|Post is a part of a larger series (Advanced concepts):|
Anyone who has ever tried to design box pleated origami models knows that those models have rather narrow flaps. This feature doesn’t have to be a problem, quite a contrary, it can even be an asset. But when wider flaps are needed, gadgets known as level shifters have to be used. Level shifters, in their essence, allow us to change the elevation of axial creases. In other words, they make flaps wider.
In a model where no level shifters exist, all axial creases or folds are in two levels. Valley folds are at elevation zero, while mountain folds are at elevation one. But, if we introduce level shifters, the elevation of axial creases could be at higher levels. Thus the name: level shifters. But to make the whole idea more understandable, let me show you a simple example of a level shifter.
I know that most of you will argue that accordion in figure 1 is not a real origami model, but it is. This model is, by all means a simple one, but it is also sufficiently complex so we could introduce a concept of level shifters.
As I have already mentioned since there are no level shifters all axial creases on the initial model are in two levels (or elevations). On the other hand, the second model has one level shifter, so due to its implementation, one axial crease is elevated to level 2. Quite an interesting feature, isn’t it?
Before we dive into the nuts and bolts of level shifters, we have to establish some rules, and frankly speaking, the rules are simple.
First of all, level shifters affect only axial creases. Other types of creases are not affected by level shifters whatsoever.
Second, to implement level shifters of any kind, ridge creases must be present since all level shifters, even though they change axial creases elevation, are in fact, implemented on the ridge creases.
Of course, implementation of the level shifters does not change the fact that all other basic origami rules still stand.
Level shifters classification
All level shifters can be divided into two distinct groups based on ridge crease type they are implemented on. So broadly speaking, there are:
- evel shifters on horizontal or vertical ridge creases, and
- level shifters on the ridge creases at a 45-degree angle
Level shifter on a ridge crease at a 45-degree angle
Level shifters on ridge creases at a 45-degree angle are the easiest to recognize, so let’s begin with them. In figure 2, marked with a green rectangle, you can see the most iconic level shifter of all. By the way, I hope you can see the underlining ridge crease the level shifter is located on.
To fully appreciate its value, it is necessary to compare the model in figure 2, with the basic version of the same model shown in figure 3.
If we compare these two models, we will realise that everything is the same except for the width of two flaps. These two flaps (A and B) are twice as wide because the axial crease marked in red is at elevation 2.
Isn’t that nice? Unfortunately, this level shifter has one not-so-nice property. It does not coincide with the grid. At least not in all its points.
Fortunately, the problem could be solved easily.
You see, the same result could be achieved, with a very similar level shifter shown in figure 4. This one fully coincides with the grid, while at the same time having all desirable properties of the original one.
The question is: how to construct the level shifter? The answer is, strangely enough, quite simple. There is a more or less straightforward procedure. You see, you could look at figure 2 and ask yourself, what is supposed to be inside the green rectangular. Of course, we know that the level shifter is supposed to be inside, but for the sake of argument, let’s assume we do not know that. In that case, only what we have is a bunch of creases outside the green rectangle that should be connected somehow respecting all the origami rules.
So let’s try to do so.
First, what we have to do is to extend these creases until some of them collide. In figure 5, two creases collide at 45 degree-angle. At that point, two additional creases must be added. Why?
Well, two colliding creases cannot end at that vortex. They must propagate further. But, there is more. According to the Maekawa-Justin theorem, the number of creases in every vertex must be even. Therefore, at least, two creases must propagate from that vortex. Meaning, at least two additional creases are needed.
Second, the angle between these two additional creases must be 135 degrees in order to satisfy Kawasaki-Justin theorem. Now, things become interesting. Since we only know the angle between them, the number of possible solutions is infinite. You see, this pair of creases can rotate around their initial vortex as long as the angle between them is maintained. That is precisely the reason why so many different level shifter designs exist.
So let’s move on, and let’s choose the one shown in figure 6. I can not stress enough that the choice is fully arbitrary.
