In this blog post, I would like to address the problem of unused parts of a paper.
This problem is not uncommon in origami. As a matter a fact, you will stumble upon it quite a lot, and in this regard box pleating technique is no exception. Simply, when polygons and rivers are arranged on a square piece of paper, it is very likely that part of a paper will be left unused.
Simple example of unused paper problem
To make this problem more understandable, let me show you one simple example.
As you can see, sixteen squares in the central part of a paper do not belong to any polygon or a river. They are simply not used. This is, of course, a problem since in origami all parts of a paper must be used. Paper cannot disappear, it has to be used somewhere.
So, what we can do about it?
Let’s analyse the different options at our disposal.
The simplest solution is to incorporate those unused squares into the adjacent river. You can see such a solution in the following picture.
As you can see the river is somewhat more complex, it starts to meander a bit, but it is still continuous. Therefore, everything is according to the rules. What is more, newly formed ridge creases (lines that run at 45 degree angle) are also continuous and unbroken, which was not the case before we rearrange a river a bit.
This idea in which unused squares are hidden into the river represents the basic approach to solve this kind of problem. Everything else is only a variation on a theme.
But nevertheless, let’s analyse those variations. You see, hiding unused paper into a river can be a bit tricky. The problem arises from the fact that the river itself holds more than a few paper layers. So, adding additional paper into it will only increase already considerable river thickness. This is not necessarily a problem, but it might be.
More advance solution
So, as an alternative, we might consider incorporating unused paper into one of the flaps. In general, it is favourable to do so, since the additional paper will be less apparent inside a flap, especially if a flap is long enough. Therefore, let’s try adding unused paper into the longest flap, one in the lower-left corner.
To do so, we have to use a little trick. We have to replace ten-units-long flap with two elements: a shorter eight-units-long flap and a two units wide river that separates it from the rest of the model.
Look at figure 4. Do you see what we have just done? We have just divided one element, a flap, into two elements, a flap and a river. But note that a river that encircles only one flap is for all practical purposes a part of that flap. So, technically, nothing has really changed and yet again, everything has changed.
Now that we have a new river, we can incorporate unused squares into it. Doing so, we have technically managed to incorporate unused squares into the largest flap.
I hope you understand why at the beginning I have told you that unused paper is always incorporated into the river and that everything else is a variation on a theme.
We can also incorporate unused squares into the lower right polygon. Approach is literally the same (look at a figure 6.).
Real life example
Before I end this post, I would like to show you an interesting example from Robert Lang’s book “Origami design secrets”. There is a crease pattern of a model called Salt Creek Tiger Beetle. Now, if you pay close attention to its crease pattern, you will soon realise that in the middle of a crease pattern there is a central polygon with a quite strange shape (look at figure 7).
Its shape is a direct consequence of adding additional unused paper into it. As you can see in figure 8 this polygon is de facto a central polygon with additional unused squares on both sides (Unused squares are marked in yellow). To incorporate those unused squares into the blue central polygon, first, we have to divide the polygon into two elements – a polygon and a river that encircle it. Again, a river that encircle only one polygon is for all practical purposes a part of that polygon so technically nothing has changed. Now, we can make this newly formed river to meander a bit, in order to include an unused paper on both sides of the central polygon.
I believe the theory behind this unused paper hiding technique is not that hard, and that you will be able to use it in your models or at least recognise it on other people crease patterns.
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