## Can central flaps be free?

How many times have you heard that central polygons/flaps are notoriously difficult to collapse because they are not free? This statement is correct by all means. Central flaps are indeed hard to collapse in comparison to edge polygons. The problem lays precisely in their lack of freedom. In this post, I would like to explain what it means and why this knowledge is so important for the successful collapsing of models based on the box pleating technique.

I guess it’s clear how edge polygons should be collapsed. If not, I would suggest you read the blog post “Elias stretch“. In this post, I will deal exclusively with central polygons and I will show you that, unlike edge polygons (flaps), they are indeed very rarely free. In order to explain what it means and how this fact affects the collapsing procedure of the central polygons, I have to start from the beginning, from the crease pattern that contains central flaps (polygons).

Figure 1 shows a crease pattern that contains central polygons. It seems complicated at first glance but trust me it is very simple. Figure 2 shows all polygons clearly designated next to the collapsed base. All polygons are identically marked on the collapsed base and the crease pattern. The central polygons (E and F) are also clearly marked in green.

Here, I would like to point out that the model shown does not fully correspond to the crease pattern, since the crease pattern shows a fully flattened base, which is clearly not the case here. Figure 2 shows the base that is not flattened. This is done on purpose so all its flaps could be clearly visible. What ‘s more, when presented this way all flaps look like if they can easily rotate around their hinge creases.

But the real problem is not that the base is not flattened. You see, even if the base was flattened, it would still not correspond to the initial crease pattern because it would be twice as wide as it supposed to be. For crease pattern to fully correspond to the shown base, its central horizontal axial line should disappear. That is the difference I was talking about (look at Figure 3).

If we return the central axial line so that the base corresponds to the original crease pattern, it will have additional consequences about which I would like to say a few words.

Nevertheless, if we go back to our original crease pattern, one with central horizontal crease, a folded base should be little bit different. It should be additionally folded along its central crease, as can be seen on figure 4.

The result of such move is clearly visible on figure 5.The modal and the corresponding crease pattern in Figure 5 show that all edge polygons marked with the letters A, B, C, D, G, H, I, and J are free, while the central polygons (flaps) E and F are trapped and barely visible. They obviously cannot rotate freely in that position.

Here we come to the first rule. The central flaps are almost always hidden (trapped) between the axial lines of one of the adjacent polygons/flaps and as such they are NOT free. Since central polygons typically have multiple adjacent polygons logical question arise: how do we know within which polygon/flap the central flap is hidden? Also, what does it mean that the central flaps are “almost” always hidden within the adjacent polygon/flap? Is there a situation when this is not the case? The answers to all these questions are very exact and it is clearly written in the crease pattern. You only have to know how to read it correctly.

Have you noticed that polygon boundaries always coincide with hinge crease positions? Of course, you have. When I marked the polygons in Figure 2 no hinge creases remained visible. They are simply covered by the lines that represent polygon edges.

By definition, hinge creases are always situated on polygons’ edges and as such hinge creases are lines around which flaps can rotate. But, and this is very important to remember, not all polygon edges are marked by hinge creases. Some polygon edges could be missed out due to the flat-foldability constraints imposed on the crease pattern. Meaning hinge creases do not have to cover all polygon edges (although they could).

Sound complicated. Maybe, so let me show you this on a previous example. Consider, for example, the central polygon E. Although the polygon has 4 sides, only two are visible in the form of a hinge crease. The other two are missing, they are not visible. The visible edge of the polygon, one that represent a hinge crease shows us how the flap is positioned in relation to all other elements of the model when model is completely flattened. Remember this, crease pattern shows us only creases that exist when a model is completely flattened.  For example, on figure 5, a crease between flaps/polygons H and I are not visible since in flattened form (one showed on figure 5) there is no crease in-between. Shared surface is fully flat.

On the other hand, flap A has all the edges of the polygon defined by the hinge creases. If you look at the collapsed model, the edges (hinge creases) toward the adjacent polygons/flaps are clearly visible because both flaps are oriented in the same direction. Likewise, polygon C has a clearly visible edge (hinge crease) towards the flap G (they are similarly oriented) but the edge (hinge crease) toward the flap B is not visible (opposite orientation). Precisely the visibility of the hinge crease clearly defines the orientation of each element of the model. So, the rule is follows; If the elements are oriented in the same direction the hinge crease between them is visible. On the other hand, if they are oriented into the opposite direction, the hinge crease between them is not visible.

