Uniaxial base


What is a uniaxial base?

Post is a part of a larger series (Uniaxial base):

Every origami model, no matter how complex, has a base in its core. Those bases can vary in their complexity, from simple traditional origami bases to complex custom made bases that can be seen today. 

Nevertheless, one class of origami bases stand out: a uniaxial base.

A uniaxial base is a class of origami bases in which, when collapsed all its flaps lay on a single axis, and its hinge creases are perpendicular to that axis. That is its definition, but what does this definition mean? 

I will try to explain, but before that, I would like to underline once again that uniaxial bases are not something unusual and rare. Quite contrary. These bases are ever-present. 

You already come across such bases, since all four well known traditional origami bases are uniaxial. In figure 1, you can see one of them – a traditional fish base. It is fairly simple since it has only four flaps, and all of them lay on the same axis.

Frog base
Figure 1

Furthermore, axial creases are clearly visible on the crease pattern.  You see, by definition, axial creases are lines that connect circles’ centers, thus forming two distinctive triangles (in figure 1, axial creases are marked in green). These triangles are known as axial polygons. Now, before we move on, it is important to emphasize that axial polygons do not necessarily have to be triangles. Any convex shape will do, but triangles and quadrangles are the most common. 

If you look at figure 1 once again, you will see that upon collapsing all these green lines forms a single axis.

Nonuniaxial base

Even though uniaxial bases are ever-present, that does not mean that all bases are necessarily uniaxial. Quite contrary. For instance, very prolific origami artist John Montroll has developed and subsequently used several non-uniaxial bases. A great example of such a base is shown in figure 2. That base was developed by Montroll while designing his dinosaur models (diagrams for those dinosaurs’ models were published in his book – Prehistoric Origami: Dinosaurs and Other Creatures). As you can see in figure 2, the base is obviously not uniaxial

Nonuniaxial base
Figure 2

Why then, most of the people believe that uniaxial bases are the only ones. Well, first of all, the fact that almost all traditional origami bases are uniaxial could be a dominant reason. But there is something else. You see, there is a software such as TreeMaker (developed by Robert Lang), that is based solely on the circle packing method and the uniaxial base theory. Those two facts led to the widespread belief that a circle packing method can produce only a uniaxial base and vice versa that the uniaxial bases could be made only by using the circle packing method. I have to admit that this belief could seem logical and accurate and that it has very sound foundations. However, that line of thinking is not entirely true. Uniaxial bases could be made using other techniques as well. Namely, we can make one using the box plating method.

Box pleated uniaxial base

Even though uniaxial bases are in most cases a result of a circle packing technique it is surprisingly easy to construct box pleated uniaxial bases. One such base can be seen in figure 3.

Box pleated uniaxial base
Figure 3

There is no doubt that the base is uniaxial. You only have to flatten the base and look at all flap’s tips. The fact that the base is uniaxial will become apparent immediately. Besides, all hinge creases are perpendicular to the prime axis, which is also one of the preconditions for uniaxiality.  

We could come to the same conclusion if we analyse the crease pattern only. At first glance, it is apparent that all circle centres are at the paper edges. Meaning all of them are at elevation zero. That also means that the flaps’ tips are at that same elevation.

On the other hand, if we design a base in which some of the circle centers are at different elevations, the uniaxiality of the base will be lost. In figure 4, you can see such a base.

Box pleated nonuniaxial base
Figure 4

As you can see one circle center is at different elevations consequently making the base non-uniaxial.

Importance of uniaxial bases

Origami is vast uncharted territory. Maybe such a statement is not something that you would expect from me, but the truth has to be told. But, due to the work of Robert Lang, Takahashi Meguro, and few other origami masters one segment of origami is “charted” to the point that software can be developed. For example, an already mentioned TreeMaker, developed by Robert Lang. Obviously, to make the development of such software possible, a comprehensive theory has to exist.

The theory behind the uniaxial base

The first step in the crease pattern design process, as you probably know, is to pack a needed number of circles and rivers on a square piece of paper. The process itself is the most creative part of a base design even though it can be done automatically by software such as TreeMaker.

In figure 5, an example of one simple uniaxial base is given.

Uniaxial base
Figure 5

The next step in the procedure asks for adding creases between centers of all adjacent circles. These new lines, marked in green in figure 6, are axial creases that form axial polygons.

The base in figure 5, is a nice example since the two most often used types of axial polygons are present: triangles and quadrangles. Higher-order polygons are hardly ever used since they could always be reduced to triangles and quadrangles by introducing an additional circle or flap.

Since that is supposed to be a uniaxial base, all these axial creases must coincide when the base is collapsed. But, to enable that, we have to add additional creases in every single axial polygon. These additional creases are most commonly known as molecules.

Molecules
Figure 6

Triangle molecule

The simplest of all and most widely used type of molecule is the triangle molecule. That molecule is used whenever we encounter axial polygons in the shape of a triangle. Triangle molecule is so versatile that traditional origami bases (Kite, Fish, Bird, and Frog base) have nothing but triangles and triangle molecule known as rabbit ear. To form the rabbit ear, we only have to draw bisectors of all triangle’s angles. They will undoubtedly meet at a single point. Finally, we have to add a crease from that point perpendicular to one of the edges of the triangle. In figure 6. you can see how it is done.

Quadrilateral molecules

Although triangles are the most common, they are not the only ones. Sometimes axial polygons are quadrangles. When we encounter such a polygon, we have to rely on more complex molecules. There are a handful of molecules that can be used, like Waterbomb molecule, Arrowhead molecule, Sawhorse molecule, etc. Some of them are specialized while some are more versatile, but the most versatile of all is the so-called Gusset molecule. It is so versatile that all other molecules, except the Arrowhead molecule, are in fact its special cases. An example of such a molecule is shown in figure 6.

Conclusion

The uniaxial base is the most advanced origami concept developed so far. Very few areas of origami are scrutinized as the Uniaxial base. 

Even though uniaxial bases are quite versatile and as such widely used, they are not universally applicable. The technique is best suited for bugs with lots of legs and tentacles. Unfortunately, such a large number of flaps are not always needed, so the origami authors have to resort to some other techniques and ideas.

Nevertheless, studying uniaxial bases is a must for everyone who is striving toward origami mastery. Uniaxial bases can give you great insight into the origami theory simply because they are well studied, documented, and explained.