## Hatsune Miku and colour change

Hatsune Miku by Itagaki Yuichi is a fascinating model. Not only that, it accurately depicts the original Anime character, but it also implements a nice colour changing method that deserves a detailed explanation.

The Crease pattern (figure 1) of this model is freely available on the Itagaki Yuichi Twitter profile , so there is no reason not to try to fold the model itself.

If you look at a crease pattern (figure 2), the first thing you would notice is the fact that the crease pattern is asymmetrical even though the model itself is fully symmetrical. But do not be intimidated by that. You see, asymmetrical crease patterns tend to be more optimal, meaning models, for the same amount of paper, tend to be larger. So, this is a good thing. I hope you agree.

Anyhow, we are not going to analyse the model any further. Instead, the main topic of this blog post will be a colour changing method used by the author. So let’s begin.

## Colour change

In different forums, you can find various comments about the colour changing method implemented on this model. Indeed, the implemented colour changing method is probably the most interesting part of the model. So, in this blog post, I will try to shed additional light on this colour changing method.

To begin with, I would like to remind you that there is a nice set of lectures on origami theory given by Prof. Erik Demaine (MIT). For anyone interested in origami theory, those lectures are a must.

But, in the context of the colour changing method, especially interesting is Episode 2 (the Simple fold), where Prof. Demaine has demonstrated, on a long strip of paper, the same colour changing method as one used in the Hatsune Miku model.

If you look at figure 3, You will see the very same method demonstrated by Prof. Demaine.

As you can see, the approach is simple and easy to implement if used on a long strip of paper. But, the question is; can this simple colour changing method be implemented together with all other elements of the model? Can it be implemented on something that is not a long strip of paper? Well, yes, it can. But, let me explain the whole idea.

### Level shifters

Even though it is not apparent immediately, the easiest way to implement the just shown colour changing method is to do it simultaneously with level shifters. After all, this colour changing method is, in its essence, just a clever way of using change in axial creases elevation in order to expose another side of the paper.

If you are unfamiliar with level shifters, I would strongly suggest reading an article on level shifters. It is a bit long, but I believe you will find it worth reading.

So let’s start with a simple level shifter, also known as diamond. Look at figure 4. Here we have a simple model and its crease pattern with a diamond-shaped level shifter.

Now, look at figure 5. The crease pattern is a bit different, but not much. If you compare the models in figures 4 and 5, you will immediately realise that the model in figure 5 is created by lifting the right paper edge of the model shown in figure 4. The result is self-evident; exposure of the differently coloured side of the paper. But, what is more, on a crease pattern (figure 5), we got a distinctive “X” shaped pattern. That pattern will appear whenever we try to implement this method. That is something to remember.

So, what did we have learned from this simple example?

• First, the colour changing method can be implemented only on the paper edges. Only on the paper edge, you can expose another side of the paper.
• Second, the just shown colour changing method introduces axial crease elevation change and as such is unavoidably linked to the concept of level shifters. In other words, the “X” shaped colour changing element introduces disruption, which manifests as an axial crease elevation change. This disruption must be compensated somehow, hence we have to use level shifters.

## Reversed approach

The previously described concept is valid no matter which type of level shifters we use. Therefore, a level shifter on a 45-degree ridge crease is equally acceptable as a means of disruption compensation.  As a matter of fact, on the Hatsune Miku model, precisely this kind of approach was used.

So, I will try to explain this approach in more detail. But before I do, I would like to emphasise that the approach, that will be shown, is a reversed one. Meaning, we will introduce the “X” shaped colour changing element first. Only then we will construct a suitable level shifter that will compensate for the disruption that unavoidably appears.

With this in mind, let’s start with a simple example. In figure 6, the complete model consists of only one flap in the paper corner. There is nothing more to it.

If we want to implement our colour changing method on that flap, several rearrangements are in order. First, we must add a distinctive “X” shaped element I have mentioned before (figure 5). To do that, we have to split the original circle into two parts, a smaller circle and a river that completely encircles it. From an origami point of view, a river that encircles only one circle is effectively part of that circle, so we did not change anything. We still have a flap of the same length.

Now we are ready to add our “X “ shaped element. We will place it in-between the circle and the river (figure 8). Be aware that if we want our flap to be of the same length as before, we will need a larger paper because we have to place an “X” shaped element somewhere, while at the same time avoiding any circle and river disruption. Simply put, the river and the circle must be of the same width as before if we want to retain the initial flap’s length. Look at figure 8.

If you analyse figure 8, you will notice that the smaller circle is inscribed into a somewhat irregular polygon (enclosed by a purple line). But that is OK. According to the theory, the shape of a polygon does not necessarily have to be a square. As a matter of a fact, the shape of a polygon can be highly irregular as long as the circle of an appropriate size can be inscribed in it. If you are unfamiliar with the just mentioned concept, then it would be good to read the blog post on the subject (“How to hide an unused paper in your Origami model?”)

The next step is to add ridge and axial creases onto the river. That should not be that hard. Look at figure 9.