Let’s return to the central flap E. It has two hinge creases, towards polygons D and B. What does this tell us? The fact that there is a clearly visible hinge crease between these polygons means that these polygons are oriented in the same direction when the base is fully flattened. Look at figure 3. It shows that polygons A, B, D, E, and H are oriented on one side and polygons C, F, G, I, and J on the other. If we draw the demarcation line between these two groups of polygons (yellow line) it is clear from Figure 5 that the demarcation line is the line between polygons where no hinge creases are present.

Now, that we have managed to clarify few important facts about hinge creases, we could go back to our main topic: Central flaps. At the beginning, I have told you that the central flaps are hidden between the axial lines of adjacent polygon/flap. With this in mind let’s analyze the central flap E. Because of its position regarding the demarcation line, only polygons A, B, D, and H are potential candidates for hiding central flap E. This is logical because only these five flaps are pointing in the same direction.

If you take a closer look at the crease pattern, you will see that polygons A and H do not touch polygon E, therefore central flap E cannot be hidden inside these polygons/flaps. So, you are left only with the flaps B and D.

At first glance, you can see that hinge creases between these polygons look different. Namely, the hinge crease between polygons D and E is symmetrical while hinge crease between polygons B and E is not. This is very important difference. You see, for the central polygon to be able to roll over its hinge crease and hide in-between the axial lines of the adjacent polygon, the hinge crease between them must be symmetrical. Knowing that, it is easy to see that central flap E is hidden inside polygon/flap D because only a hinge crease between polygons E and D is symmetrical.

Now, additional question arises. Is it possible for a polygon to have more than one symmetric hinge crease? Of course, it is, but even in that case, the situation is quite clear. With this in mind let’s consider once again the central polygon E in Figure 6.

The difference between crease pattern in Figure 6 and in Figure 5 is the fact that the demarcation line puts both central flaps (E and F) on the same side. This means that both central flaps are pointing in the same direction. Consequently, polygon E got one additional hinge crease. What’s more, two of its hinge creases are symmetrical. So, the question is, how do I know between which axial lines the polygon E will be hidden. Luckily, the answer is obvious.

If we want to hide something in-between the axial lines of a polygon then its central axial line in direction in which you want to hide another flap must form a valley. This is logical. I hope you can agree on that. In Figure 7, only the polygon D has a central valley in which you can hide the polygon E. On the other hand, if you want to flip the flap E into the opposite direction into the polygon F, it will encounter a mountain there (red line). This mountain will prevent it from doing so. Therefore, it is obvious that the flap E must be hidden inside flap D. What about flap F. Well, polygon F has only one symmetrical hinge crease, so direction is obvious. Flap F will be hidden between the axial lines of polygon E. To be sure that this is proper direction just look at a central horizontal axial crease of polygon E in which a flap F is supposed to be hidden. It is a valley as it supposed to be.

From everything mentioned, it is clear that when a base is fully flattened, the central polygons will not be free simply because they will be hidden inside some other flap. Finally, there is just one question or dilemma if you like that I would like to address. Is there any configuration of our model that produce free central flaps? Is a free central flap possible at all?

Answer is yes, but, and I would like you to remember this, only one central flap could be free.  Remember that as a rule.

Look at Figure 8. In this example demarcation line divides only two central flaps from the rest of a model. This means that two central flaps are pointing in one direction while all other flaps are pointing into the opposite direction.

As a result, central flaps have only one hinge crease, one between them.

If we reduce the base to a unit width or draw the missing central axial line as shown in Figure 9, it would become quite clear that the central flap E is free and can rotate independently while the flap F is trapped inside flap E as expected. How do we know that it is not vice versa? How we can be sure that flap F is hidden inside flap E and not flap E inside flap F. It is easy. You just have to find out which flap will not encounter an axal crease designated as mountain.

From above mention examples, it is clearly visible that central flaps will be in most cases locked inside some other flaps. Obviously, such a situation is not overly favorable, as we usually need free central flaps rather than the locked ones. Of course, freeing central flaps could be achieved by opening the axis on which they are located. Needless to say, some restrictions exist. The axial line will disappear, and consequently the base will be twice as wide. Also, all the central flaps that are not on that axis will remain trapped. And finally, if the axis along which you are opening the base is not an axial line along its entire length as in our example, there will be a significant change in the configuration of the base.

That would be all. Bye …