These two additional creases will inevitably collide with the other two creases. Look at figure 6. At the points of collision, we again have the same problem. Two additional creases must be added, but this time, our decision will not be completely arbitrary. You see, to keep the number of creases under control, newly formed vertices, marked in green, should have one crease in common. Advantage of such an approach is that these two vertices are then fully defined. Three out of four creases are known. Adding forth crease shouldn’t pose a problem, since there is only one possible solution. So let’s add forth crease at both vertices (look at figure 6). Again these two creases will collide, there is no doubt about it. But what is interesting is that the collision point will be positioned exactly on the original ridge crease. The very same on which the level shifter is positioned on.
Finally, we must extend the last axial crease that was initially out of the green rectangle. It too will end up in the last collision point conveniently being forth crease at the vortex. With this, our level shifter is fully constructed (figure 7).
During this analysis, I have never mentioned how the mountain-valley assignment is done. I have never bothered to talk about why some creases are blue while others are red? The reason is that the procedure itself is quite easy, so I didn’t see the need to explain it in detail. You see, whenever a new crease is added at a certain vortex, we have to pay attention that three out of four creases at that vortex have the same colour. And, that’s it.
It seems that two previously presented level shifters do not have any shortcomings. But they do. If you look at figures 2 and 4, you will most certainly realize that in both figures level shifters are in pairs. There is a reason for that. You see, if the level shifter raises the elevation of a particular axial crease, the axial crease will stay at that elevation until it goes out of paper or another level shifter brings it down. That feature can be seen in figure 4. Once elevated, axial crease propagates across flap B all the way to the flap C. There it is finally brought down by another level shifter. This propagation is important because it affects every single flap on its way. If you look at figure 4 more closely, it is obvious from the crease pattern, that flap B has nothing to do with the level shifters. Yet, it is affected by them. It is wider than you would expect.
Asymmetric level shifters
Level shifters can be both: symmetric and asymmetric. The asymmetric type shown in figure 4 coincides with a grid which is, of course, a desirable property. That is why asymmetrical level shifters are so popular. But since it is asymmetrical, it could be designed differently. In figure 8, a mirror version of the level shifter in figure 4 is presented.
As you can see, there is almost no difference. But the difference though subtle still exists. If you take a good look at figure 8, you will notice that the flap that holds a level shifter does not reach its new desired elevation immediately. In other words, level shifters have to use part of the flap’s length to reach the desired elevation. If you compare crease patterns in figures 4 and 8, it is easy to notice that flaps upper ridge (marked in red) are of different length.
With this in mind, let’s compare two versions of level shifters in figures 4 and 8 (figure 9).
Figure 9 clearly shows that there is a difference in the slope between these two similar yet different level shifters. Basically, the first level shifter needs two basic units to reach the desired elevation, whereas the second one needs three. If you compare them to the symmetrical version, you will notice that the slope of the symmetrical level shifter is somewhere in between. It needs a little bit less than two and a half basic units to reach the desired elevation (figure 2). So it’s up to you, the designer, to choose the best-suited version.
Higher level shifters
Rising an elevation of an axial crease by just one level is not the only possibility. Higher elevations are also possible. The only constraint is the flap length because the level shifter needs some space to raise the elevation to a certain point. The higher the elevation, the more space is needed. In figure 10, you can see an example of the symmetrical level shifter, and how it is adjusted for higher elevations. In this particular case (figure 10) at least 6 units log flap is needed.
Similarly, asymmetrical level shifters can be used to reach desired elevations as well (look at figure 11).
The some applies to other asymmetrical level shifter (look at figure 12). Difference between these two asymmetrical level shifters is explained in the previous chapter.
The same principle can be used to reach even higher elevations if needed. What you have to remember is that flap must be longer in order to incorporate a larger level shifter. And that is it. I strongly encourage you to construct and fold at least one of them. It will give you more confidence in newly acquired knowledge.
Level shifters on a horizontal or vertical ridge crease
Level shifters presented so far have two drawbacks. First is that there is no clean-cut between two elevations. In other words, some form of slope always exists.
And secondly, in most cases, another flap should also have a visible level shifter to bring the elevation down, to the previous level.
With this in mind, let me show you another kind of level shifter, the one that has distinctly different properties. Maybe it is better to show you this in a simple example. In figure 13, you can see one such level shifter.
For now, disregard its crease pattern and take a good look at the folded base. It has four flaps, out of which one is distinctly wider. But what is exceptional about this example is there is no slope between two elevations. Here we have a clean cut. Another distinctive feature is the fact that only one flap is made wider seemingly without affecting the others. Unfortunately, such desirable properties are not for free. But let’s start from the beginning.
