How to Collapse Box Pleated Crease Pattern?


Post is a part of a larger series (Advanced concepts):

Introduction

One of the most stated questions on the WEB regarding Origami is: How to collapse a crease pattern based on a box pleating method. Surprisingly the answer is very simple.

But before I answer this question, I would like to talk about the reasons why crease patterns are used in the first place. The reason is obvious, diagrams for the majority of models are simply not available and we have to rely on crease patterns only. To be completely honest, drawing step by step diagrams is by no means a simple job but it is not that complicated either. Real problem is that not all models can be folded in a step-by-step fashion and that is especially true for models designed using the box pleating method. The vast majority of models for which diagrams are available are based on the circle packing method, with few notable exceptions like Dragonfly by Satoshi Kamiya or Violinist by Hoyo Takashi.

Procedure

In this blog post, I will try to demystify the collapsing procedure and show you one simple but effective approach that can be applied to almost any crease pattern that does not include special features like Pythagorean stretch elements or level shifters. For that purpose, I have designed a rather small origami base that contains all distinctive features of a box pleating method. This base was designed for academic purposes only and as such does not represent a foundation for any future model.

Crease pattern

To successfully collapse a crease pattern, you have to follow a simple procedure. And the procedure is:

  1. Analyze the crease pattern
  2. Find all central polygons if they exist
  3. Find an axial line that connects the majority of central polygon
  4. Simplify CP by disregarding most of the ridge creases and do the collapsing of simplified CP
  5. Introduce one previously disregarded ridge crease and collapse the base one step further
  6. Repeat step 5 until the crease pattern is collapsed

Crease pattern analysis – Step 1

Crease pattern analysis is a way of finding all polygons on a crease pattern. Quite a simple task, believe me. Unfortunately, the full explanation of this procedure is beyond the scope of this blog post. If you are unfamiliar with the procedure I would strongly suggest consulting the blog post “Origami and circles”. Most of the knowledge needed for crease pattern analyses can be found there.

Crease pattern and polygons
Figure 2 – Crease pattern and polygons

In figure 2 all polygons are clearly designated. As you can see there are 16 flaps and one river.

Central polygon problem – Step 2

What we are looking for during crease pattern analysis is the position of all central polygons since they are by far the most difficult to collapse. Only the central polygons have fully visible circles. All other polygons have circles that are only partially visible. Unlike edge polygons that are easily collapsible using the “Elias stretch” maneuver, central polygons are far more problematic. The problem lies in the fact that the central polygons are rarely free. They simply cannot be rotated around their hinge crease because they are almost always hidden between the axial lines of one of the adjacent polygons.

This sounds complicated but it is not. We just have to be aware that central polygons/flaps have to be hidden inside one of the adjacent polygons. That is all. With that in mind, it would be wise to collapse central polygons as early as possible. Most probably they should be collapsed first. Also, if all central polygons are supposed to be hidden, then their collapsing is by no means an easy task. So, to make the whole procedure easear, we have to make them free by opening the base along one of its axial creases. One that connects several central polygons. The full explanation can be found in the blog post “Why is central fold opening so popular?

If we have a CP without central polygons, the whole crease pattern can be collapsed using Elias stretch. Of course, CPs without central polygons are very rare and not overly interesting.

Axial line – Step 3

Since we have established the fact that our not overly complex crease pattern has four central flaps we can proceed to the third step of the procedure. What we are looking for is an axial line or crease that connects several central polygons. This is important because if such a crease exists more than one central polygon can be set free and collapsed at once. As you can see in figure 3 there are two such lines.

Axial lines that connect several central polygons
Figure 3 – Axial lines that connect several central polygons

They are designated as A-A and B-B. You have to choose which one would be used. There is no good or bad choice. Either can be chosen and everything would be fine. For the purpose of this blog post, I have chosen line A-A which is also the line of symmetry of the model. You can argue that line B-B is a better choice since it would cover three central polygons at once. I have to admit that such reasoning is more than reasonable, there is no doubt about it. Problem is that we can choose only one and I have chosen A-A. If I had made a different choice, the procedure would still be the same.

