Relationships between basic elements of an origami model


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

All basic elements of an origami model (flaps and rivers) must be in the same relationship, both on the stick figure and the crease pattern. Reason is quite simple. Both stick figure and crease pattern are a graphical representation of the same future model. Therefore, it is not possible for a crease pattern to show one thing and for a stick figure to show something else.

Nevertheless, even though this statement is absolutely correct, sometimes it seems, and I repeat “it seems” that this is not always the case.

Stick figure

But, before I start analysing one such a seemingly odd and incorrect case, I believe it is necessary to give a clear and fully understandable definition of a stick figure.
A stick figure is a very simplified representation of a base of an origami model. A stick figure shows all flaps and rivers, also it shows relations between them. Also, a stick figure shows lengths of all flaps as well as widths of all rivers. What a stick figure doesn’t show is flaps width.
Nevertheless, to be sure we you understand what a stick figure is, let’s look at one very simple example and compare it with its stick figure (look at Figure 1). As you can see relation between the model and the stick figure is more than obvious.

Abrashi Origami stick figure
Figure 1: Simple model and its stick figure
Abrashi Origami crease pattern
Figure 2: Crease pattern

Simple rule

If you take a good look at the crease pattern of our simple model (Figure 2) you will see that the polygons (e.g. A and B) that define flaps on the crease pattern touch each other. The same can be seen on the stick figure. Flaps A and B touch each other as well. It is logical, isn’t it. Also, a river that divide flaps A and B from the flaps C, D and E, is doing the same.
So, I believe that the logic behind this is quite clear: elements of an origami models (flaps and rivers) must be in the same relationship to each other both on a stick figure and on a crease pattern.

Seemingly odd example that does not follow the rules

Now, that we managed to define what a stick figure is, we can proceed further by examining one interesting example. On Figures 3 and 4 one such an example is shown.

Abrashi Origami stick figure
Figure 3: More complex origami model and its stick figure
Abrashi Origami stick figure
Figure 4: Crease pattern

If you thoroughly examine the stick figure, you will notice something interesting. In central part of the stick figure there are five small flaps that touch each other (flaps A, B, C, D and E). Nothing too odd about that, except that on the crease pattern their polygons are not next to each other. As matter a fact they are grouped in two distinctly separated groups on the left and on the right side of the crease pattern. Between these two groups of polygons two rivers can be seen.
Now, a logical question arises. How is it possible? How is it possible to have polygons that do not touch each other while at the same time to have a model or a stick figure where these flaps do touch each other.
Explanation is quite simple. If you think a bit about what is drawn on the crease pattern, we will soon realise that these rivers (red and green on figure 5) do not separate these polygons.
All five polygons are on the same riverbank. They are all on the same riverbank no matter which river we are looking at. Remember this. For a river to be able to separate polygons, the polygons must be placed on the opposite riverbanks, which here is not the case. Look at Figure 5. One riverbank of the red river is clearly marked in purple. Do you now see that all five polygons are on the same riverbank? They are all next to the purple line.

Abrashi Origami crease pattern
Figure 5: Red and green river with red riverbank being marked in purple

Conclusion

So, these rivers indeed meander between these five polygons leaving us with the impression that they separate these polygons. But truth is far from that. These five polygons are always on the same riverbank no matter which river you are looking at, meaning for all practical purposes these five polygons can be consider next to each other. Exactly as it can be seen on a stick figure.
That would be it. I hope I have managed to show you that even though sometimes it is not so obvious, a stick figure and a crease pattern always show the same configuration. As they should.