Learnéd Man applies
himself to the fundamentals; for once fundamentals are there, system comes
into being.
-- From the Analects of Confucius
By Leong Cheng Chit
No person is too young for origami. No person is too old. I embarked on my journey into the fascinating world of paper folding at the age of fifty-five.
I did have an advantage over most beginners to start with. My undergraduate training was in naval architecture. The most important structure of a ship is, of course, the hull. It is made from steel plates welded together. As you may have observed, much of the hull surface is curved. Steel plates come stacked in flat sheets, like a ream of paper. To transform them into the shape of a hull, individual sheets have to be press rolled into the correct curvature by a method of triangulation or defining a spatial form by triangles. The greater the curvature, the greater the number of triangles.
There are several similarities, as well as differences, between a ship hull and an origami model. (Origamians call their finished works models.) Both are formed from essentially flat material. In origami, you start with one piece of material, usually square shaped. In a ship hull, you require several sheets of flat material. In origami, you start and end with the original material, no more and no less. No cutting is allowed. In a ship hull, the steel sheets are first cut into the various shapes before forming. They are then joined together.
The common link between origami and a ship hull is triangulation. To appreciate this point, all you need is to unfold an origami model to its original flat form, and you will notice that the folds described are triangles. In the model, the folds are generally prominent. The fold lines of a ship hull, on the other hand, are subtle.
Incidentally, many of us in our origami youth would have tried folding paper boats. So much for ship hulls.
Like many serious folders in the West, I came to origami through mathematics. Though not a mathematician myself, mathematics forms the basis of engineering and has also found its way into non-technical disciplines, like economics and sociology. Mathematics is often referred to as the science of patterns and I was most fascinated by the folding patterns of origami. So I thought, if I could understand the fundamentals of the folding patterns, I could fold complex origami models and even design a few myself. Mathematics would shorten the learning curve.
When I started origami, I read all the origami books that I could get hold of, especially those by authors, who take the trouble of explaining the fundamentals of the folding patterns rather than merely show how the models are folded. Among my favourite origami authors is Peter Engel. He not only explains the fundamental folds, but also compares them to patterns in nature, great musical pieces and fractals, a branch of chaos theory.
I also read all the books I could get hold on geometry and the polyhedra; on the relationship between mathematics, the arts and life; and on tiling and tessellations. Geometry helped me to explain why some folds can be collapsed onto a plane and some cannot. Studying the arts and life from a mathematical perspective provided me with an appreciation of the concepts of beauty, truth, complexity and simplicity. Tessellation and tiling, excellent samples of which can be found in Islamic architecture, added to my understanding of periodic and non-periodic patterns.
The art of paper folding is understood to have begun in China, not long after the discovery of paper. It soon spread to Japan and along the silk route to the Europe. The word origami itself means folding paper in Japanese and it was in Japan, that the traditional fundamental folding bases were first formalised. The person who did most to popularise origami is a Japanese by the name of Akira Yoshizawa. He elevated the art of paper folding to greater heights.
Paper folders are often initiated into the art of paper folding by practicing to fold models based on the four fundamental bases. These fundamental folding bases are kite, fish, bird and frog, and are related geometrically. The kite base has two repeated patterns or folding modules, one is the mirror image of the other. The fish base has four modules, the bird base eight, and the frog base sixteen.
We can, of course, repeat a module at smaller and smaller scale, and continue to increase the number of modules geometrically. This is done by blintzing the four corners of the square. However, the four folding bases, arising from a simple pair of mirror image modules, should be sufficient for developing a very wide range of complex origami models.
The concept of simplicity giving rise to complexity has, in fact, a parallel in life itself. At the molecular level, all life forms are built from surprisingly only twenty amino acids and four nucleotide bases. And, if you take the trouble to examine complex things and even events in detail, you will find that their causes are basically simple ones. A few simple principles and phenomena can, therefore, have and amazing complexity when they operate in many combinations and situations. This is the essence of origami.
As I started to fold the origami models using each of the four folding bases, I learnt that there are other folding patterns and folds. The most fundamental fold, in flat folding which is folding where the model can be collapsed onto a plane, is what is known as the reverse fold. To fold a paper along a line, you are causing the two planes, divided by the folding line, to face each other. The minimum number of fold lines meeting at a point that allows the paper to collapse on a plane is four. The folding pattern consisting of four fold lines at a point is our reverse fold. Three of them are mountain folds and one is a valley fold. In particle physics, the quark is a more fundamental particle than the neutron, proton and electron. The reverse fold, therefore, is to the four folding bases what the quark is to the neutron, proton and electron.
Using only reverse folds, interesting and beautiful models can be made. In fact, my journey in origami began with experimenting with reverse folds. I created several cylindrical and conical models, by forming on a rectangular piece of art paper, a row of reverse folds, columns of reverse folds or other patterns of reverse folds. These models make interesting vases and lampshades. Again they demonstrate the concept of simplicity from complexity.
I ask myself, “Are my reverse fold models origami?” Origami purists often limit themselves to folding from a square piece of paper. Folds are developed as one goes along in making a model. Gluing is frowned upon and cutting is absolutely forbidden. My reverse fold models are formed from rectangular pieces of paper. I use a sharp object (an ink dried ballpoint pen) to score the folding lines before folding. I join the two edges together by glue to form a cylinder or cone, and in one model, I cut small parts of the base to allow the model to sit better.