In figure 14, you can see the level shifter on a vertical ridge crease. Even though it looks quite simple, it is quite useful. Its main purpose is to facilitate the disappearance of two axial creases. Namely, upon reaching points A and B two mountain folds (a and c) no longer propagate to the end of the paper, effectively allowing the central axial crease (c) to change its orientation from a valley fold to a mountain fold. By the way, I hope you understand that Kawasaki and Maekawa’s rules are satisfied in both points.
Unfortunately, part of the paper is lost, and that is the price we have to pay. In our simple example, the lost region, marked in pink, propagates from one side of the paper to the other, but it does not have to be like that. In most cases, the lost region is significantly smaller due to its interaction with other elements of the origami model.
I would like to point out that the appearance of level shifters is allowed to be different as long as all basic roles in points A and B are satisfied.
Look at figure 15. The level shifter is no longer a square. Nevertheless, basic origami rules (Meakava and Kawasaki) still stand. Unfortunately, we manage to waste twice as much paper compared to the previous example (figure 14). It seems that the square level shifter is the optimal one.
I do not know if you have ever tried to fold origami models with level shifters. But if you have, then you probably know that every level shifter could have more than one orientation. If you look at figure 14, you will see that all creases on the level shifter gadget are valley folds (all creases are marked in blue). But if we turn everything upside down, making all creases inside the level shifter gadget a mountain (this time all creases are red), the level shifter will become quite different (figure 16). It is still a perfectly legal level shifter. It is only a little bit different, that is all.
As you can see, the level shifter in figure 16, is different in the sense the level shifter gadget is partly visible. It is not completely buried in-between paper layers.
So which approach is better? Well, it depends on what you need. Different authors prefer different approaches. However, the usefulness of the second approach becomes apparent when higher elevations are needed.
Higher level shifters
By now you know enough to make an educated guess, about the possibility to construct level shifters that could reach higher elevation. And if you think that any elevation could be reached, at least theoretically, if there is enough paper at our disposal, you would be right. But there is a difference. Look at figures 17 and 18. Like in the previous example, there are two versions of more or less the same level shifter. The only difference is their crease’s orientation.
The version in figure 17 is the one in which the level shifter is completely hidden. Please pay attention to the fact that four basic units of paper are lost (marked in pink on the crease pattern). Consequently, such a configuration piles five layers of paper. Larger level shifters piles even more and you should be aware of that.
But if you look at figure 18, you will see a level shifter that reaches the same elevation and at the same time piles up only three layers of paper. Or in the other words, twice as much paper is preserved. Of course, nothing is for free. The level shifter in figure 18 is clearly visible. It is no longer completely hidden.
Sometimes you will come across oddly looking level shifters that for some reason confuse people. Look at figure 19.
On the left side is this so-called “unusual” level shifter. It is the level shifter for sure, no doubt about it. After all, a bunch of axial creases are missing. But the truth is that this level shifter is by no means special. The difference is immediately evident if you look at a crease pattern on the right side of figure 19. Part of the so-called “Regular” level shifter is missing (the part marked in grey). Fortunately, the explanation is exceptionally simple. So far, in all previous examples, we always tried to elevate all axial creases from one elevation to another. We never tried to elevate two axial creases from different initial elevations. We never tried to take axial creases at, for example, levels one and two, and raised them to elevation three (look at figure 20). Exactly that kind of combination is achieved with the so-called “unusual” level shifter in figure 19.
Even higher level shifters
If we go one step further, the number of combinations increases dramatically. In figure 21, you can see the basic level shifter. There is nothing special about this level shifter. It is just extrapolation of previous concepts (figure 17). Only difference is the height to which the central axial crease is elevated
Now, if we reconfigure our model as it was shown in figure 22, meaning, if we introduce one axial crease at the higher elevation, then the level shifter should change too.
You see, by introducing axial creases at different elevations, we are staging the situation in which one level shifter has to simultaneously raise axial creases from two different initial elevations. As in the previous example, the solution is simple. Only what we have to do is to remove part of the initial level shifter as it was shown in figure 22.
