Rabbit ear fold


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

Origami rabbit ear is one of the primary folds in origami. In its essence, a rabbit ear is an elegant way to force three lines or folds that forms a triangle to coincide.

Folding procedure

Before we start to talk about the importance of rabbit ear, it would be wise to know how to fold one. So, to show you a procedure, let’s take a square piece of paper and let’s fold it along the diagonal (figure 1A).
What we got are two identical triangles. And since by definition, a rabbit ear is a way to force triangle sides to coincide, we will do exactly that with one of these triangles. We will make its sides to coincide; we will make them all appear on the same mutual axis.

To do so, first, we have to define or fold angle bisectors of all three triangle’s corners (Figure 1B). These three bisectors will intersect in a single point which is more than convenient. But from a flat foldability standpoint, this point (or node) does not satisfy one of the basic origami rules, namely the Meakawa-Justin theorem. All three creases that converge in that node are valleys. Additionally, the number of folds at that node is an add number which is not allowed. A solution to this problem is obvious and very simple; we have to add one additional crease at that node. In other words, we have to add a mountain fold that is perpendicular to the paper side (Figure 1B).

Rabbit ear folding procedure
Figure 1

As you can see in figure 1 upon folding, all triangle’s sides are aligned. They all coincide with one another, which is what we wanted in the first place.

Rabbit ear and traditional origami bases

Now that we know how to fold a rabbit ear, I would suggest analyzing a few traditional origami bases to see if we could find rabbit ear there.
First, the obvious starting point is a so-called fish base. If we look at figure 1, we will see that we have already folded the first half of the fish base. Only what is left to do is to fold another triangle in the way we folded the first one (figure 2).

Rabbit ear and Fish base
Figure 2

Isn’t that interesting?

But, if we go further and analyse the bird base, we will soon realise that the same pattern repeats. If you look at figure 3, you will see that the bird base consists of four rabbit ears and nothing else.

Rabbit ear and Bird base
Figure 3

If we go even further and analyse even more complex bases we will see that the same pattern repeats again and again. In figure 4, you can see that the frog base consists of eight rabbit ears and nothing else.

Rabbit ear and Frog base
Figure 4

Rabbit ear and the circle packing method

Now that we have established the link between the rabbit ear and traditional origami bases let’s dig deeper into the origami theory to see the real importance of the rabbit ear. Namely, we will discuss the connection between the circle packing method and rabbit ear.

I have mentioned this in a few of my previous posts, but I will repeat it once again: the circle packing method is all about packing circles on a piece of paper. That is more or less obvious,  but the concrete procedure is not that simple. You see, in this method, one of the first steps is to connect centres of all adjacent circles with additional lines or folds. These lines are called axial creases, but what’s more, these lines form close concave polygons known as axial polygons. 

Why am I telling you this? Well, because according to the circle packing method, the next step upon defining axial polygons, is to find a way to force all polygons’ edges to coincide. How it could be done when the axial polygon is of complex shape is beyond the scope of this blog post. Only what you need to know is that the set of folds that have to be added inside the axial polygon is called a molecule. Remember this term: the molecule.
Now, something interesting. Even though these axial polygons can be of any concave shape, most often they are simple triangles. And here comes our rabbit ear. The rabbit ear is nothing but the molecule, or a set of creases that allows us to fold any triangle axial polygon that appears as a product of the circle packing method. That is its main value.

Rabbit ear in more complex example

Before I end this episode, I would like to show you the whole theory on a more elaborate example. In this particular case, I will show you the rabbit ear on a whitetail deer model designed by Robert Lang.

Whitetail deer model designed by Robert Lang
Figure 5

I have chosen this example because it is simple yet interesting enough. Also, there are no special features whatsoever. Therefore there is nothing that can blur the understanding of a basic concept.

In figure 6, you can observe the first few steps of the design process. As always everything starts with positioning of the needed number of suitably sized circles (figure 6A). 

Upon defining the position of all circles, we have to add axial lines. These are lines that connect the centres of all adjacent circles (figure 6B).

These axial lines form so-called axial polygons (figure 6B), among which, the vast majority are in the shape of a rectangle (marked in yellow). Now, as I have mentioned before, inside each axial polygon, a set of creases, known as a molecule, must be added, allowing the polygon’s edges to coincide upon folding.  

Here, only axial polygons in the shape of a triangle are of interest to us, since their molecule always has a shape of a rabbit ear, In figure 6C and 6D you can see creases that form the rabbit ears in each and every polygon that has a shape of a triangle. With this, our story about rabbit ears is over. Of course, the definition of the crease pattern is not complete. As could be expected, the final solution (figure 6E) is significantly more complex since we have to add creases inside non-triangle axial polygons too (marked in green). 

Whitetail deer model designed by Robert Lang - design procedure.
Figure 6

Conclusion

I hope I have managed to shed additional light on the rabbit ear fold. I hope, now, you can fully appreciate the importance and ingenuity behind something so simple as the rabbit ear. You see, the internet is full of various tutorials explaining what rabbit ear is and how it is folded. But, hardly any of those tutorials even remotely tried to explain its main purpose: to be the molecule inside a triangle axial polygon.