As I read more origami books and folded the models in them, I came to the conclusion that not using a square piece of paper should not be an objection. After all, a rectangle or, for that matter, or other shapes, can easily be formed from a square. This only adds more layers of paper to fold. David Brill, a notable British origamian, in fact, uses a triangle for several of his animal models. What about marking the folding lines beforehand? This also should not be an objection, as many origamians use the technique of marking the folds beforehand by folding and then unfolding them for later refolding. This makes later folding easier. They also use simple implements to form the folds. Scoring the folding lines only makes folding easier.
The objection to using glue is more difficult to dismiss. Granted it is neater, origamically speaking, to tuck in a fold inside another fold than using glue to hold it in place. However, there is a technique known as “wet folding” which is pioneered by the Japanese master, Akira Yoshizawa, himself. Essentially, the technique involves wetting the paper slightly before folding. This gives the completed model greater rigidity. This also allows the paper to be moulded slightly and for the different layers to adhere (glue?) together better. So, some gluing is already permissible. I rest my case.
What about cutting? Although I did cut one of my reverse fold model, I do tend to agree that cutting will open up a Pandora’s box in origami. If cutting is allowed, what is there to prevent cutting a paper into a particular shape to facilitate folding or into the shape of the model itself? There would not, therefore, be any challenge left in folding. So, a line has to be drawn.
So as not to wander to far from the mainstream of origami, I decided that it was time for me to create my first model, which should be an animal and which should immediately be recognised as an origami model in the more traditional sense. That was three months after I took up seriously the paper folding art form. Many folders, I learned, spent years folding models from books or from more experienced origamians before coming up with their own creations. Even the process of developing a creation may take many months. The task I set for myself appeared daunting, but it is worth a try. In any case, although I did enjoyed folding the models from the books that I have or from the Internet origami websites, I felt I needed a greater challenge.
M. C. Escher, a Dutch graphic artist, who lived from 1902 to 1972, described the creative process as crossing of the divide. Escher created some of the most intellectual stimulating drawings of all time. Many of them exude a sense of paradox, illusion and double-meanings, which are also manifestations of origami models.
In 1936, Escher visited the Alhambra, the 14th Century Moorish palace in southern Spain. The walls and floors are decorated with colourful and intricately carved tessellations. Islam forbids the making of images, so the tessellations are restricted to figures with abstracted geometrical shapes. Escher’s crossing of the divide was in making the bold move of using concrete recognisable figures such as birds, fish, reptiles and human beings in many of his tessellation-like drawings.
One of such drawings of Escher provided me with the inspiration for my first origami animal creation. Entitled “Liberation” the drawing starts with black and white triangular tiling patterns at the bottom. Progressing upwards, these triangles transform themselves into ghost-like figures, into pterosaurs, cranes, geese, etc. Finally, high up in the sky are the seagulls or what look like seagulls. One of the most popular and traditional origami models, that we first learned to fold, is the crane, built from the bird base folding pattern. So, I asked myself, “Can the crane be transformed into a seagull?”
That question was not quite crossing the divide. It only points to the way, but does not tell you how to get there. So, what does it take to make a crane into a seagull? I had to find an answer, if I were to create my seagull origami model.
The idea, which finally allowed the crossing of the divide, came from a famous biologist, D’Arcy Wentworth Thomson. In 1917, Thomson published his monumental work, “On Growth and Form” which showed how different species could all be generated from a given archetypal species just by changing the coordinate system. The coordinate system he drew resembles the force field in physics. By varying the coordinates, a species of fish, say, the sea bass could be transformed into the lengthy prehistoric coelacanth or into the rounded discus fish. So, Eureka! To transform the archetypal origami crane to a seagull, all I had to do is simply vary the proportions of the coordinates or basic folding patterns of crane. Origami is after all the art of illusion.
Compared to the crane, the seagull has a shorter neck, a shorter and wider tail and a longer wing spread. The beak of the seagull is also shorter and the wings more slender. To achieve the longer wing spread and shorter beak to tail length, I hit on the idea of using a diamond shaped paper, formed by two triangles as in M. C. Escher’s “Liberation”. The diamond, or rhombus, is actually a square with one of its diagonals shortened or, if you like, lengthened. The longer diagonal gives the elongated wing spread, and the shorter diagonal the compressed beak to tail dimension. The folding base remains the same as that of the crane, that is, the bird base. The rest of the folds are quite similar to that of the crane, except that certain folds are varied, added, or subtracted to suggest the features of the seagull.
The best way to produce a rhombus shaped paper is to fold a rectangular paper along a diagonal and cutting off the two triangular sections that do not overlap. This is the maximum size that can be make from a rectangle and has the same area as the maximum size square from the same paper. Incidentally, for those who are mathematically inclined, the rhombus made in this way from an A4 size paper (8¼ in. x 11¾ in.) is a very close approximation of the fat golden rhombus. The ratio of the longer diagonal to a side of this rhombus is the golden ratio. This golden ration is commonly found in nature. The thin golden rhombus has a side to its longer diagonal in the golden ratio. The fat and thin golden rhombi form a pair of what are called Penrose tiles. Tessellation with tiles of these two shapes produces non-periodic patterns.
I have digressed. The seagull was, in a sense,
my first origami creation. Soon after, I used the same transformation trick
to turn David Brill’s rhinoceros into a tyrannosaurus rex. Dave’s model
starts from a square. Mine starts from a kite base. How this shape can
transform the tiny tail of the rhino into the large one of the T-rex is
indeed amazing. After the T-rex, I was well on my way to transforming myself
from being an origami folder to being a creator.