not to take away from its sentimental value, of course. In the months after the bombs were dropped in Hiroshima and Nagasaki, everything was still in ashes and the shell-shocked economy had no way of providing toys for the kids of the region. At that time, the community had to return to their comforting roots and make toys out of origami. 

Sadako Sasaki was one of those kids that found comfort in the therapeutic activity of paper-folding. Sasaki was only an infant when the bombs fell that fateful day, and as the dust settled, she grew into a world that was torn apart physically and emotionally. Nearly 10 years later, Sasaki fell seriously ill. Her lymph nodes were swollen, and she was taken to the doctors. She was diagnosed with radiation-induced leukemia. Her parents tried keeping the news from her, but Sasaki knew all too well. Sasaki remembered the ancient Japanese legend that “if you were to fold a 1,000 paper cranes, you will be granted a wish.” Sasaki’s only wish was to live. In an unrelenting battle to save her own life, Sasaki began folding cranes right from her hospital bed.

Her friends couldn’t sit around and just be bystanders, so they started folding as well. On October 25th, 1955 Sasaki passed away having folded well over 1,000 paper cranes. Sasaki’s life came to a premature ending, but her efforts and those of her friends were not in vain. Today, her paper crane is recognized as a symbol of peace, harmony, and healing. Some of the cranes Sasaki built have been exhibited around the world as a gesture of peace. It is a testament to how something as simple as origami can hold meaning beyond its folds.  

After all these years, Origami is still a very traditional form of art that retains a lot of its originality in the depths of its creases. The use of anything other than a square sheet of paper is still frowned upon in some origami communities, gluing is a carnal sin, and scissor use is utterly laughable. Origami, unlike other forms of art, does not add or subtract from the medium. It transforms the medium. The lack of tools has made it resistant to changes for a long time, and that doesn’t look like it’s about to change anytime soon. The process of folding is as therapeutic today as it used to be centuries ago. The details have certainly changed though; with the addition of computer software, it is now aiding the tedious task of folding and allowing users to marvel at the ridiculous levels of complexity achievable.

However, the fundamentals have largely remained the same. The same square sheet of paper. The same 4 rules. What is changing, however, is our admiration for this art form. From its inclusion in engineering to architecture, from the operating room to the docks of telescopes probing the edge of the universe, from the hands of children to those of Nobel Peace Prize winners, origami has found its place in a lot more places than we realize. It has also become a graceful symbol of peace with stories such as that of Sadako Sasaki’s. 

If paper is indeed the language, then Origami is its poetry.

- MA, MM