Simplification and collapsing – Step 4

Before I proceed, I would like to go one step back and shed additional light on the proposed procedure. According to the procedure you have to tackle the collapsing problem from the center of the crease pattern toward its edges. If you look at the crease pattern more closely you should realize that, wherever the axial crease comes across the ridge crease it changes its direction. Consequently, it means that the whole crease pattern is divided into regions in which all axial creases are parallel to each other. Since all ridge creases are at a 45-degree angle only areas of horizontal and areas of vertical axial creases exist.

 Areas of horizontal and areas of vertical axial crease
Figure 4 – Areas of horizontal and areas of vertical axial crease

In figure 4 these areas are designated in different colors. And now comes the important part. Out of all ridge creases that belong to the central polygons on the chosen axial line, you must pick those that form a closed area like the one in figure 5. Pay attention to the fact that the chosen axial line is missing since we want to make central flaps on that line free and easily collapsible.

A simplified crease pattern
Figure 5 – A simplified crease pattern

I believe that you are familiar with the fact that ridge crease cannot appear or disappear in the middle of the paper. For that reason, ridge creases that form a closed area must be chosen. This closed area is important because everything inside that area must be folded according to the original crease pattern. Everything outside that area must be disregarded. Meaning, outside chosen central areas (designated in green) we have to construct a simplified crease pattern by adding new and disregarding old creases. New creases should be a direct consequence of an interaction between lines inside the closed area and its boundaries. In other words, all axial lines inside our chosen closed area are simply reflected across the ridge creases disregarding all the original creases. 

This very simplified crease pattern can be easily collapsed by pushing down the line that separates two central polygons (look at figure 6). This line is nothing but a hinge crease. 

If we for some reason chose to start the collapsing procedure from axial line B-B we would have to push down four hinge creases since there are five polygons on that line.

Pushing down a hinge crease
Figure 6 – Pushing down a hinge crease

Note that we will also have to perform an open sink since the central flaps should be much thinner. In other words, we have to collapse all axial lines on our simplified crease pattern (look at figure 7). Result fully corresponds to the simplified crease pattern in figure 5.

A simplified crease pattern collapsed
Figure 7 – A simplified crease pattern collapsed

Ridge crease selection – Step 5

After we have collapsed the first ridge crease, we need to select the next one. We simply select the next ridge crease based on the rule that the new ridge crease must touch the already collapsed one. Only in the case when such a crease does not exist, we can choose any other ridge crease as long as no other ridge crease exists between then. But, let me show you this rule in action.

The next ridge crease selection
Figure 8 – The next ridge crease selection

There are several possibilities here. In all these cases a new ridge crease (designated in purple) touches, already collapsed one. Therefore, no ridge crease exists between them. 

Why does such a restriction exist? Its purpose is to ensure that once set axial lines between collapsed ridge crease and the new one remain unchanged. If somehow a ridge crease exists in between, it will definitely change the direction of axial lines in some of the future steps and we will again have to pay attention to the already collapsed parts of the crease pattern. And this is not desirable.

So we will choose the simplest one (the first one).It is important to realise that the axial lines between the first ridge crease, the one we have already collapsed, and this new one remain vertical, while all other axial creases located in the space separated by the new ridge crease become horizontal (green area). Look at figure 9. Collapsing such a crease pattern shouldn’t pose a problem.

Collapsing
Figure 9 – Collapsing

Of course, an open sink is necessary to fold the remaining axial creases and narrow down the resulting flaps.

An open sink
Figure 10 – An open sink

In the next step, we need to consider another ridge crease. This one is more complicated, but the rules still apply. All the axial creases separated by the ridge crease from the rest of the model would become horizontal like in the previous example.

Collapsing
Figure 11 – Collapsing

Here I would like to point out one important detail. If you look closely at the simplified crease pattern in figure 11 you will notice that we have used only a half of the ridge crease of the central polygon that is not on the axis of symmetry. What does this mean? It means that in the current simplified configuration we do not have a central polygon. Instead, we have an edge polygon. Look at figure 12. This edge polygon is clearly marked with a green line.