We can go even further and reconfigure the model by introducing two axial creases at the higher elevation (figure 23). As you can guess, the level shifter has to be reconfigured even further by removing yet another part of the initial level shifter (figure 23). As you can see, the obtained level shifter is surprisingly simple. But, do not be mistaken. This is still our basic level shifter with two squares omitted.
Finally we come to the most extreme case.
I believe that a picture is worth a thousand words and that further explanations about how this level shifter is constructed are not needed.
Nevertheless, I would like to show you something else. I would like to deepen this discussion by showing you how to construct so-called universal level shifters. The level shifters capable of bridging any combination of elevations.
Universal level shifter
In all previous examples, we were only changing the initial configuration of the level shifter by changing only one of its sides. Another side of the level shifter always stayed the same.
So, the main question is as follows: is it possible to somehow bridge or connect an arbitrary combination of input and output elevations. The answer is, of course: yes.
But, and I cannot stress this enough, one specific rule has to be obeyed.
Namely, both input and output elevation must start and end at some elevation, since the paper can not be physically discontinued at its edges.
But let me show you one universal level shifter in action and how that rule is applied. In figure 25, a very small level shifter is shown.
What we got here. On the front side of the level shifter, axial creases’ elevations are 0,1,0,2,1,4. On the backside, we have a completely different configuration. Axial creases’ elevations are 0,4,3,4,3,4. Please, pay attention to the fact that the first and the last elevations are identical. In our example, the first elevation marked in purple is 0, on both sides. The same applies to the last elevation. It is 4 on both sides (marked in green).
You see, this rule, even though it sounds like some kind special and imposing rule, is in fact very obvious. Simply put, the level shifter can bridge any difference in elevation if there is enough paper at its disposal. But edges must be on the same level because otherwise, the paper will have to be discontinued, which is not physically possible.
Consequently, this constraint implies that the level shifter defining line always has closed contour.
Since it is clear that it is possible to bridge any combination of axial creases’ elevations as long as we stick to the already mentioned rule, let’s try to construct one universal level shifter from the beginning. In figure 26, you can see the initial configuration of the problem at hand.
It is obvious that some kind of gadget is needed to properly connect these lines. But, before we begin, we have to check if all prerequisites are satisfied. First, far left and far right axial lines should have the same orientation, and we should be able to connect them immediately. These lines are distinctly thicker in figure 26. Next, we have to check if the total elevation difference is the same. But how do we know that?
First, we have to draw two green lines which make all mountain and valley assignments more intuitive. Using these two green lines, we can measure total elevation difference. As you can see, there are the same.
Now, since all prerequisites are satisfied, we can construct our universal level shifter. The procedure is simple. To begin with, we have to merge these two green lines in the middle of the paper (look at figure 28).
Then we have to draw all the horizontal creases that intersect the level shifter. These creases define the area of the paper that will be lost, due to the level shifting.
The number of horizontal creases must be even, and, please pay special attention to that, these horizontal creases are not allowed to go inside the level shifter. Why? Because all the lines inside the level shifter will be perpendicular to these horizontal lines, so for now, leave them unassigned.
Next step is to extend all the vertical creases that formed the initial configuration. Look at figure 29. These creases are not allowed to enter the level shifter as well. On the other hand, those vertical lines that evade the level shifter should proceed all the way to the other end of the paper. But, please be careful, when they come across the horizontal crease, they have to change their orientation. Mountain folds should become valley folds and vice versa in order to satisfy basic rules (Meakava’s and Kawasaki’s rules).
At this moment, everything is resolved except the level shifter itself. More precisely, all that is left is to define creases inside the level shifter itself.
To do this, only what we have to do is to extend all the horizontal creases into the level shifter. But while doing so, we have to turn them at 90-degree angle whenever these horizontal creases hit the level shifter (look at figure 24). While doing that, the orientation of these horizontal creases should stay the same. And that’s it. Final result can be seen in figure 30.
I hope you will agree that implementing the universal level shifter is not that hard. Still, that kind of level shifter is quite a powerful and useful gadget when various elevation has to be somehow connected.
In the previous discussion, I have shown you that construction of universal level shifter is not so complicated, at least for level shifters on the horizontal or the vertical ridge creases. Can the same logic be applied to other types of level shifters? Namely, is it possible to construct a universal level shifter on a 45-degree angled ridge crease? Theoretically yes, but the procedure is not that simple.
