Maekawa-Justin and Kawasaki-Justin theorems are not enough


Post is a part of a larger series (Design process):

I believe that most of you understand what flat-foldability of an origami model means. But if you do not, I will give you short introduction into the topic.
Flat foldability of an origami model means that whole model could be flattened into the single plane without getting any additional creases or worse being wrinkled.

In other word a model could be considered flat foldable if we could put it into the book and then close that book without damaging the model itself.
I know, most of you probably thinking: Origami models are flat-foldable, are they? Well, most of them are not but theirs bases certainly are. Believe me, no matter how complex certain model looks, its base is almost always flat-foldable.
Therefore, if you intended to design an origami model, you better be sure that its base is designed as flat-foldable. But, since the theory behind flat foldability of a model is quite complex, I will limit this discussion to something simpler. I will talk about flat foldability of a single node/vortex and its folds. But do not be disappointed. Flat foldability of nodes is a main prerequisite for flat foldability of a whole model.

So, if you want to be sure that folds that converge into the single node form a flat-foldable structure, two famous origami theorems must be satisfied. In other words, you must insure that so called Maekawa-Justin and Kawasaki-Justin theorems are satisfied.

Maekawa-Justin theorem

Let’s start with Maekawa-Justin’s theorem. Maekawa-Justin theorem states that the numbers of valley and mountain folds in a single node or vertex always differ by two.

M – V = ± 2

In plain English it means that if a node consists of six folds four of them must be valley and two a mountain. Or vice versa, two must be a valley and four a mountain. In any case, difference between them must always be two (Figure 1).

Maekawa-Justin theorem example
Figure 1. (valleys are marked in blue, while mountains are marked in red)

Kawasaki-Justin theorem

Kawasaki-Justin theorem is little bit more complex but nothing to hard to understand. Let’s see what Kawasaki-Justin theorem states. It states that a crease pattern is flat-foldable only if the alternating sum of incident angles at any given vertex is always zero.

α1 − α2 + α3 − · · · − α2n = 0

This might sound complicated, but actually it’s not.
To make things more understandable, please take a look at Figure 2. So, simply put, Kawasaki-Justin theorem states that the sum of all green angles between folds is equal to the sum of all red ones. As simple as that.

Kawasaki-Justin theorem example
Figure 2.

Are these theorems enough to insure flat foldability?

If you follow almost any discussion about flat-foldability feature of an origami model, you will constantly hear only about these two theorems. But is satisfaction of these theorems enough to ensure flat foldability? Unfortunately, it is not. To show you what I mean, let’s analyse one super simple example. On figure 3 you can see four very similar examples. All four satisfy both Maekawa-Justin and Kawasaki-Justin theorems, but unfortunately, only two are truly flat-foldable.
So, by looking at figure 3, what do you think which two examples are correct?

Four simple examples (flat foldability)
Figure 3.

Well, correct are examples B and C, simply because a mountain fold (red one) is next to the smallest angle (15°). Remember that, if a vortex consists of only four folds, the fold that is different from the rest (in this case a mountain fold) must be next to the smallest angle.
OK, this example was super simple. But what you supposed to do in case you have more than four folds in a single vortex. For example, what to do if you have six folds (figure 4).

Flat Foldability (example with six folds)
Figure 4.

Again, you must ensure that Kawasaki-Justin and Maekawa-Justin theorems are satisfied. As far as Kawasaki-Justin theorem is concerned, I can assure you that in this example this theorem is satisfied. Believe me, just count the angles. Now, if you ensure that four folds out of six are valleys and two are mountains then the Maekawa-Justin theorem will be satisfied too.
But as you probably suspected, to be sure that the model is flat-foldable, you must know which two folds are mountain folds.
So let’s find out.

Step 1

To choose the position of the first mountain fold you simply must find the smallest angle. The first mountain fold is without any doubt next to it. So, first mountain fold is either fold C or D since the smallest angle is between them. Which one you choose is fully arbitrary. Both options are possible and correct. So, for example, let’s choose fold C to be a mountain. Since this is smallest angle, fold D must be a valley. Remember this, smallest angle must always be defined by folds of different orientation (Big-Little-Big Angle (BLBA) Theorem).

Flat foldability  - example with six folds (step 1)
Figure 5.

So, finding position of the first mountain and first valley fold was easy. I hope you agree on that.

Step 2

Now, before proceeding further, you must literally fold these two folds that form the smallest angle (look at Figure 6). As you can see, by doing so you will get something that resembles a cone. But what is more important you will hide folds C and D. Now, what is left are folds A, B, E and F and angles between them. But be aware of the fact that angle between fold A and D has change. It became smaller as you can see on figure 6. I hope this is clear.

Flat foldability  - example with six folds (step 2)
Figure 6.

Step 3

Now again, between these four folds you must find smallest angle and then choose a fold next to it to be a mountain. In this example angle between folds A and F is smallest one, so one of the folds A or F must be a mountain. Which one you chose is fully arbitrary. This way you managed to define your model that consists of only one vortex and six folds as fully flat-foldable (figure 7).

Flat foldability  - example with six folds (step 3)
Figure 7.

Of course, if you have a vortex with more than six folds, procedure is literally the same except it will have few additional steps. That’s all.

Conclusion

I hope I manage to convince you that Maekawa-Justin and Kawasaki-Justin theorems even though very important are not enough to insure a flat foldability of a model. Specially Maekawa-Justin theorem is not enough. It states number of folds of a different orientation correctly, but it still does not define actual arrangement of them.
For those who wants to know more on this topic I gladly suggest mr. Meguro Toshiyuki twitter account (@meguro77) or Robert Lang’s book “Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami”.

It would be all. If you enjoyed this blog post, share it with a friend!
Bye!