A hinge crease serves two purposes. By definition, a Hinge crease is a line that defines polygons by the mere fact it surrounds it. Also, a hinge crease is a line around which a flap can rotate. Hence the name. It looks similar to a door hinge.
In figure 1 we can see the same hinge crease both on a crease pattern and on a fully folded base (red line). You can see that the hinge crease fully surrounds its polygon. Also, there is something very interesting about hinge crease. You see, if we take scissors and make one complete straight cut along the hinge crease (on a fully folded base), the whole polygon on the crease pattern will be completely cut out (figure 2). In other words, no matter how complicated a polygon boundary may be, it will always fold along a single line.
This well-known property of a hinge crease is known under the name of “The Fold-and-Cut Problem” (for more information on “The Fold-and-Cut Problem” see Erick Demaine personal web page).
If you take a closer look at the crease pattern (figure 3) you will notice that the hinge crease unlike the other two types of creases (ridge and axial creases) is not always complete. To make things worse, we successfully cut out the polygon even though its hinge crease is completely invisible. Sometimes, due to the constraint imposed by flat foldability rules some parts of the polygon are not fully defined by the hinge crease. Part of the hinge crease is simply missing.
The hinge crease visibility problem
The problem I am addressing here is the visibility of the hinge crease when the model or the base is fully flattened. In other words, I would like to show you that when the model is flattened into the single plane, the parts of the hinge crease will not be visible. What’s more, if we change the position of at least one flap, the arrangement of the hinge creases will also change. Some will disappear while others will appear. So, the topic of this article can be formulated as a simple question. Based on the position of individual flaps, is it possible to know in advance which part of the hinge crease will be visible and which part will be hidden? The answer is, of course: yes.
I hope you have noticed that the base in figure 3 is not flattened. Which means, the crease pattern on the left does not fully correspond to the base configuration on the right. If the crease pattern would somehow correspond to the base on the right it would have all the hinge creases visible, without exception. Still, the crease pattern does not have all hinge creases visible since by definition crease pattern represents the model in a flattened form. Look at figure 3, edges of the polygons on the crease pattern are marked in gray, but not all of them have corresponding hinge crease. The actual base configuration that corresponds to this crease pattern is shown in Figure 4.
Above the base, an accompanying stick figure is drawn, just to make it easier to identify parts of the base. It is important to notice the orientation of every single flap. For example, look at the flaps E and D. Since they are oriented in the opposite direction no fold appears in-between. The same applies to the crease pattern. There is no hinge crease between them either. The same is true for flaps C and F. There is no hinge crease between them too. On the other hand, the flaps A and D are oriented in the same direction so there must be a fold between them. Consequently, a hinge crease must appear between them on the crease pattern too.
The hinge crease visibility rule
A unique rule for determining hinge crease visibility on the crease pattern is as follows: Polygons that touch each other will have a common fold, therefore a hinge crease in-between, only if they are oriented in the same direction. But remember, the same rule applies to rivers as well. For example, flap E and river G do have a common hinge crease because they are oriented in the same direction while flap D and river G do not.
Also, if you mark all hidden hinge creases in yellow, you will obtain a demarcation line between the differently oriented flaps in each node. Notice that these lines are continuous.
Application of the rule
Now that the rule is established, we will apply it in reverse order. On a crease pattern that has an only ridge and axial creases, we will first determine the demarcation line. Since we know that the flaps C and F and the river G have a node in common, we will draw a demarcation line between them. Remember the line must be continuous. We will also draw a demarcation line between the flaps A, B, D, and E and the river G (Figure 5).
It does not matter where it goes as long as it is continuous, and it follows the edges of the polygons. Once the demarcation line is determined all remaining polygon edges represent the visible hinge creases (Figure 6). That simple.
In the end, one important question remains to be answered. If the hidden parts of the hinge crease are a direct result of flap orientation, can we orient the flaps so that the demarcation line disappears entirely, and all hinge creases become visible? The answer is again: yes.
What would happen if we remove one demarcation line and replace it with the hinge creases? This simply means that flaps F and C should be oriented in the same direction as the river G. Look at figure 7.
We can apply the same logic to flaps A, B, D, and E. Unfortunately, there is a problem. The problem is that flaps C and F have taken all the space and there is no way flaps A, B, D, and E could be turned in the opposite direction.
To solve this problem, it is necessary to reconfigure the model as shown in Figure 8, thus turning the flap F to the other side, consequently opening the space for the flaps A, B, D, and F to be rotated into the opposite direction. In such a configuration complete Hinge crease for every polygon are visible. So, the demarcation lines no longer exist.
Hinge crease is one of three types of Origami folds. Although hinge creases may help determine various polygons positions, their true purpose is something else. It is a known fact that only the ridge creases are needed in determining the polygon positions during Crease pattern analysis, the flaps mutual orientation, when the model is fully flattened, is determined by the hinge creases alone. And that is its true nature.