I hope you remember how to construct a basic level shifter on a 45-degree angle ridge crease. The universal level shifter could be constructed in a similar way. But there is a small problem. Maybe not a problem but certainly an inconvenience. You see, if we compare the construction procedure between level shifter on a 45-degree angled ridge crease and the level shifter on a horizontal/vertical ridged crease, we will see a significant difference. Namely, the construction procedure of the former is not entirely straightforward. Meaning we will have to make some decisions along the way.
Universal level shifter on a 45 degree angle ridge crease
Let me show you the complete procedure.
In figure 31, you can see the initial configuration of all axial creases at the paper edges. Problem is that the number of creases, as well as their positions, are not the same. To make the problem understandable, green lines are added on both sides. As you can see, the green lines are not the same. Therefore they cannot be connected easily, without some kind of level shifter.
Before I show you how to construct a level shifter that can successfully bridge these elevation differences, one prerequisite must be met. Total elevation difference on both sides must be the same. Otherwise, there would be no solution to the problem. Luckily, the elevation difference is the same on both sides, so we can begin with level shifter construction.
As I have already told you, there are multiple solutions, therefore we will have to make certain decisions during level shifter construction. Every decision we make, will inevitably result in different level shifter configuration. In order to show you how far-reaching these decisions can be, I will constrict three different level shifters at the same time. So let’s begin.
The first two pairs of axial creases can be connected easily (look at figure 32). The problem begins with the third axial crease on the left side. That axial crease does not have its pair, so obviously, a level shifter of some kind is needed.
So, what we have to do is to extend that axial crease up to the ridge crease. There, axial and ridge creases will intersect at 45-degree angle. At that vortex, we have to add two additional creases with 135-degree angle in-between, in order to satisfy basic origami rules (Meakawa’s and Kawasaki’s rule). How these two creases will be positioned is the subject of arbitrary choice. For explanatory purposes, in each example, we will choose different positions (rotations). Look at figure 33.
These two creases are extended until they reach the next set of axial creases (figure 33). Please pay attention to the fact that in the third example, one axial crease is skipped. I did it on purpose, only to show you an additional concept. Just to show you that something like this is also possible.
In all three examples, we end up with two additional vertices that, again, need two additional creases each. It seems that by solving one problem we get two more. Fortunately, there is a very elegant solution. We just have to connect these two vertices effectively forcing them to have one crease in common. In this way, both vertices become fully defined since three out of four creases are known.
Again, these two creases have to be extended until they reach another set of axial creases. Unfortunately, only one crease has an axial crease it could be extended up to (Look at figure 35)
Also, be aware of the fact that in the third example, the skipped axial crease is the one our crease should be extended up to. Something like this is perfectly legal.
So far, everything seems to go smoothly. But what about the second crease, the one that does not have the crease it could be extended up to.
Well, in that case, we have to extend it up to some imaginary point. Technically, we could choose this imaginary point fully arbitrarily, but this would not be a particularly smart move. The better choice would be the one shown in figure 36. As you can see, the new imaginary point A is positioned in such a way to make newly added crease b, parallel to the similar crease marked with c.
What have we accomplished with such a decision? First, one vertex has become fully defined so it can be connected to the next axial crease. And second, we have found the solution for the second crease that did not have an axial line with which it could form the next vertex.
Now again, we have a standard problem of two vertices that should have one crease in common. So let’s make that crease.
his crease will make both vertices fully defined, which will finally give us the final solutions.
As you can see, all three solutions are distinctly different. It is hard to say if one solution is better than others. But a good measure of quality is grid coincidence. Meaning, solutions with more vertices on the grid are preferred.
Level shifter on a 90 degree angle ridge crease – revised
So far we have learned that level shifters on horizontal/vertical ridge creases can be constructed for virtually any combination of elevation if the prerequisites are satisfied. Unfortunately, there is one problem I would like to address here. Universal level shifters can pile up large numbers of layers in the same place, which can pose a problem. Let’s examine that problem in more detail.
