## Colour change – part 1

Sometimes it is hard to believe that multi-coloured origami models are folded from a single piece of paper. For some people, “origami colour change” seems like magic.

However, like anything else in origami, there is no magic whatsoever. Quite contrary, the complete colour-changing process is governed by strict and clearly defined rules. So, if you are interested in learning and understanding yet another technique concerning an origami colour change, then this blog post is just right for you.

## Basic rules

The colour change technique, in its essence, has only two simple but fundamental rules.

The first rule states that the colour change can be realized only on the paper edges. That is obvious since only on the paper edge, another side of the paper can be exposed. In other words, there is no way to accomplish the colour change in the middle of the paper. The second rule is less obvious but equally important. It states that the developed crease pattern design must satisfy the so-called perimeter theorem. Sometimes that can be tricky, but there are pretty solid guidelines that can help.

So, let’s see the rules in action.

## Paper edge calculation

We will start with a simple example, a 2×2 checkerboard. Most probably, you have already seen this design (figure 1). But, we will use it anyway in order to show you all steps of the design process. You see, even though the model is simple, it is, at the same time, complex enough to reveal all the challenges of the design process involving colour change.

Since colour-change can be achieved only on the paper edge, it is no-brainer to realise that in its essence the whole colour-changing technique revolves around the paper edges. Therefore, first what we need to do is to calculate how long the paper edge should be. In other words, what we need to do is draw the line that would encircle both yellow squares. In figure 2, such a line is shown. Again, it is not hard to realise what the length of that line represents. The line length defines the minimal perimeter of the paper. Or, to be more exact, line length determines minimal lengths of paper edges needed, in order to  achieve a given amount of colour change.

Now, here comes the most exciting part of the design process. This step involves taking a narrow strip of paper in order to cover the previously defined line that encircles all yellow squares. Of course, by doing so, we must always follow several simple rules (see next chapter). Nevertheless, since the example is a simple one, there is no need for an extensive explanation here. The procedure is more or less self-explanatory (see figure 3).

Now, the next step is to unfold the narrow strip. What we got is a paper edge with a few additional creases. These creases are a direct consequence of strip folding. In other words, these creases are necessary so the narrow paper strip can follow the yellow square edges. In figure 3, you can see folded and unfolded versions of a paper strip. Additionally, paper corners are clearly marked as well.

The next step is to add all additional creases inside that hole (figure 4). In this simple example, this shouldn’t be overly complicated. We merely have to connect the creases, and the creases pattern will be completed. So let’s see what we got (figure 4).

In figure 4, you can see the crease pattern and a final model. It seems that the final model did not meet expectations. If you look closely, you will realize that white squares are not complete. Some parts are missing. What does it tell us? It tells us that there is not enough paper, that the paper is too small. Or, to be more accurate, the line that defines the edges between the differently coloured fields is simply too short.

## Perimeter of the model

The easiest way to deal with this problem is to use a longer strip of paper in order to reach all the corners. By encompassing all corners of the model with a narrow paper strip, we can be pretty sure that none of the corners on the final model will be missing.

As you can see in figure 5, the new encircling line is longer. Consequently, the paper perimeter is longer as well, meaning the paper is larger. With this in mind, let’s compare figures 2 and 5. In figure 2, an 8-unit long encircling line is used, whereas, in figure 5, a longer 12-unit encircling long line is used. In other words, the paper sizes needed are 2×2 and 3×3, respectively.

Again, we must take a narrow strip of paper and fold it along the just defined line. The final result can be seen in figure 6. It may seem simple, but in reality, it is not. The major problem arises from the fact that the correct positioning of the corners is not a trivial problem. In other words, all four paper corners must appear on the folded strip of paper in the appropriate positions. Meaning, the distances between all four corners on the folded strip of paper must be the same. So, if you compare corners positions in figure 5, you will surely realise that they are positioned correctly.

Finally, what is left is to fill in the missing creases in the middle of the paper and, of course, to fold the model itself.