On the other hand, adding ridge and axial creases onto the irregular polygon that holds a circle is more complicated. The problem arises from the fact that we have to connect all ridge creases (angle bisectors of the central irregular polygon – figure 9) while at the same time respecting Maekawa-Justin and Kawasaki-Justin rules. The final result can be highly irregular, though meaningful. You see, the final result will produce a structure that will inevitably include some sort of level shifter. A shape of a newly formed level shifter could be highly irregular, but it will be a level shifter nevertheless (figure 10).

To make things even worse, the crease pattern in figure 10 is not the only one possible. There are other possibilities too. Look at figure 11. All of these crease patterns are acceptable.  And all of them include some level shifters (shaped more or less regularly). Unfortunately, the complete theory behind the construction process of such complex crease patterns is way out of the scope of this blog post (maybe one day, I will write a blog post on that topic).

But let’s go back to our initial solution (figure 10) and let’s analyse the final result (figure 12). As you can see, the shape of the flap is identical to the initial one (figure 6), except for the nice colour-changing feature on one side.

Let’s analyse this example a little bit more. I believe this is important.

You see, due to the “X” shaped element insertion in between the river and the circle, a visible disruption occurs. This disruption primarily manifests as a broken ridge crease that runs across the crease pattern (look at figure 13 – magenta line). What is more, introduced disruption has a profound negative effect on the axial creases too. Because the ridge crease is broken, existing axial creases on both sides of the ridge crease cannot be connected easily. Look at figure 13. Two axial creases, marked in green, collide at the ridge crease. They should be at the same elevation (colour), but they are not, and as such render this configuration potentially illegal.

So, to solve this problem, a level shifter is introduced (look at figure 12). It may look strange, and it could be highly irregular, but this is the level shifter nevertheless. An element that changes the elevation of one axial crease, making its connection to the other axial crease possible, is by definition, a level shifter.  It is not a standard level shifter, like the one you can find in my blog post on level shifters, but it is a level shifter nevertheless since its whole purpose is to change the elevation of an axial crease. And this is precisely what it does. On the other hand, this level shifter does not change the width of a flap. Nothing like it. The flap is of the same width before and after the “X” shaped colour changing element insertion. To be sure, compare the models in figures 6 and 12. So, let me say it again: the whole purpose of this particular level shifter is to handle the mishmash that appears inside the folded flap. That is all.

### Position of colour change

Now, yet another logical question. Is it possible to change the position where the colour change appears on the flap? The answer is, of course, YES. To change it, you only have to reposition the “X” shaped colour changing element, and a different irregular polygon holding the circle will emerge. Of course, the river width will adjust accordingly. That way, we will obtain the very same flap but with a different colour disposition (figure 14).

### Colour change on both sides

n most cases having colour change implemented on only one flap’s side is adequate, but sometimes we need a colour change on another side as well. To get a colour change on both sides additional “X” shaped colour change element has to be added, making a crease pattern even more complex.  Look at the example in figure 15.

As you can see, the formed crease pattern looks quite complex but do not be overly discouraged by its strange appearance. What we have here is just a combination of two “X” shaped colour change elements (examples 1 and 7 in figure 14). The fact that they are not symmetrically positioned significantly contributes to the complexity of the crease pattern. Nevertheless, both “X” shaped colour changing elements, as well as corresponding level shifters are present.

On the other hand, if we need a flap identical on both sides, the crease pattern becomes symmetrical and surprisingly simple (figure 16).

So, to make a story short, all combinations are possible. There are no limits whatsoever.

## Final thoughts

I didn’t mention it before, but I presume that you have already noticed that the colour changing element used in the Hatsune Miko model is twice as large, when compared to the grid size.

In other words, whenever we introduce an “X” shaped colour changing element, the affected flap will have to be twice as wide (at least in the vicinity of the “X” shaped colour changing element). That is why all previous examples show flaps that are double in width. But that raises an additional question. You see, if the colour change is implemented only on one side, as it is in figure 12, does that imply that another side, the one without an “X” shaped colour changing element, does not have to be twice as wide? Well, the answer is NO. You see, only one “X” shaped element is enough to force the complete flap to be doubled in size. The fact that colour change appears only on the one side, changes nothing. The flap will have to be double in width anyway.

Therefore, if we want to narrow our flap down to the basic grid size, we will have to apply additional level shifters. Simple as that. The fact that we already have one level shifter should not confuse you since that level shifter is there only to deal with a mishmash caused by the “X” shaped colour changing element. It has nothing to do with the flaps width.

Finally, I would like to show you where, the colour changing method I was talking about, appears on the Hatsune Miku crease pattern. Look at figure 17. There are only two such places, but their influence propagates through the whole model leaving the impression of a significantly larger number of colour changing method implementations. Also, if you take a good look, you will most certainly realise that additional level shifters were added to the edges of the polygons inside which colour change is implemented (marked by yellow square). These level shifters are there to facilitate a smooth transition from a flap that is double in width to the rest of the model.