In figure 40, you can see two almost identical level shifters. The only significant difference is the number of layers each level shifter has piled up. The level shifter on the left is the classical one, the one that we have used so far. The one on the right is slightly modified, but the final result is the same. The elevation shift is the same. Of course, the new crease pattern is a bit more complex, I can agree on that, but at the same time the number of piled up layers are significantly smaller too.
We can even play with different initial crease elevations, as we did with the classical version of this level shifter. The result will be almost identical. Just for comparison, look at figure 41. Here we have the first modification of the basic concept. The result is quite elegant.
Even more, a similar result is obtained if all one-unit elevations are replaced with two-unit elevations (as it was done in figure 23). Solutions are almost indistinguishable, except for the fact that paper layer pile up is now significantly smaller.
The most interesting is to compare solutions where initial three-unit elevation is used (as it was done in figure 24). It seems these solutions are completely different, but they are not. Close inspection will reveal many striking similarities. Above that, the most important feature of this new gadget is the fact it also coincides with the grid in all its vertices. The handy property, I would say.
Finally, before I close this topic I would like to point out that this new concept can be extended for any elevation. In figure 44 you can see the same gadget but for different elevations.
Putting everything together
By looking just at the crease pattern, can you recognize two level shifters? They are here for sure.
You see, it is obvious that in between a number of flaps two rivers exist (look at a folded model). But what is most interesting is the fact that both of them are wider than any other flap. So the level shifters must be somewhere.
Problem with this example is that its elements are very tightly packed. That is done on purpose, only to obscure level shifters from untrained eyes.
So, where are they?
In figure 46, one level shifter is marked by the grey rectangle. A clear indication that the observed element is indeed a level shifter is the fact that the number of axial creases on its sides is different.
Since this is a level shifter on vertical ridge crease, part of the paper will be inevitably lost. Do you remember when I have shown you the level shifter on a basic accordion? In all examples, a certain amount of paper was always piled up and was lost for good.
The same applies here. Since we are introducing level shifters a certain amount of paper will be lost, no doubt about it. But the question is: where?
With this question in mind, let me show you the main idea.
You see, this kind of level shifter (horizontal/vertical level shifter) always needs some kind of host. It needs an element with horizontal or vertical ridge crease it could be constructed on. But since this element will be inevitably lost, it is wise to create a dummy element with a vertical ridge crease. I call it dummy because it serves no other purpose but to host a level shifter. In the final model, this element is invisible. It would not exist.
So, if this is the case, then it would be wise to make this element as small as possible but at the same time large enough to host a level shifter. In figure 46, the dummy element is clearly marked in green.
I would like to stress that this is the normal classical element. If this element doesn’t have a level shifter inside, it would form an ordinary central flap. But since the level shifter is present on one of its ridge creases, the element will inevitably disappear. In other words, we have to dedicate a certain amount of paper to level shifters, and this is precisely what we have done. I hope this is more or less clear. I hope you see where the level shifter is and I hope you understand why it is created inside the dummy element.
Next, what I would like to talk about the impact the level shifter has on the other origami elements.
If you remember, when the axial crease is raised to a certain elevation, it will stay at that elevation until some other level shifter brings it down. Consequently, all the elements on its path will be wider because all of them will have at least one axial crease at a higher elevation. Therefore, if we want to limit its impact we have to add another level shifter so we could bring the elevated axial crease down. This way, by introducing another level shifter, we can effectively limit the number of elements affected by the elevated axial crease.
That being said, we can analyze our crease pattern once again to see how level shifters are used, how they are incorporated into the crease pattern and which elements are directly affected by its usage?
In figure 47, all flaps and rivers are clearly marked. Rivers are green while flaps are blue. What is the purpose of the elements marked in pink is probably clear by now. They serve as hosts for level shifters. Therefore they are dummy elements.
In other words, the first level shifter rises the axial crease (highlighted red line) while the second level shifter brings it down. But what is even more important to realise is the fact that elevated axial crease belongs only to the rivers K and L. Therefore only rivers are wider. No other element is affected by level shifters.
Before I end this blog post, I would like to add something else. Unlike level shifters on a vertical/horizontal ridge creases, a level shifters on a 45-degree angled ridge creases do not need dummy elements since their implementation does not result in paper loss.
The blog post is quite long, but the theme is more than interesting and valuable. I believe that now you have sufficient insight into level shifters to at least recognize them when you see them on a crease pattern.