As you can see in figure 7, the final model is precisely what we wanted. There are no missing corners.

## Potential problem

What we accomplished so far can be, to a certain extent, attributed to the simplicity of the model and its crease pattern. In other words, since the model is simple, we have managed to construct the crease pattern that produces a satisfactory final result with ease. But, the problem is hardly ever that simple.  According to the Perimeter theorem, even this simple example can allow more than one perfectly legal crease pattern. In figure 8, one such crease pattern is shown. It is made more complex on purpose only to show you that the appearance of all white squares on a final model is by no means guaranteed. That is more than evident if you look at figure 8. As you can see, the model is incomplete even though all colours are correctly positioned.

## Completeness guaranteed

The question is, how to be sure that the final model is complete? Well, the answer is surprisingly simple: All sides of all squares have to be covered by a narrow paper strip. Figure 9 shows one possible solution.

As you probably expected, the line is longer, and therefore the initial paper is bigger as well. This time a 4×4 grid is needed. The rest of the procedure is the same. Meaning, we have to take a narrow strip of paper in order to cover the just defined line that encircles all squares. Again, paying particular attention to the positions of the corners. That is very important.

The final result should be in line with our expectations (figure 11).

Now, the logical question arises. Was this additional complexity really needed? Well, it was not. You see if you can find a satisfactory solution that can accomplish the same result using less paper, then, by all means, use that solution.

Nevertheless, it is necessary for you to understand that model incompleteness is not an uncommon problem. It arises almost always when the ratio between paper size and final model size is relatively small. Or, to put it in a different way: if the final model is much smaller than the paper then it is much easier to make a model complete. So, in general, we can say that more complex models tend to have a higher ratio between paper size and final model size, therefore, the possibility that they will experience incompleteness problems is highly unlikely.

## Building blocks

While covering the model with a long strip of paper, we have to follow a few simple rules: First, we have to be aware of the fact that the strip is following the line that is, in its essence, a border between two colours. Since our strip of paper has two differently coloured sides, it is essential to position it, paying attention to the model colours underneath. In other words, strip colour and model colour underneath must be the same.

With this in mind, situations when a strip of paper changes direction is of special interest. You see, when the strip of paper has to change direction, two solutions are available. In figure 12, you can see the most basic one. The strip of paper makes a turn by changing the colour, effectively making the cross from one field to another. I hope you can see how the strip colour resembles the colour of the fields underneath.

Making a turn with colour change is not the only option. We can make a turn without changing the colour too. Look at figure 13. Here, again,  we have more possibilities. The difference is not in their function. From a functional point of view, they are the same. The real difference is in their crease patterns. Since the difference is not functional, we can change one crease pattern for another if we find it useful.

The ability to choose one solution, or another, can be indispensable during model crease pattern design. More about it in one of the following chapters.

## Possible mistakes

There are two common mistakes that are worth mentioning. The first mistake arises when one tries to make a turn involving colour changing from the wrong side. If you look closely at the first example in figure 14, you will immediately realize the problem. Even though the turn seems rational, it was made using the valley fold instead of the mountain fold, which is not correct. This way, opposing colours appear over each other, consequently making the final result unsatisfactory. The right implementation of the 90-degree turn is shown in figure 12.

The second mistake is more subtle but equally problematic. If you look at the second example in figure 15, there is a good chance you would not immediately figure out what is the problem. Here we have a turn without colour change. At first glance, everything seems fine, but actually, it’s not. The problem is that making a turn, as shown in figure 15, that goes through three different squares, asks for an additional paper, which is not available. So, let’s look again at the solutions shown in figure 13. It is easy to recognise that boundary-crossing should not be allowed while performing a turn that does not include colour change. Remember that as a rule.

## Putting everything together

Let me show you the whole procedure using one simple and yet sufficiently complex example. In figure 15, a white letter P on green canvas is shown. Our task is to make a crease pattern and consequently fold the model that resembles the one in figure 15.