Clearly marked edge polygon
Figure 12 – Clearly marked edge polygon

Central polygon outside the axis of symmetry

As you can see from figure 12, this is for now an edge polygon that has stacked a lot of paper inside itself.

Now, we come to the hardest part. The next part of the ridge crease you need to collapse is shown in figure 13.

As you can see, by adding another part of the ridge crease, the central polygon will be fully defined. For that reason, you need to open up previously collapsed edge polygons to get to the paper buried inside. The purpose of this maneuver is to redefine axial lines according to our new simplified create pattern (figure 13). As in all previous steps, the newly added ridge crease separates a piece of paper from the rest of the model. In our case, the axial lines inside the area bounded by the ridge crease become vertical. They become the same as one on an original crease pattern (look at figure 1).

Central polygon collapsing
Figure 13 – Central polygon collapsing

At this point, collapsing procedure becomes a little bit tricky because it is necessary to collapse the central flap (A) and press it in between the axial lines of the central flap located immediately above (C).

Important polygons clearly marked
Figure 14 – Important polygons clearly marked

Edge flaps and Elias stretch

By collapsing the central flap located outside of the axis of symmetry we have completed the most difficult part of the crease pattern. What is left is to collapse edge polygons, for which we have to apply the already mentioned procedure or a maneuver known as the Elias stretch.

So, we only have two ridge creases left. One on the left and the other on the right. We will start from the left one.

Edge polygons
Figure 14 – Edge polygons

I hope you understand that the observed ridge crease can be divided into several parts since it touches the edge of the paper. But, since we are dealing with a very simple example, we will collapse the complete left side of the model at the same time. First, as in all previous cases, we have to define a new simplified crease pattern. Again all axial creases in the newly formed area (green area) have to correspond to the original crease pattern. In other words, axial creases are now vertical even though they used to be horizontal in the previous simplified crease pattern.

Elias stretch
Figure 15 – Elias stretch

An identical procedure must be carried out on the other side.

Elias stretch
Figure 16 – Elias stretch

Finally, the model is fully collapsed. No ridge crease was left unused.

Final narrowing of the model

In the end, it remains to narrow down the model to fully match its crease pattern. The current configuration does not have a central axial line and is twice as wide.I hope you remember that we open the model along the central axial line in order to set two central flaps free. What we have to do now, is to close the model. By doing that, a central axial line is formed. As a result, all central flaps/polygons will be hidden.

Alternative approach

In the beginning, we had a choice, to collapse central flaps along line A-A or line B-B. We have chosen A-A. What would happen if we had decided that central flaps along the line B-B should be collapsed first. Nothing much. Collapsing order would be different but the final result would be the same. I strongly encourage you to try an alternative approach starting from line B-B. I will show you how to begin. Everything else is up to you. 

So, in the beginning, it is necessary to find a closed area formed out of ridge creases that belong to the central flaps. There is more than one such area so we will collapse them all at once. A simplified crease pattern is shown in figure 17

A simplified crease pattern
Figure 17 – A simplified crease pattern

Collapsing will be done in two steps. Look at figures 18 and 19. First, we push down all hinge creases between the polygons. By doing so, all central flaps will become clearly visible.

Pushing down a hinge crease
Figure 18 – Pushing down a hinge crease

In the second step we have to perform an open sink, just to fold the remaining axial creases and to narrow down all the flaps.

An open sink
Figure 19 – An open sink

OK, I just show you how to begin. Everything else should be easy ;-). And remember, a fully collapsed base should be the same no matter which approach you take.

Conclusion

I hope it is now clear why it is difficult to make a diagram for models based on the box pleating method, even though collapsing it is more or less straightforward. Of course, advanced box pleating techniques such as level shifters or Pythagorean stretch were not used.

How they are collapsed will be discussed in some other post.

Until then bay.