Since we know from the previous chapters that colour change can be accomplished only on the paper edges, our first task is to define a continuous line that would serve as a paper edge. One possible solution is shown in figure 15. What we try to accomplish here is to create a continuous line that would cover all the boundaries between the two colours. If you compare the two models in figure 15, you will most certainly realise that few lines are added. These additional lines allow us to construct one continuous line. Now that we have the line, we know for sure how big the paper should be. Just count the number of unit lines, and you will get the perimeter of the paper.  In this particular case, the line is 48 units long which means that the needed paper is 12×12 units in size.

The next step is to strategically place all the corners. Since the unfolded paper has four natural corners,  the same number of corners must be placed along the line, respecting, of course, the positions of the turns (look at figure 16). I believe that the logic behind such a procedure is more or less obvious.

To position the first corner is easy. We just have to pick one point on the line where the line makes a turn, and that’s it. Then we must count 12 units along the line and place another corner there. Of course, the corner must also be on the line’s turning point. If it is not, we have to start all over again by choosing another position for the first corner. Or, we have to reconfigure the line itself. Luckily, in this case, the location of the second corner is just fine. Then we move along the line for an additional 12 units and place the third corner. Again, we have to pay special attention to the fact that the newly chosen point is indeed on a turning point. Finally, the position of the fourth corner has to be defined. Unfortunately, this time, the location is not at the turning point as it is supposed to be. Meaning, the proposed solution is not valid. So, what can we do about it? I can assure you that no matter which point we chose for the first corner or no matter how we reconfigure the initial line, there is no valid solution. Our only choice is to make the line longer. Since the proposed paper was 12×12, the first larger one is, of course, 13×13, meaning the perimeter of the new paper is 4 units longer.

Our task is again to place the line along the border between two colours. A longer line allows us to create an additional turning point, where we need one. One such solution is shown in figure 17. The whole structure looks more complex, but do not be intimidated by its strange appearance. Additional 4 units (purple line) were used only to make a new turning point at an appropriate location. That turning point will accommodate the fourth corner.

What you have to have constantly on your mind is the fact that the new interval between corners is now 13 units. I believe that you have realized by now that line configuration and corner placement is an iterative procedure.

The solution shown in figure 17 is satisfactory. Again, it is not the only possible solution. There are other legal solutions as well.

## Strip of paper

Placing the strip of paper along the just defined line is the next step. This step is important since folding the strip of paper alongside the line will produce a partial crease pattern. As you know, in our example the letter “P” looks like it is positioned on the green card (7×5). Be aware of that. Why? Well, because when you fold the strip of paper along the outer card perimeter, you always have to be inside the card. Going out of the card, meaning positioning paper strip on the outer side of the card perimeter, would imply that there is actual paper outside the card, when in fact there is nothing there. Contrary to that, when folding a paper strip along with the letter “P”  perimeter, you are free to choose the side of the line you are about to place the paper strip. That is possible because, on both sides of this line, actual paper exists. The paper is only differently coloured.

Next, all strip natural corners must be so-called “inside turn”. We have already mentioned that in previous chapters. Look at figure 18, and observe how the paper corners are placed (corner 4 is hidden behind another strip of paper).

When we unfold the strip, the initial crease pattern emerges. This crease pattern represents the base for implementing the Perimeter theorem. Going from the narrow strip of paper to a fully constructed crease pattern is by no means easy, but it is not that hard either. After all, you have some tricks at your disposal that can help you accomplish the task more easily. You see, some creases can be altered, if it is necessary. In the chapter “Building blocks”, it is mentioned that certain crease patterns are interchangeable. Specifically, the way you handle a “turn without colour change” can be changed without consequences.

Precisely this kind of change has been performed in order to make the crease patterns simpler. Look at figure 19. A fully developed crease pattern is shown with the changed creases clearly marked.

Compare figures 18 and 19 to see the difference.

The presented crease pattern complies with Maekawa-Justin and Kawasaki-Justin theorems and, therefore, can be folded. As you can see from the folded version in figure 20, it fully resembles the template from figure 15.